#  The Geometry of Isochrone Orbits
from Archimedes’ parabolae to Kepler’s third law

Paul Ramond†‡111Email<EMAIL_ADDRESS>(contact author) and
Jér$\hat{\text{o}}$me Perez‡222Email<EMAIL_ADDRESS>†Laboratoire
Univers et THéories
Observatoire de Paris, PSL Research University, CNRS, Université Paris
Diderot, Sorbonne Paris Cité
5 place Jules Janssen 92190 Meudon, France  ‡Laboratoire de Mathématiques
Appliquées
ENSTA Paris, Institut Polytechnique de Paris
828 Boulevard des Maréchaux 91120 Palaiseau, France

###### Abstract

The Kepler potential $\propto{-1/r}$ and the Harmonic potential $\propto
r^{2}$ share the following remarkable property: In either of these potentials,
a bound test particle orbits with a radial period that is independent of its
angular momentum. For this reason, the Kepler and harmonic potentials are
called isochrone. In this paper, we solve the following general problem: Are
there any other isochrone potentials, and if so, what kind of orbits do they
contain? To answer these questions, we adopt a geometrical point of view
initiated by Michel Hénon in 1959, in order to explore and classify
exhaustively the set of isochrone potentials and isochrone orbits. In
particular, we provide a geometrical generalization of Kepler’s third law, and
give a similar law for the apsidal angle, of any isochrone orbit. We also
relate the set of isochrone orbits to the set of parabolae in the plane under
linear transformations, and use this to derive an analytical parameterization
of any isochrone orbit. Along the way we compare our results to known ones,
pinpoint some interesting details of this mathematical physics problem, and
argue that our geometrical methods can be exported to more generic orbits in
potential theory.

Keywords : Classical gravity; Potential theory; Orbital mechanics; Isochrony;
Kepler’s laws

## Introduction

The concept of isochrony in physics can be traced back to Galileo’s pendulum
and his discovery of isochrone oscillations: The period of (small)
oscillations of a simple pendulum is independent of its initial conditions. In
modern classical mechanics, the one-dimensional motion of an oscillator or
small oscillations of a pendulum are characterized by harmonic potentials
$\psi(q)=\omega^{2}q^{2}/2$ where $\omega$ is the common pulsation to all
orbits and $q$ represents the varying amplitude of the oscillation through
time. In general or theoretical physics the concept of isochrony is
crystallized around this fundamental potential: This notion is reducted to
potentials $V$ which offer constant periods for all solutions of the ordinary
differential equation $\ddot{q}+\partial_{q}V(q)=0$ (see Sfecci (2015) and
reference therein). Applications for such problems range from scalar field
cosmologies Hawkins and Lidsey (2002) to quantum mechanics Dorignac (2005); in
the former the isochronous property is often associed to regularly spaced
discrete spectra. In all cases isochrony is appreciated for providing exact
models and explicit analytical formulae.

In this paper we are interested in a more general paradigm which include the
historical and classical previous one. It comes from gravitational potential
theory applied to astrophysics. It was coined by Michel Hénon in 1959 to
qualify a gravitational potential meant to describe globular clusters. As the
core of such spherical clusters of stars is roughly homogeneous, their mean
field potential is harmonic at small radial distances $r\ll 1$. By opposition,
stars confined to the outer parts only feel a Kepler potential
$\psi(r)=-\mu/r$ associated with a point mass distribution, seeing the cluster
from far away $r\gg 1$. In these two celebrated potentials, bound test
particles orbit along ellipses; and their associated orbital period exhibit
the striking feature of being independent of the angular momentum of the
particle. Michel Hénon then proposed looking for a general potential
characterized by this property, in order to describe globular clusters as a
whole.

In his seminal paper Hénon (1959a) (in French, for an English version see
Binney (2014)) he succeeded in solving this ambitious problem and found what
he called isochrone potential: $\psi(r)=-\mu/s$, where
$s:=b^{2}+\sqrt{b^{2}+r^{2}}$ and $b$ is a size parameter closely related to
the half-mass radius of the system. While having the requested dynamical
properties, the corresponding mass density distribution, obtained by solving
the Poisson equation, was in good agreement with some of the observed globular
clusters available in 1959. Although the recent refinement of observations has
actually revealed a wider diversity, Hénon’s isochrone model remains at the
center of cluster modeling for at least two reasons. As the harmonic and
Kepler potentials, this potential is fully integrable and its action-angle
formalism provides a fundamental basis for both the modeling and simulations
of stellar systems (see e.g., McGill and Binney (1990)). More recently, a
detailed numerical analysis Simon-Petit  _et al._ (2019) showed that the
isochrone model could be associated with the initial state of the evolution of
singular stellar systems (e.g., globular clusters and/or Low Surface
Brightness Galaxies); a result that followed an involved extension of many
aspects of Hénon’s work on isochrone potentials Simon-Petit  _et al._ (2018).

The modern version of the isochrony proposed by Simon-Petit  _et al._ (2018)
extended many mathematical aspects of the work pioneered by Hénon on isochrone
potentials. In particular, other kinds of potentials with the isochrone
property were found and classified using elements of group theory and
Euclidean geometry. In the present paper, building on these results, we go a
step further in two directions. On the one hand, we provide a fully
geometrical treatment of the problem first posed by Hénon, namely: Finding all
isochrone potentials. We shall see that with a geometrical treatment, one
family of potentials was left aside in Simon-Petit  _et al._ (2018).
Therefore, we complete and exhaustively classify all isochrone potentials,
based on their physical properties. On the other hand, we study in details the
shape, properties and conditions of existence of isochrone orbits, i.e.,
bounded orbits in isochrone potentials. In particular we generalize Kepler’s
third law to all isochrone orbits, providing a synthetic analytic formula for
both the radial period and the apsidal angle. We also detail and fulfill a
geometrical program that leads to an analytic parameterization of any
isochrone orbit, completing the program started in Simon-Petit  _et al._
(2018).

This paper’s main content is the solution to a problem of mathematical
physics: Finding the complete set of isochrone potentials and describing the
isochrone orbits. It is remarkable that it can be solved analytically and that
everything is expressible in terms of elementary functions. Furthermore, these
solutions can be obtained using elementary Euclidean geometry. We stress that,
physically speaking, the isochrone potentials with interesting properties are
the Kepler, the harmonic and the Hénon one, as was already found by Hénon. All
other potentials are necessary to get the complete picture of isochrony, but
present somewhat unfamiliar physical properties that shall be discussed. They
may nonetheless be of some interest as toy-models for astrophysics or
electrodynamics, and also for academic purposes. Many of our results and
geometrical methods are relevant to orbits in any central potential, as shall
be pointed out in the text. Throughout the paper, we emphasize on the geometry
of the problem, fill in some gaps that may be found in Simon-Petit  _et al._
(2018) and pinpoint some interesting mathematical physics details.
Computations that are not central to the results are left in the appendices,
while the main text is organized in four main sections, as follows:

• In Sec. I we briefly mention well-known results about bounded orbits in
central potentials, along with our notations and conventions (Sec. I.1). We
define the notion of isochrony for potentials and the Hénon variables (Sec.
I.2) that shall be used throughout the paper.

• The aim of Sec. II is twofold: First we derive an explicit formula for the
radial period in an arbitrary isochrone potential in terms of geometrical
quantities (Sec. II.1), and second, we use this formula to give a geometrical
proof that isochrone potentials are parabolae in Hénon’s variable (Sec. II.2).
This proof is inspired by the findings of Archimedes.

• Based on these results, in Sec. III we first sum up some generalities on
parabolae (Sec. III.1) in the plane. We then discuss the physical and
mathematical properties of the associated potentials (Sec. III.2) and draw the
bifurcation diagram that ensures the existence of periodic orbits, in terms of
the energy and angular momentum of the test particle. This is necessary to
give an exhaustive classification of isochrone potentials (Sec. III.3).

• This leads naturally to Sec. IV where our main and new results are stated.
We provide a generalization of Kepler’s third law (Sec. IV.1) for all
isochrone orbits, both for the radial period and the apsidal angle. We discuss
their geometrical meaning in various context. We then show how to
geometrically derive an analytic parameterization of any orbit in any
isochrone potential (Sec. IV.2). Lastly we depict some isochrone orbits,
analyze their properties and classify them (Sec. IV.3).

##  I Periodic orbits in central potentials

In this first section, the aim is to lay down the definitions and notations
that shall be used in this paper. First, in Sec. I.1, we derive some standard
results regarding periodic orbits of test particles in a given central
potential. In Sec. I.2, we define the qualifier isochrone for a central
potential, as well as the Hénon variables that shall be used throughout the
paper.

###  I.1 Basic definitions

Let us consider the three-dimensional Euclidean space and an inertial frame of
reference equipped with the usual spherical coordinates $(r,\theta,\varphi)$
and the associated natural basis
$(\vec{e}_{r},\vec{e}_{\theta},\vec{e}_{\varphi})$. We assume that around the
origin $O=(0,0,0)$ lies a spherically symmetric distribution of matter with
mass density $\rho(r)$. This system generates a gravitational potential,
denoted $\psi(r)$, that obeys Poisson’s equation

$\Delta\psi(r)=\frac{1}{r^{2}}\frac{\mathrm{d}(r^{2}\psi^{\prime})}{\mathrm{d}r}=4\pi
G\rho(r)\,,$ (1)

where a prime ′ denotes a differentiation with respect to $r$ and $G$ is the
universal gravitational constant333In electrostatics, $G$ must be replaced by
the Coulomb constant $-1/4\pi\varepsilon_{0}$, where $\varepsilon_{0}$ is the
vacuum electric pe rmittivity. In this context, $\rho(r)$ is the charge
density and is not required to be positive as in the gravitational case.. We
shall also use the usual dot $\dot{r}$ for the time derivative
$\mathrm{d}r/\mathrm{d}t$.

Let us now consider a test particle of mass $m$ orbiting this system, with
position vector $\vec{r}$ and velocity vector
$\vec{v}:=\mathrm{d}\vec{r}/\mathrm{d}t$. From the spherical symmetry, the
angular momentum $\vec{L}:=m\vec{r}\times\vec{v}$ of the particle is
conserved. Its norm can be computed explicitly and is given by
$|\vec{L}|=mr^{2}\dot{\theta}$, with the usual notation
$\dot{\theta}=\mathrm{d}\theta/\mathrm{d}t$ for the time derivative. The total
energy $E$ of the particle, sum of a kinetic term $m|\vec{v}|^{2}/2$ and a
potential term $m\psi$, is conserved as well. Let us introduce $\xi:=E/m$, the
(total) energy of the particle per unit mass ; and $\Lambda:=|\vec{L}|/m$, the
(norm of the) angular momentum per unit mass. The explicit computation of the
energy in terms of $r$ yields the following energy conservation equation

$\xi=\frac{1}{2}\biggl{(}\frac{\mathrm{d}r}{\mathrm{d}t}\biggr{)}^{2}+\frac{\Lambda^{2}}{2r^{2}}+\psi(r)\,.$
(2)

Since $E,\vec{L}$ and $m$ are conserved quantities, $\xi$ and $\Lambda$ are
two constants of motion for the particle. In a given potential $\psi$, the
quantities $(\xi,\Lambda)$ are sufficient to know everything about the
dynamics of a particle, up to initial conditions. Accordingly, we may abuse
notation and speak of $(\xi,\Lambda)$ as a particle. Along with some initial
conditions, Eq. (2) is a nonlinear ordinary differential equation for the
function $t\mapsto r(t)$. We are interested in orbits and therefore will
consider bounded solutions to Eq. (2).

Since the orbit is bounded and the function $r$ continuous, we may define
$r_{P}$ and $r_{A}$ as the minimum and maximum values of $r(t)$. Accordingly,
we will sometimes use the notation $[r_{P},r_{A}]$ for an orbit since,
physically, $r_{P}$ is (the radius of) the periapsis, i.e., the point along
the orbit closest to $O$ ; and $r_{A}$ that of the apoapsis, the one farthest
from $O$. At these turning points, the radial velocity $\dot{r}\vec{e}_{r}$
vanishes and $\dot{r}$ changes sign. Consequently, by Eq. (2), $r_{P}$ and
$r_{A}$ are two solutions to the following algebraic equation444When there are
more than two solutions to Eq. (3)., say $(r_{1},\ldots,r_{n})$ for some
$n\geq 3$, the orbit is selected on the graph by the initial radius $r_{0}$:
its periapsis and apoapsis being $r_{P}=r_{i}$ and $r_{A}=r_{i+1}$, where $i$
is such that $r_{0}\in[r_{i},r_{i+1}]$.

$\xi=\frac{\Lambda^{2}}{2r^{2}}+\psi(r)=:\psi_{e}(r)\,,$ (3)

where we have introduced the effective potential $\psi_{e}(r)$, sum of the
potential $\psi(r)$ and the centrifugal term $\Lambda^{2}/2r^{2}$. Note that
when there is unique solution $r_{C}$ to Eq. (3), the associated orbit is
circular, of radius $r=r_{C}$. This can always be seen as the degenerate case
$r_{P}\rightarrow r_{A}$.

It is customary to use the effective potential to study geometrically the
orbit of a particle, depending on its energy. As depicted in Fig. 1, one plots
the function $\psi_{e}$ for a given value of $\Lambda$, and then draws a line
of height $\xi$. By construction, any choice of initial conditions will yield
$\xi\geq\min{\psi_{e}}$. When there are two intersections between the line
$y=\xi$ and the curve $y=\psi_{e}(r)$, the orbit is non-circular and one can
read the periapsis $r_{P}$ and apoapsis $r_{A}$ as the abscissae of the
intersection points. When there is only one intersection, its abscissa is the
orbital radius $r_{C}$ and the orbit is circular. Furthermore, at $r=r_{C}$
the tangent to the curve is the horizontal line $y=\xi$, and therefore,
$\psi_{e}^{\prime}(r_{C})=0$. Two exemples of this well-known construction are
depicted in Fig. 1 for two particles with same energy, but different angular
momenta.

Figure 1: The graph $y=\psi_{e}(r)$ corresponds to the effective potential
$\psi(r)+\Lambda^{2}/2r^{2}$. Two $\psi_{e}$ are depicted, associated with two
particles with different angular momenta : $\Lambda$ (bottom curve, red) and
$\Lambda_{C}>\Lambda$ (top curve, light red). The vertical line $y=\xi$
defines two orbits associated with the same energy $\xi$. Particle
$(\xi,\Lambda)$ is on a generic, non-circular orbit $[r_{P},r_{A}]$ and
particle $(\xi,\Lambda_{C})$ is on a circular orbit of radius $r_{C}$. Note
that they both orbit in the same potential $\psi$.

We stress that, by virtue of Eq. (2), the quantity
$\xi-\psi_{e}(r)\propto\dot{r}^{2}$ should always be strictly positive when
$r(t)\in]r_{P},r_{A}[$, and vanish at $r_{P}$ and $r_{A}$, by definition. This
remark is important for later, so we summarize it geometrically as

$\text{On an orbit
$[r_{P},r_{A}]$}\,\,:\,\,\begin{cases}\,\,y=\xi\,\,\,\text{intersects}\,\,\,y=\psi_{e}(r)\,\,\,\text{at}\,\,\,r=r_{P},r_{A}\,,\\\
\,\,y=\xi\,\,\,\text{is
above}\,\,\,y=\psi_{e}(r)\,\,\,\text{when}\,\,\,r\in]r_{P},r_{A}[\,.\end{cases}$
(4)

####  I.1.1 Radial Period

It is a remarkable result of Hamiltonian dynamics that any bounded and
continuous solution to Eq. (2) must be periodic Arnol’d (1995). In other
words, if an orbit is bounded in a central potential, it is necessarily
radially periodic. We shall denote by $T$ the radial period (the period in
short hereafter), i.e., the smallest $T\in\mathbb{R}^{\star}_{+}$ such that
$r(t+T)=r(t)$ for all $t\geq 0$. Note that $T$ always exists for bound orbits,
and it should not be confused with the period of motion of the particle (i.e.,
the period of $t\mapsto\vec{r}(t)\in\mathbb{R}^{3}$), which only exists if the
orbit is closed555For example, in a harmonic potential the period of motion is
twice the radial period, and in a Kepler potential, both periods coincide. in
real-space (to be discussed below).

For a generic, non-circular orbit666We do not define $T$ for circular orbits,
since the radial motion for the latter is $r(t)=r_{C}$ where $r_{C}$ is a mere
constant, therefore any $T\in\mathbb{R}$ is a radial period. , one can get a
formula for $T$ by first isolating the variables $t$ and $r$ in Eq. (2). This
yields

$\mathrm{d}t=\pm\frac{\mathrm{d}r}{\sqrt{2\xi-2\psi(r)-\Lambda^{2}/{r^{2}}}}\,.$
(5)

In this formula, the $+$ sign corresponds to an increasing radius $r(t)$,
i.e., when the particle goes from $r_{P}$ to $r_{A}$, whereas the $-$ sign
corresponds to a decreasing radius, i.e., when the particle comes from $r_{A}$
back to $r_{P}$. Integrating Eq. (5) over a full period and taking into
account the two different signs provides the following integral formula for
the period

$T:=2\int_{r_{P}}^{r_{A}}\frac{\mathrm{d}r}{\sqrt{2\xi-2\psi(r)-\Lambda^{2}/{r^{2}}}}\,.$
(6)

Notice that the bounds of the integral $r_{P}$ and $r_{A}$ are precisely the
values making the denominator vanish, by virtue of Eq. (3). The fact that
$x\mapsto 1/\sqrt{x}$ is integrable near $0$ ensures the convergence of the
integral.777Indeed, using Eq. (3), we have the Taylor expansion
$\xi-\psi_{e}(r)=\psi_{e}^{\prime}(r_{P})(r_{P}-r)+o(r_{P}-r)$, and
$\psi_{e}^{\prime}(r_{P})\neq 0$ since the orbit is non-circular. The
integrand in Eq. (6) is thus equivalent to $(r_{P}-r)^{-1/2}$, which is
integrable at $r_{P}$. The same holds at $r_{A}$..

####  I.1.2 Apsidal angle

Let a particle $(\xi,\Lambda)$ be at position $(r(t),\theta(t))$ on its orbit
at some time $t$ (red point on the right of Fig. 2). The radial period $T$
corresponds to the time taken for the particle to go back to the radius $r(t)$
(with sign of $\dot{r}$). This does not mean, however, that the orbit itself
is a closed curve in real space. It will be the case only if after a period
$T$, the new angle $\theta(t+T)$ is equal to $\theta(t)+q\pi$, for some
$q\in\mathbb{Q}$. The orbit then closes after a number of radial periods equal
to the denominator of $q$.

To quantify this, let us define the quantity $\Theta:=\theta(t+T)-\theta(t)$.
It is a constant angle along the orbit888Since $\Lambda=r^{2}\dot{\theta}$ we
have $\dot{\Theta}=\Lambda/r(t+T)^{2}-\Lambda/r(t)^{2}$, which vanishes by
$T$-periodicity of $r(t)$. that corresponds physically to the angle difference
between two positions, a period $T$ apart. In orbital mechanics it is
customary to take the angle difference between two successive periapsis, as
depicted in Fig. 2. Therefore, we shall call $\Theta>0$ the apsidal angle.
When $\Theta$ is a rational multiple of $\pi$, the orbit depicts a closed
curve in real space. Otherwise, the orbit densely fills the shell region
$r\in[r_{P},r_{A}]$.

An integral formula can be obtained for $\Theta$, by using the conservation of
angular momentum $\Lambda=r^{2}\dot{\theta}$. This equation gives
$\mathrm{d}\theta=\Lambda\mathrm{d}t/r^{2}$, which, when combined with Eq. (5)
and integrated over one period, gives easily

$\Theta:=2\Lambda\int_{r_{P}}^{r_{A}}\frac{\mathrm{d}r}{r^{2}\sqrt{2\xi-2\psi(r)-\Lambda^{2}/{r^{2}}}}\,.$
(7)

Once again, we stress that Eq. (7) is valid for a generic, non-circular999As
for the radial period $T$, any $\Theta\in\mathbb{R}$ is an apsidal angle for
circular orbits, since $r(t)=r_{C}$ for all $t$ and, therefore, the particle
is in some sense always at periapsis. orbit $[r_{P},r_{A}]$, the convergence
of the integral (7) being justified by the same argument that was used for $T$
in Eq. (6).

####  I.1.3 Radial action

The Hamiltonian formulation of a test particle orbiting in a central potential
allows us to define the so-called radial action
$A(\xi,\Lambda)\propto\int_{r_{P}}^{r_{A}}\dot{r}(t)\mathrm{d}r$, see e.g.,
Binney and Tremaine (2008). Explicitly, using Eq. (2), this action is defined
for any orbit $[r_{P},r_{A}]$ and reads

$A(\xi,\Lambda):=\frac{1}{\pi}\int_{r_{P}}^{r_{A}}\sqrt{2\xi-2\psi(r)-\Lambda^{2}/{r^{2}}}\mathrm{d}r\,.$
(8)

The radial action acts as a generating function for the radial period
$T(\xi,\Lambda)$ and the apsidal angle $\Theta(\xi,\Lambda)$. Without going
into too much detail, which can be found, e.g., in Sec. 6 of Binney and
Tremaine (2008), one can think of $T$ and $\Theta$ as the frequencies
associated with the angle-action variables of the dynamics. In particular, we
can set

$\frac{T}{2\pi}:=\frac{\partial
A}{\partial\xi}\quad\text{and}\quad\frac{\Theta}{2\pi}:=-\frac{\partial
A}{\partial\Lambda}\,,$ (9)

and this coincides with the definitions (6) and (7), respectively. The
definitions (9) give an important result that we shall keep in mind: $T$
depends only on $\xi$ if and only if $\Theta$ depends only on $\Lambda$. This
is immediate from Eq. (9) since
$\partial_{\Lambda}T\propto\partial_{\xi}\Theta$ by swapping the order of
derivatives using Schwartz’s theorem.

Figure 2: A typical orbit in a central potential (solid black), centered on
the origin $O$, during $\sim 2$ periods $T$. At some initial time $t$, the
particle is at a radius $r=r(t)$ (red dot on grey circle, right). At times
$t+T$ and $t+2T$ it comes back to that same radius, crossing the grey circle
with the same sign of $\dot{r}$. During the first period $[t,t+T]$, the
particle reaches the periapsis (inner dashed circle) and then the apoapsis
(outer dashed circle). During the second period $[t+T,t+2T]$, the process
repeats. $\Theta$ is the angle between two successive periapsis, but also
between any two successive positions a period $T$ appart.

###  I.2 Isochrony and Hénon’s variables

####  I.2.1 Hénon’s definition of isochrony

For a generic central potential $\psi$, the radial period $T$ and the apsidal
angle $\Theta$ are functions of both $\xi$ and $\Lambda$, as should be clear
in view of Eqs. (6) and (7). In this paper, we are particularly interested in
the class of isochrone potentials. A central potential $\psi$ is called
isochrone if all periodic orbits it generates are such that $T$ is a function
of the energy of the particle only, i.e., $T=T(\xi)$. In an isochrone
potential, particles with the same energy share the same period, whence the
name. As we have mentioned above (see Eq. (9)), we thus have an alternative
characterization of isochrony, namely that $\Theta$ depends on $\Lambda$ only
(and not on $\xi$). In the end, one should keep in mind the following result

$\psi\text{ is isochrone}\quad\Leftrightarrow\quad
T=T(\xi,\Lambda\\!\\!\\!/)\quad\Leftrightarrow\quad\Theta=\Theta(\xi\\!\\!\\!/,\Lambda)\,.$
(10)

Isochrone potentials were introduced and studied first by Michel Hénon in
1959. In a series of three papers Hénon (1959a), Hénon (1959b) and Hénon
(1960), he studied their physical properties, the orbits they generate and
their application to astrophysics, respectively. Historically, the interest of
Hénon in the isochrone property was motivated during his study of globular
clusters (a particular spherical collection of stars). He knew that these
systems were isolated and had a homogeneous core. He also knew that a constant
density profile is associated with a harmonic potential ($\psi\propto r^{2}$),
and that outside any isolated spherical system, the potential is Keplerian
($\psi\propto-1/r$). Consequently, Hénon wanted to find a potential that could
interpolate these two. He noticed, quite remarkably, that one common feature
of the Kepler and the harmonic potentials was isochrony, and this led him to
try and find other isochrone potentials. After quite a remarkable analysis, he
succeeded in finding a third potential with this property, nowadays commonly
known as the101010In this paper as in Simon-Petit  _et al._ (2018), this one
isochrone potential is reffered to as the Hénon potential, isochrone being a
qualifier used here for the whole class of potentials such that
$T=T(\xi,\Lambda\\!\\!\\!/)$. isochrone potential Binney and Tremaine (2008)
that would help describe the density profiles of some stellar systems Hénon
(1960).

####  I.2.2 Hénon’s variables for central potentials

The effective potential method, presented in Fig. 1, mixes the properties of
the potential $\psi$ with that of the test particle $(\xi,\Lambda)$, as
$\psi_{e}$ includes the centrifugal term $\Lambda^{2}/2r^{2}$. In particular,
it is unpractical to draw and compare the orbits of two particles with
different $(\xi,\Lambda)$ in a given potential $\psi$. In other words, a line
$y=\xi$ crossing the curve $y=\psi_{e}(r)$ does not characterize a unique
particle, as $\Lambda$ is encoded in $\psi_{e}$ and not in that line. We
define in this section Hénon’s variables, which provide a way of working
around this problem.

In his seminal paper on isochrony Hénon (1959a), Michel Hénon introduced a
change of variables in order to compute some complicated integrals. These
variables have a much broader use that we shall exploit here. Instead of
working with the physical radius $r$ and the physical potential $\psi(r)$, let
us introduce the Hénon variables $x$ and $Y(x)$ defined by

$x:=2r^{2}\quad\text{and}\quad Y:=2r^{2}\psi(r)\,.$ (11)

Since $Y(x)$ and $\psi(r)$ are in a one-to-one correspondence through Eq.
(11), the $x$ variable can still be thought of as a radius and $Y$ as a
potential, and we shall sometimes abuse and speak of the radius $x$ and the
potential $Y$, always referring to this duality. In terms of the Hénon
variables, the energy conservation (2) can be rewritten in the following
evocative form

$\frac{1}{16}\biggl{(}\frac{\mathrm{d}x}{\mathrm{d}t}\biggr{)}^{2}=(\xi
x-\Lambda^{2})-Y(x)\,.$ (12)

In view of the paragraph above Eq. (3), the apoapsis and periapsis
$x_{A}:=2r_{A}^{2}$ and $x_{P}:=2r_{P}^{2}$ in Hénon’s variables are given by
the intersection between the curve $\mathscr{C}:y=Y(x)$ and the straight line
$\mathscr{L}:y=\xi x-\Lambda^{2}$, as can be read off the right-hand side of
Eq. (12). Furthermore, it is clear that line $\mathscr{L}$ should always lie
above curve $\mathscr{C}$ since the left-hand side of Eq. (12) is always
positive. In what follows, $\mathscr{L}$ will always denote a line of equation
$y=\xi x-\Lambda^{2}$, associated with a particle $(\xi,\Lambda)$ ; and
$\mathscr{C}$ will always be the curve of equation $y=Y(x)$, associated with a
potential $Y(x)=2r^{2}\psi(r)$. We note that, in terms of Hénon’s variables,
the conservation of angular momentum reads

$\Lambda=\frac{x}{2}\,\frac{\mathrm{d}\theta}{\mathrm{d}t}\,.$ (13)

Hénon’s variables $(x,Y(x))$ take advantage of the fact that a particle is
entirely described by two numbers $(\xi,\Lambda)$, and is therefore in a one-
to-one correspondence with a line, that has two degrees of freedom (e.g., the
slope and the $y$-intercepts). The potential $Y(x)$ corresponds to a unique
curve $\mathscr{C}$, and a particle $(\xi,\Lambda)$ is associated with a
unique straight line $\mathscr{L}$. If $\mathscr{L}$ intersects $\mathscr{C}$
and lies above it, this particle orbits periodically the origin as detailed in
Fig. 3. The advantage is that for a given potential $Y(x)$, one can draw any
particles and compare their orbital properties (which is not possible with the
$(r,\psi_{e})$ variables).

Figure 3: Same situation as in Fig. 1 depicted here in the Hénon plane, with
Hénon’s variables. The curve $\mathscr{C}$ of the graph $y=Y(x)$ corresponds
to the potential $\psi$, in Hénon’s variables. Two particles are depicted as
straight lines $\mathscr{L}$ (red) and $\mathscr{L}_{C}$ (light red). They
have the same energy $\xi$ (same slope for both lines) but different angular
momenta $\Lambda$ and $\Lambda_{C}>\Lambda$ (different $y$-intercept).
Particle $(\xi,\Lambda)$ is on a generic orbit with periapsis $x_{P}$ and
apoapsis $x_{A}$ given by the two intersections at $P$ and $A$. Particle
$(\xi,\Lambda_{C})$ is on a circular orbit of radius $x_{C}$ given the unique
intersection at $C$. Notice that one can draw several different particles
orbiting the potential without changing the curve $y=Y(x)$.

##  II A geometrical characterization of isochrony

This section aims at giving a geometrical characterization of isochrony. In
Sec. II.1 we derive the Hénon formulae which give $T$ and $\Theta$ explicitly
for isochrone potentials. In Sec. II.2, we give a geometrical proof that the
Hénon formula for $T$ implies that the potential $Y$ must be an arc of
parabola. We follow the notation introduced in the last section using Hénon’s
variables: A particle $(\xi,\Lambda)$ is associated with a line
$\mathscr{L}:y=\xi x-\Lambda^{2}$, and $\mathscr{C}:y=Y(x)$ is the curve of an
arbitrary isochrone potential $Y(x)=2r^{2}\psi(r)$.

###  II.1 Hénon’s formulae

This subsection is split into three parts. In the first, we integrate Eq. (6)
explicitly, for any central potential, following a method of Hénon Hénon
(1959a). Assuming isochrony, we simplify in the second part this result to get
the Hénon formula for $T$. The third part presents more briefly this
computation for $\Theta$.

####  II.1.1 Computing the integral for $T$

As we motivated below Eq. (12), we start by performing in Eq. (6) the change
of variables $r\rightarrow x=2r^{2}$ and we introduce the potential
$Y(x)=2r^{2}\psi(r)$. We readily obtain the following expression

$T=\frac{1}{2}\int_{x_{P}}^{x_{A}}\frac{\mathrm{d}x}{\sqrt{D(x)}}\,,\quad\text{with}\quad
D(x):=(\xi x-\Lambda^{2})-Y(x)\,.$ (14)

The bounds of the integral are $x_{P}:=2r_{P}^{2}$ and $x_{A}:=2r_{A}^{2}\geq
x_{P}$. In the $(x,y)$ plane, the quantity $D$ appearing in Eq. (14) is the
vertical distance between the curve $\mathscr{C}$ and the line $\mathscr{L}$.
The fact that $D(x)\geq 0$ is ensured by the very existence of the orbit, or
equivalently by Eq. (12), as discussed in the last section.

Since the curve $\mathscr{C}$ is smooth and lies below $\mathscr{L}$ on
$[x_{P},x_{A}]$, there exists a line $\mathscr{L}_{C}$ that is both parallel
to $\mathscr{L}$ and tangent to $\mathscr{C}$ at some point $C$ of abscissa
$x_{C}\in[x_{P},x_{A}]$. This line intersects $\mathscr{C}$ exactly once, and
corresponds to a particle with a circular radius $r_{C}$, such that
$x_{C}=2r_{C}^{2}$. Moreover, $\mathscr{L}$ and $\mathscr{L}_{C}$ are parallel
and therefore associated with particles that share the same energy $\xi$.
Consequently we may write $\mathscr{L}_{C}:y=\xi x-\Lambda_{C}^{2}$, where
$\Lambda_{C}$ is the angular momentum of the other particle, on the circular
orbit.

With the help of this secondary line $\mathscr{L}_{C}$, we may rewrite the
distance $D$ of Eq. (14) as the difference $\ell^{2}-z(x)^{2}$, where
$\ell^{2}:=\Lambda_{C}^{2}-\Lambda^{2}>0$ is the vertical distance between
$\mathscr{L}$ and $\mathscr{L}_{C}$, and $z(x)^{2}:=Y(x)-(\xi
x-\Lambda_{C}^{2})>0$ is the vertical distance between $\mathscr{C}$ and
$\mathscr{L}_{C}$ (see Fig. 4). We denote the latter by a squared quantity
$z(x)^{2}$, so that we may conveniently choose $z(x)\leq 0$ on $[x_{P},x_{C}]$
and $z(x)\geq 0$ on $[x_{C},x_{A}]$. We stress that this is nothing but a
convention: The positive distance is still $z(x)^{2}\geq 0$, but the sign of
$z(x)$ depends on where we are on $[x_{P},x_{A}]$.

Figure 4: Summary of the geometrical quantities used to compute the integral
for the period $T$. Both lines $\mathscr{L}$ and $\mathscr{L}_{C}$ define an
orbit with the same period, and $\ell^{2}$ is the vertical distance between
$\mathscr{L}$ and $\mathscr{L}_{C}$. The distance between $\mathscr{L}_{C}$
and $\mathscr{C}$ is $z(x)^{2}$, such that $D(x)+z(x)^{2}=\ell^{2}$.

These new quantities are depicted in Fig. 4, and upon insertion in Eq. (14),
we obtain

$T=\frac{1}{2}\int_{x_{P}}^{x_{A}}\frac{\mathrm{d}x}{\sqrt{\ell^{2}-z(x)^{2}}}\,,$
(15)

Now, by construction, $z(x)$ varies monotonically on $[x_{P},x_{A}]$: It is
negative and increasing on $[x_{P},x_{C}]$, it hits zero at $x_{C}$ and it is
positive and increasing again on $[x_{C},x_{A}]$. We can therefore perform the
change of variables $x\rightarrow z(x)$ in Eq. (15). We readily obtain

$T=\frac{1}{2}\int_{-\ell}^{\ell}\frac{f^{\prime}(z)\mathrm{d}z}{\sqrt{\ell^{2}-z^{2}}}\,,\quad\text{with}\quad
x:=f(z)\,.$ (16)

It is now natural to perform in Eq. (16) one last change of variables, namely
$z\rightarrow\ell\sin\phi$, with $\phi$ varying between $-\pi/2$ and $\pi/2$,
corresponding to $z=-\ell$ and $z=\ell$, respectively. We then get

$T=\frac{1}{2}\int_{-\pi/2}^{\pi/2}f^{\prime}(\ell\sin\phi)\mathrm{d}\phi\,.$
(17)

We cannot, in general, compute explicitly the integral in Eq. (17), for
$f^{\prime}$ is but a generic, unspecified function that depends on the
potential and the particle. However, assuming that the potential is regular
enough, we can expand the function $f^{\prime}$ as a Taylor expansion at zero,
i.e., write $f^{\prime}(z)=a_{0}+\sum_{n\geq 1}a_{n}z^{n}$. Inserting this in
Eq. (17) and integrating term by term gives

$T=\frac{\pi}{2}a_{0}+\sum_{n\geq
1}a_{2n}W_{2n}\ell^{2n}\,,\quad\text{with}\quad
W_{n}:=\int_{0}^{\pi/2}\sin^{n}\\!\phi\,\mathrm{d}\phi\,.$ (18)

Note that only the even terms $a_{2n}$ remain since the integral of the odd
function $\sin^{2n+1}$ vanishes over the symmetric interval $[-\pi/2,\pi/2]$.
The integral $W_{n}$ is the celebrated Wallis integral, and can be given
explicitly. Notice that Eq. (18) is valid for any potential and any particle
orbiting within it. Both the coefficients $a_{n}$ and $\ell$ depend on
$(\xi,\Lambda)$ and the properties of the potential. As such, it is not that
useful. However, for isochrone potentials, it is of considerable interest.

####  II.1.2 Explicit formula for $T$

So far, what we have done does not take advantage of the isochrony property,
and Eq. (18) is valid for any particle $(\xi,\Lambda)$ in any central
potential $Y(x)$. In particular, we insist that the coefficients $a_{n}$
appearing in Eq. (18) are all function of $\xi$ and $\Lambda$, a priori. Now
let us fix the energy $\xi$ of the particle. If the potential is isochrone,
then by definition $T$ is independent of $\Lambda$, and so is the right-hand
side of Eq. (18). We may therefore choose $\Lambda=\Lambda_{C}$, i.e.,
$\ell=0$, so that the last term on the right-hand side of Eq. (18) vanishes,
and we readily find that

$T(\xi)=\frac{\pi}{2}a_{0}\,.$ (19)

Now Eq. (19) is true for any $\Lambda$. (Actually, it is independent of
$\Lambda$.) The combination of Eqs. (18) and (19) implies that the sum on the
right-hand side of Eq. (18) is a power series in $\ell$ that vanishes for any
$\ell$. By a classical result on power series, this is true if and only if all
the coefficients of the power series vanish, i.e., $a_{2n}W_{2n}=0$ for all
$n\geq 1$. Since the Wallis integrals $W_{n}$ are all nonzero, we conclude
that, if the potential is isochrone, $a_{2n}=0$ for all $n\geq 1$. In
particular, the Taylor expansion of $f^{\prime}$ now reads

$f^{\prime}(z)=a_{0}+\sum_{n\geq 1}a_{2n+1}z^{2n+1}\,.$ (20)

The last step consists in finding explicitly the coefficient $a_{0}$ appearing
in Eq. (19). To this end, we integrate Eq. (20) over
$[z(x_{P}),z(x_{A})]=[-\ell,\ell]$. On the left-hand side, we use
$f(z_{A})=x_{A}$ and $f(z_{P})=x_{P}$ (which follows from the definition
$x=f(z)$). On the right-hand side, the first term is a mere constant, and the
second term is an odd function of $z$: Its integral over $[-\ell,\ell]$ will
vanish. Consequently, the integrated result is simply $x_{A}-x_{P}=2\ell
a_{0}$. With the help of $\ell^{2}=\Lambda_{C}^{2}-\Lambda^{2}$ and Eq. (19),
we obtain the following explicit formula for $T(\xi)$

$T(\xi)=\frac{\pi}{4}\frac{x_{A}-x_{P}}{\sqrt{\Lambda_{C}^{2}-\Lambda^{2}}}\,.$
(21)

Let us make a few remarks on Eq. (21). First, and quite remarkably, we stress
that altough both $x_{A}-x_{P}$ and $\sqrt{\Lambda_{C}^{2}-\Lambda^{2}}$
depend explicitly on $\Lambda$, their ratio does not, since $T$ is independent
of $\Lambda$ by assumption.

Second, if we square both sides of the equation, we observe that the
horizontal distance $x_{P}-x_{A}$ squared is proportional to the vertical one
$\Lambda_{C}^{2}-\Lambda^{2}$, and that the constant of proportionality,
namely $16T^{2}/\pi^{2}$, is independent of $\Lambda$. We shall use this
geometrical result to prove that the curve $\mathscr{C}$ must be a parabola in
the $(x,y)$ plane.

Third, we insist that this relation is valid for all isochrone potentials,
even though their explicit form is unknown at this stage. In particular, given
an isochrone potential, the radial period of any orbit can be read simply by
drawing the line $\mathscr{L}$ intersecting the curve $\mathscr{C}$, and then
finding the secondary line $\mathscr{L}_{C}$ that is both parallel to
$\mathscr{L}$ and tangent to $\mathscr{C}$.

Lastly, let us mention that formula (21) for the period $T$ in not entirely
new: It can be found as an intermediate equation in the seminal paper of Hénon
Hénon (1959a), (with a missing factor of 1/2 there). In fact, our method here
is similar to his, altough more detailed. Hénon did not seem to be interested
in this particular equation, perhaps because his main goal was not to obtain a
formula for the period. Yet, we shall see that this equation is rather central
in the context of isochrony.

####  II.1.3 Explicit formula of $\Theta$

In the last paragraphs, we were able to obtain the explicit formula (21) for
$T(\xi)$. The recipe for the computation went in five steps that can be
summarized as follows:

  * •

fix $\xi$ and rewrite the integrand in Eq. (6) as $1/\sqrt{D(x)}$ using
Hénon’s variables,

  * •

rewrite $D(x)$ as $\ell^{2}-z(x)$ using the line $\mathscr{L}_{C}$ associated
with the circular orbit of same energy (and thus same period),

  * •

introduce $x=f(z)$, perform the change of variables $x\rightarrow z$ and then
$z\rightarrow\phi$,

  * •

perform a Taylor expansion of $f\prime$ around $0$ and integrate explicitly,

  * •

assume that $\psi$ is isochrone and thus use $T=T(\xi,\Lambda\\!\\!\\!/)$ to
constrain $f$ and conclude.

Ultimately, the effectiveness of this recipe can be traced back to the Hénon
variable $x=2r^{2}$ which isolates $\Lambda$ from the denominator of the
integrand in Eq. (6), as can be seen on Eq. (14). Knowing this, it is possible
to try and adapt the recipe to find an explicit formula for $\Theta$, starting
from its integral definition in Eq. (7). As we argued earlier, examining the
radial action (8) shows that $T=T(\xi,\Lambda\\!\\!\\!/)$ is equivalent to
$\Theta=\Theta(\xi\\!\\!\\!/,\Lambda)$. Therefore, one can apply the same
recipe provided that one uses a variable that isolates $\xi$ in the
denominator in Eq. (7). The Binet variable $u:=1/r$ turns out to be the
appropriate variable this time.

More precisely, with the Binet variables $u=1/r$ and the Binet effective
potential $\Psi_{e}(u):=\psi_{e}(1/u)$, it is possible to make a one-to-one
dictionary between what was used for $T$ and what can be used for $\Theta$.
The latter is presented in Table 1. The detailed computation is given in App.
C. At the end of the computation, for any given $\Lambda$ we obtain the
following formula in the case of isochrone potentials

$\Theta(\Lambda)=\frac{\pi\Lambda}{\sqrt{2}}\frac{u_{P}-u_{A}}{\sqrt{\xi-\xi_{C}}}\,,$
(22)

with $u_{P,A}:=1/r_{P,A}$. The value $\xi_{C}$ depends only on $\Lambda$ and
is the energy to be given to a particle of angular momentum $\Lambda$ to
obtain a circular orbit. Moreover, as we argued earlier for $T$ in Eq. (21)
despite appearances the right-hand side of Eq. (22) is independent of $\xi$.

Just as Eq. (21) will be used in Sec. IV in order to write a generalized
Kepler’s third law for all isochrone orbits, Eq. (22) will be used to find a
similar law for the apsidal angle of any isochrone orbit. We shall not use it
directly and present a more astute computation, but it is possible to derive,
without any trick, this periapsis law directly from Eq. (22).

This apsidal angle law can, in turn, be used to give a proof of Bertrand’s
theorem, a well-known result of classical mechanics that states that the only
two potentials in which all periodic orbits are closed are the Kepler and the
harmonic potentials. In fact, since these two are also isochrone potentials,
it should come as no surprise that Bertrand’s theorem is closely related to
isochrony. As demonstrated in Simon-Petit  _et al._ (2018), the theorem
actually follows from the examination of Eq. (22), once the latter is
expressed in terms of $\Lambda$. Let us mention that the equivalent of Eq.
(18) for $\Theta$ (Eq. (97)) can be used to give a proof a Bertrand’s theorem
with brute force as in Santos  _et al._ (2009) [compare Eq. (20) of Santos
_et al._ (2009) to Eq. (97)].

Isochrony $\Leftrightarrow$ |  Variable | Potential |  Curve $\mathscr{C}$ | Line $\mathscr{L}$ | Distance $D$
---|---|---|---|---|---
$T=T(\xi,\Lambda\\!\\!\\!/)$ | $x=2r^{2}$ |  $Y(x)=2r^{2}\psi(r)$ | $y=Y(x)$ |  $y=\xi x-\Lambda^{2}$ | $D(x)=\xi x-\Lambda^{2}-Y(x)$
$\Theta=\Theta(\xi\\!\\!\\!/,\Lambda)$ | $u=1/r$ | $\Psi_{e}(u)=\psi_{e}(r)$ | $y=\Psi_{e}(u)$ | $y=\xi$ | $D(u)=\xi-\Psi_{e}(u)$
Table 1:  Dictionary between the geometrical quantities involved in the
derivation of the Hénon formulae: (21) for $T(\xi)$ and (22) for
$\Theta(\Lambda)$.

###  II.2 Geometry of parabolae

In this section, we provide a geometrical proof that the curve
$\mathscr{C}:y=Y(x)$ must be a parabola111111Strictly speaking, we will show
that $\mathscr{C}$ is an arc of parabola, as it is the graph of a function.
However any given arc of parabola defines a unique parabola, so there will be
no possible confusion here. in order for the associated potential $\psi$ to be
isochrone. This result was first established by Hénon in Hénon (1959a), using
a very technical argument. It is also found in Simon-Petit  _et al._ (2018)
using techniques from complex analysis. In this paper, sticking to the
geometrical approach, we present a new, geometrical proof of this remarkable
fact, based only on Eq. (21) and a characterization of parabolae that can be
traced back to Archimedes.

####  II.2.1 Archimedean characterization

Archimedes, in his treatise Quadrature of the Parabola121212We refer the
interested reader to pp.233-252 of Heath Heath (2002) for a modern English
translation of this work, and to pp.51-62 of Stein Stein (1999), for a
pedagogical version., proved in a series of 24 propositions the following
remarkable property shared by all parabolae. On a given parabola
$\mathscr{P}$, take two points $A$ and $B$ defining a chord $AB$ and a third
point $C$ where the tangent to $\mathscr{P}$ is parallel to the chord $AB$.
Then the area enclosed by $\mathscr{P}$ and $AB$ is four thirds that of the
triangle $ABC$. Although it was not known to Archimedes, it turns out that
this property uniquely characterizes parabolae Bényi  _et al._ (2003). In
other words, we have the following theorem :

######  Theorem.

Let $\mathscr{C}$ be an arbitrary smooth curve in the plane, and $\mathscr{L}$
any line that intersects $\mathscr{C}$ exactly twice, say at points $P\\!$ and
$A$. Let $C$ be the point where the tangent to $\mathscr{C}$ is parallel to
$\mathscr{L}$. Then, if $\mathscr{T}$ denotes the triangle $P\\!AC$ and
$\mathscr{S}$ the region enclosed by $\mathscr{L}$ and $\mathscr{C}$, the
following equivalence holds

$\text{Area}\,(\mathscr{T})=\frac{3}{4}\text{Area}\,(\mathscr{S})\quad\Leftrightarrow\quad\mathscr{C}\text{
is an arc of parabola}\,.$ (23)

Notice that in order for $\mathscr{C}$ to be a parabola, the area ratio should
be $4/3$ for all of its chords. As stated above, the $\Leftarrow$ result was
the aim of Archimedes’ work.

####  II.2.2 Rewriting of Hénon’s formula

Consider the curve $\mathscr{C}:y=Y(x)$ associated with an isochrone potential
$Y(x)=2r^{2}\psi(r)$. Following the notation used so far, let us take a line
$\mathscr{L}:y=\xi x-\Lambda^{2}$, such that $P\\!$ and $A$ correspond to the
periapsis and apoapsis of the orbit of particle $(\xi,\Lambda)$. Accordingly,
the parallel line that passes through $C$ is $\mathscr{L}_{C}:y=\xi
x-\Lambda^{2}_{C}$, and defines a circular orbit with the same energy $\xi$,
and thus the same period $T(\xi)$, given by Eq. (21). If for any line
$\mathscr{L}$ the areas involved in the theorem are in proportion $4/3$, we
will have shown that $\mathscr{C}$ is a parabola. Therefore, the goal is to
find an expression for these areas, using Eq. (21).

First let us do a bit of geometry. We define $B$ to be the orthogonal
projection of $C$ on $\mathscr{L}$, and take $M$ to be an arbitrary point on
$CB$. We parameterize the length $CM$ by $h\geq 0$, with the convention $h=0$
when $M=C$, and $h=CB$ when $M=B$. Next we define a chord
$P^{\prime}A^{\prime}$ that is parallel to $\mathscr{L}$ and passes through
$M$. We denote by $L(h)$ the length of that chord $A^{\prime}$ and
$P^{\prime}$. Note that $L(h)$ varies between $0$ (when $h=0$) and $PA$ (when
$h=CB$). All these quantities are depicted in Fig. 5.

Figure 5: Initially, an arbitrary curve $\mathscr{C}$ and an intersecting line
$\mathscr{L}$ of slope $\xi$ are drawn. They define the intersection points
$P$ and $A$. The horizontal distance between $P$ and $A$ is $x_{A}-x_{P}$
(top). The line $\mathscr{L}_{C}$, parallel to $\mathscr{L}$ and tangent to
$\mathscr{C}$ at $C$, defines a circular orbit with energy $\xi$. The vertical
distance between $\mathscr{L}$ and $\mathscr{L}_{C}$ is
$\Lambda_{C}^{2}-\Lambda^{2}$ (left). The intermediary chord
$P^{\prime}A^{\prime}$ defined in the text is parallel to $\mathscr{L}$ and
defines yet another orbit with energy $\xi$.

Now let us rewrite Eq. (21) in terms of these geometrical quantities. For the
numerator, $x_{A}-x_{P}$ is but the horizontal projection of $P\\!A$, and thus
$x_{A}-x_{P}=P\\!A\cos\varphi$, where $\varphi$ is the angle that
$\mathscr{L}$ makes with the horizontal axis, i.e.,. $\varphi=\arctan\xi$.
Similarly, for the denominator, $\Lambda^{2}_{C}-\Lambda^{2}$ is simply the
vertical projection of $CB$; consequently, we also have
$\Lambda^{2}_{C}-\Lambda^{2}=CB/\cos\varphi$. Inserting these two identities
in Eq. (21) gives its geometrical variant

$T(\xi)=\frac{\pi}{4(1+\xi^{2})^{3/4}}\frac{P\\!A}{\sqrt{CB}}\,,$ (24)

where we used the trigonometric identity
$\cos(\arctan\xi)=(1+\xi^{2})^{-1/2}$. Now, formula (24) has been obtained for
any chord $P\\!A$ of the curve $\mathscr{C}$, corresponding to a particle of
energy $\xi$. However, by construction, for any $h$ the chord
$P^{\prime}A^{\prime}$ is parallel to $P\\!A$ and thus corresponds to an orbit
with the same energy $\xi$. Therefore, the potential being isochrone, all
parallel chords $P^{\prime}A^{\prime}$ generated by varying $h$ correspond to
orbits with the same energy $\xi$, and therefore the same period $T(\xi)$. The
conclusion is that Eq. (24), which corresponds to the case $h=CB$, is also
verified for any value of $h$ when the potential is isochrone. In other words,
for any $h\in]0,CB]$, we have

$T(\xi)=\frac{\pi}{4(1+\xi^{2})^{3/4}}\frac{L(h)}{\sqrt{h}}\,.$ (25)

Of course, Eq. (24) is just a particular case of Eq. (25), when $h=CB$ and
$L(h)=P\\!A$.

####  II.2.3 Computing the areas

With Eq. (25) at hand, we can now turn to the compution of the areas involved
in the theorem. For the triangle $P\\!AC$, we have the basis $P\\!A$ and the
height $CB$. For the area between $\mathscr{C}$ and $\mathscr{L}$, we can
simply integrate à la Lebesgue the infinitesimal area $L(h)\mathrm{d}h$ while
$h$ varies between $0$ and $CB$. We thus have, respectively

$\text{Area}\,(\mathscr{T})=\frac{1}{2}P\\!A\times
CB\,\,\quad\text{and}\quad\,\,\text{Area}\,(\mathscr{S})=\int_{0}^{CB}\,L(h)\mathrm{d}h\,.$
(26)

Now we compute these areas and we show that they are in proportion $4/3$. For
the area of the triangle $\mathscr{T}$, we use Eq. (24) to express $P\\!A$ in
terms of $CB$ and plug the result in Eq. (26). We obtain the following
expression

$\text{Area}\,(\mathscr{T})=\frac{2T}{\pi}(1+\xi^{2})^{3/4}CB^{3/2}\,.$ (27)

In a similar manner, the area of the region $\mathscr{S}$ can be found by
isolating $L(h)$ from Eq. (25) and expressing it in terms of $h$. Plugging the
result in the area formula for $\mathscr{S}$ in Eq. (26) and computing the
integral explicitly give easily

$\text{Area}\,(\mathscr{S})=\frac{8T}{3\pi}(1+\xi^{2})^{3/4}CB^{3/2}\,.$ (28)

Comparing Eqs. (27) and (28) shows that, indeed,
$\text{Area}\,(\mathscr{S})/\text{Area}\,(\mathscr{T})=4/3$. By virtue of the
theorem, the claimed result follows: If a potential is isochrone then the
curve $\mathscr{C}:y=Y(x)$ in Hénon’s variables is a parabola.

####  II.2.4 Final remarks

We end this section by answering a question: Why is it that $\mathscr{C}$
should be a parabola, and not any other type of curve, when the potential is
isochrone ? What is so special about parabolae ? To understand this, let us
focus our attention on a point $M$ of a generic curve $\mathscr{C}$ (i.e.,
non-necessarily a parabola).

Close enough to $M$, $\mathscr{C}$ always looks like a parabola, as can be
seen by writing its Taylor expansion131313Qualitatively speaking, at least.
The particular case when the second derivative vanishes at $M$ is not a
problem, it corresponds to $\alpha=0$ in Eq. (29). around $M$. To see this,
consider the particular frame $(x,y)$ centered on $M$ where the tangent to
$\mathscr{C}$ at $M$ is horizontal. These two conditions indicate that
$y(0)=0$ and $y^{\prime}(0)=0$ respectively. Therefore the curve has an
implicit equation of the type

$y=\alpha x^{2}+o(x^{2})\quad\text{for some constant }\alpha\,.$ (29)

For a generic curve, the $o(x^{2})$ in Eq. (29) corrects the local
parabolicness of the curve as one moves away from $M$. However, Eq. (25) shows
that the $o(x^{2})$ terms vanishes identically in the case of isochrony.
Indeed, in Fig. 5, this particular frame $(x,y)$ we are considering is
precisely the one centered on $C$ equipped with coordinates $(x,y)=(L,h)$. Now
Eq. (25) may be rewritten as

$h=\alpha L^{2}\,,\quad\text{with }\alpha=\pi^{2}/16T^{2}(1+\xi^{2})^{3/2}\,.$
(30)

Since this should be true for all $h$, or equivalently any $x$, comparing Eqs.
(29) and (30) shows that in the case of isochrony the $o(x^{2})$ vanishes
identically as claimed.

To summarize, in Hénon’s variables, any potential always looks, locally, like
a parabola, a universal mathematical property encoded in its Taylor expansion.
However, isochrony, through Eq. (25), propagates this local property to the
global level, constraining the curve to be a parabola, in addition to locally
look like one.

Following the same logic, one could ask whether Eq. (22) for $\Theta$ could
not be used to reach the same result. The central difference is that Eq. (22)
is to be read in the $(u,y)$ plane, where orbits correspond to horizontal
lines $\mathscr{L}:y=\xi$, whereas Eq. (21) is to be read in the $(x,y)$
plane, where orbits correspond to straight lines $\mathscr{L}:y=\xi
x-\Lambda^{2}$. Yet, the Archimedean characterization of parabola requires the
areas ratio to be $4/3$ for any chord, not just horizontal ones. Therefore,
Eq. (22) cannot be used to conclude that $\mathscr{C}$ should be a parabola,
at least not with the Archimedean characterization.

##  III Isochrone parabolae

The result of the last section implies that the curve $\mathscr{C}$ of an
isochrone potential $Y$ corresponds to (at least an arc of) a parabola.
However, not all parabolae will contain the potential of a physically
realistic system. The aim of this section is to classify, based on their
geometrical properties, the isochrone parabolae, i.e., these that contain the
curve $\mathscr{C}$ of a well-defined, isochrone potential $Y$. From now on we
shall always use the notation $\mathscr{P}$ for an arbitrary parabola in the
plane.

###  III.1 Generalities on parabolae

We start by a potpourri of algebraic and geometrical properties of parabolae,
and derive some general results that shall be used throughout the next
sections. The implicit equation of a parabola $\mathscr{P}$ in the $(x,y)$
plane is

$\mathscr{P}:(ax+by)^{2}+cx+dy+e=0\quad\text{with}\quad\delta:=ad-bc\neq 0\,,$
(31)

and where $(a,b,c,d,e)$ are five real numbers. The quantity $\delta$ is the
discriminant of $\mathscr{P}$ and is taken to be nonzero; otherwise, Eq. (31)
degenerates into a pair of parallel lines. Without loss of generality, we will
assume from now on that $\delta>0$.141414If $\delta<0$ we can always replace
$(a,b)$ by $(-a,-b)$. This leaves Eq. (31) unchanged and thus corresponds to
the same parabola, but changes the sign of $\delta$.

As an algebraic curve, a parabola is not, in general, the graph of a function.
It is, however, always the union of such graphs. To see this, let us take a
point $(x,y)$ on the parabola $\mathscr{P}$ given by Eq. (31). Its ordinate
$y$ can be given as a function of its abscissa $x$ by solving Eq. (31) for
$y$. There are two cases depending on the parameter $b$:

• when $b=0$, the whole parabola $\mathscr{P}$ is the graph of a function. Its
equation is given by

$\mathscr{P}:y=-\frac{e}{d}-\frac{c}{d}x-\frac{a^{2}}{d}x^{2}\,,$ (32)

where $d\neq 0$ since the discriminant $\delta=ad\neq 0$. For $d<0$ this
parabola opens upwards and we shall say that it is top-oriented. When $d>0$,
it opens downwards and we will say bottom-oriented. Any top- or bottom-
oriented parabola crosses the $y$-axis once, and the ordinate of this point is

$\lambda:=-\frac{e}{d}\,.$ (33)

• when $b\neq 0$, the curve $\mathscr{P}$ is the union of two branches, which
are actual graphs of a function. Indeed, for a fixed $x$, Eq. (31) is a
quadratic-in-$y$ equation. Its solutions are easily found to be

$\mathscr{P}_{\pm}:y=-\frac{ax}{b}-\frac{d}{2b^{2}}\pm\frac{1}{2b^{2}}\sqrt{4b(\delta
x-be)+d^{2}}\,,$ (34)

where the condition $4b(\delta x-be)+d^{2}\geq 0$ ensures the positivity
inside the square root. The support of the parabola is the set of $x$ such
that $4b(\delta x-be)+d^{2}\geq 0$. In particular, there is a unique value

$x_{v}:=\frac{4b^{2}e-d^{2}}{4b\delta}\,$ (35)

that makes the square root in Eq. (34) vanish. The quantity $x_{v}$ is the
abscissa of the common point between $\mathscr{P}_{+}$ and $\mathscr{P}_{-}$,
where the branches meet and the tangent to $\mathscr{P}$ is vertical (cf. Fig.
6). Note that $\mathscr{P}_{-}$ is always convex and always below
$\mathscr{P}_{+}$ which is concave.

The sign of $b$ controls the orientation of the parabola. If $b>0$, we will
say that the parabola is right-oriented. Its support is $[x_{v},+\infty]$, and
the parabola crosses the $y$-axis if and only if $x_{v}\leq 0$. If $b<0$, we
say that it is left-oriented. Its support is $[-\infty,x_{v}]$ and it crosses
the $y$-axis if and only if $x_{v}\geq 0$.

A left or right-oriented parabola may not always cross the $y$-axis. When it
does, the ordinates of the intersection points are obtained by setting $x=0$
in Eq. (34). In particular, the convex branch $\mathscr{P}_{-}$ crosses the
$y$-axis at ordinate

$\lambda:=-\frac{d+\sqrt{d^{2}-4b^{2}e}}{2b^{2}}\,.$ (36)

This implies that $d^{2}-4eb^{2}$ should always be positive for parabolae
crossing the $y$-axis. In particular, we have $d^{2}-4b^{2}e>0$ when there are
two intersections. The case $d^{2}-4eb^{2}=0\Leftrightarrow d^{2}=4eb^{2}$
happens when the two intersections degenerate into one, and its ordinate is
simply $-d/2b^{2}$.

Figure 6: geometrical properties of a right-oriented parabola $\mathscr{P}$,
with its two branches $\mathscr{P}_{\pm}$ that are actual graphs of functions.
The parabola itself is $\mathscr{P}=\mathscr{P}_{-}\cup\mathscr{P}_{+}$. On
the left, the point of abscissa $x_{v}$ given by Eq. (35) belongs to both
$\mathscr{P}_{+}$ and $\mathscr{P}_{-}$, and the domain of the parabola is
$[x_{v},+\infty[$ (highlighted in grey on the $x$-axis).

To summarize, the graph $\mathscr{C}$ of an isochrone potential $Y(x)$ must be
contained within a parabola $\mathscr{P}$ in the plane. From the preceding
generalities on parabolae, it follows that $\mathscr{C}$ is to be looked for
in any of the following families:

  * •

top- and bottom-oriented parabolae, whose whole curve $\mathscr{P}$ is that of
a function defined on $\mathbb{R}$, cf. Eq. (32).

  * •

left-oriented parabolae, whose curve $\mathscr{P}$ is the union of a convex
branch $\mathscr{P}_{-}$ and a concave branch, $\mathscr{P}_{+}$, each of the
two being the graph of a function defined on $]-\infty,x_{v}]$, cf. Eq. (34)
with $b<0$.

  * •

right-oriented parabolae, whose curve $\mathscr{P}$ is the union of a convex
branch $\mathscr{P}_{-}$ and a concave branch, $\mathscr{P}_{+}$, each of the
two being the graph of a function defined on $[x_{v},+\infty[$, cf. Eq. (34)
with $b>0$.

###  III.2 Physical portion of the graph

In this section we examine under which conditions the curve $\mathscr{C}$ of
the (so far arbitrary) potential $Y(x)=2r^{2}\psi(r)$ is a physically and
mathematically well-posed, isochrone potential. This will be done in four
steps, each consisting on imposing a geometrical hypothesis $H_{i}$,
$i=0,...,3$, on the curve. After each step, some curves will be discarded. At
the end of the reduction process, we obtain the complete set of isochrone
potentials.

In light of the results of Sec. II.2, the first hypothesis is

• $H_{0}$: $\mathscr{C}$ must be an arc of parabola. This geometrical
requirement ensures that the potential $\psi$ is isochrone, regardless of its
mathematical and physical properties. The next hypotheses are therefore
concerned with the parabola $\mathscr{P}$ that contains the curve
$\mathscr{C}$.

####  III.2.1 Existence of orbits and well-posedness around origin

• $H_{1}$: $\mathscr{C}$ must lie on the right half plane. From a purely
mathematical perspective, an isochrone potential is a function $\psi(r)$
defined on some subset of $\mathbb{R}_{+}$ (since $r$ is a positive radius).
Since $x=2r^{2}>0$, we only keep parabolae that exhibit a portion on the right
half plane $x>0$. The only parabolae that do not are the left-oriented ones
not crossing the $y$-axis (Eq. (34) with $b<0$ and $x_{v}\leq 0$).

• $H_{2}$: $\mathscr{C}$ must be convex. Indeed, a particle orbits
periodically when the line $\mathscr{L}$ intersects $\mathscr{C}$ twice and
$\mathscr{C}$ is below $\mathscr{L}$. The geometrical equivalent of this is
that $\mathscr{C}$ should lie under its chords, and therefore be convex. Since
this is not possible on the concave branch of a parabola, we only keep
parabolae that exhibit a convex branch. these that do not are the bottom-
oriented ones151515Strictly speaking, these bottom-oriented parabolae will
contain a unique, circular, unstable orbit. (Eq. (32) with $d<0$.). Therefore,
we discard the bottom-oriented parabolae, and stress that on the right- and
left-oriented ones, the curve of the potential will be an arc $\mathscr{A}$ of
the convex branch $\mathscr{P}_{-}$.

At this stage, with only the three hypotheses $H_{0},H_{1}$ and $H_{2}$, it
turns out that all remaining parabolae are isochrone, in the following sense :
Any parabola with a convex portion on the right half plane defines an
isochrone potential $\psi(r)$, i.e., such that Eq. (2) has a periodic solution
for some $\xi,\Lambda$ with $T(\xi)$. The following sections will be dedicated
to the detailed analysis of these periodic orbits, and serve as a proof of
this result. Nonetheless, we shall focus on parabolae that verify one more
hypothesis.

• $H_{3}$: $\mathscr{C}$ should cross the $y$-axis. This hypothesis discards
right-oriented parabolae that do not cross the $y$-axis. Indeed, these define
a potential $\psi$ on an interval $[r_{v},+\infty[$, where
$x_{v}=2r_{v}^{2}>0$. The associated potential $\psi$ is therefore undefined
in the region $r\in[0,r_{v}[$ surrounding the physical origin, as is the force
$\vec{F}\propto\vec{\nabla}{\psi}$ and the mass density
$\rho\propto\Delta\psi$. We shall coin such potentials “hollow potentials” and
leave them aside, stressing, however, that all our isochrone results apply to
this type of potentials.

After this reduction process, summarized in Fig. 7, all the remaining
parabolae verify the four hypotheses and define the set of isochrone
parabolae. We recover, in particular, the algebraic classification of
isochrone parabolae found in Simon-Petit  _et al._ (2018).

Figure 7: Tree showing the reduction process of isochrone parabolae. Starting
at the top with the implicit equation (31), the reduction consists in
exploring the properties of parabolae associated with the sign of
$\delta,b,d$. Red horizontal lines correspond to a discarded parabola (that
does not satisfy one of the hypotheses $H_{i}$). Black, downwards arrows lead
naturally to the four families $\mathscr{P}_{i}$, remain at the bottom. They
are all associated with isochrone potentials.

In this work, we shall group the isochrone parabolae into four families,
according to their orientation in the $(x,y)$ plane and their number of
intersections with the $y$-axis:

  * •

$\mathscr{P}_{1}$: top-oriented (Eq.(32), with $d<0$),

  * •

$\mathscr{P}_{2}$: left-oriented, crossing the $y$-axis twice (Eq. (34) with
$b<0,4b^{2}e<d^{2}$),

  * •

$\mathscr{P}_{3}$: right-oriented, crossing the $y$-axis twice (Eq. (34) with
$b>0,4b^{2}e>d^{2}$).

  * •

$\mathscr{P}_{4}$: right-oriented, crossing the $y$-axis once (Eq. (34) with
$b>0,4b^{2}e=d^{2}$).

####  III.2.2 Finite mass and attractive nature

Before going further, we would like to discuss some physical properties of the
isochrone potentials associated with the four families
$(\mathscr{P}_{i})_{i=1,2,3,4}$. We start with a (non-necessarily isochrone)
central potential $\psi$ and the Poisson equation (1), from which we can
easily infer the mass contained within a (spherical) shell surrounding the
origin. We choose the units so that $G=1$ in order to simplify the equations.

Let $\epsilon>0$ be the inner radius of such a shell, and $R>\epsilon$ be its
outer radius, so that $\psi(R)$ is well-defined, and we let $M_{\epsilon}(R)$
be the mass contained within this shell $[\epsilon,R]$. By definition,
$M_{\epsilon}(R)$ is given by $\int_{\varepsilon}^{R}\rho(r)4\pi
r^{2}\mathrm{d}r$. Multiplying Eq. (1) by $r^{2}$ and integrating over the
shell $[\epsilon,R]$ readily give

$M_{\epsilon}(R)=R^{2}\psi^{\prime}(R)-\epsilon^{2}\psi^{\prime}(\epsilon)\,.$
(37)

From this equation, it is clear that the total mass $M(R)$ contained within
the sphere or radius $r=R$ is simply given by the $\epsilon\rightarrow 0$
limit of $M_{\epsilon}(R)$. Therefore, for any radius $R$, $M(R)$ is finite if
and only if the rightmost term $\epsilon^{2}\psi^{\prime}(\epsilon)$ in Eq.
(37) remains bounded as $\epsilon\rightarrow 0$. If this limit is infinite,
the potential is sourced by an infinite amount of mass at the physical origin.

With the Hénon variables, it is very simple to see geometrically if $M(R)$ is
infinite or not. Indeed, if we differentiate $Y(x)=2r^{2}\psi(r)$ with respect
to $r$ we obtain $r^{2}\psi^{\prime}(r)=(xY^{\prime}(x)-Y(x))/r$. Evaluating
this at $r=\epsilon$ and Taylor-expanding around $\epsilon=0$ give easily

$\epsilon^{2}\psi^{\prime}(\epsilon)=-\frac{Y(0)}{\epsilon}+o(\epsilon)\,.$
(38)

It is clear from Eqs. (37) and (38) that the mass $M(R)$ is finite if and only
if $Y(0)=0$; a result true for any central potential $\psi$. In other words,
we have the following geometrical result: A potential $\psi$ is sourced by a
finite mass at the origin if and only if its curve $\mathscr{C}$ in Hénon’s
variables passes through the origin. In the isochrone context, this means that
any isochrone parabola whose convex branch does not cross the origin is
associated with an infinite mass at the origin. Moreover, we see on Eq. (38)
that if $Y(0)>0$, i.e., the convex branch crosses the $y$-axis above the
origin, then the central mass is infinite and positive.161616We shall see this
feature at play in the classification of isochrone orbits, later in Sec.
IV.3..

Now if we focus on a potential satisfying $Y(0)=0$, the mass $M(r)$ is finite
within any sphere, and it can be read off of the curve $\mathscr{C}$ as
follows. Plugging $r^{2}\psi^{\prime}(r)=(xY^{\prime}(x)-Y(x))/r$ into Eq.
(37) allows us to write $M(r)$ in the following evocative form

$M(r)=-\frac{Y^{\prime}(x)(0-x)+Y(x)}{r}\,.$ (39)

Notice that the numerator in Eq. (39) is nothing but the $y$-intercept of the
tangent of $\mathscr{C}$ at the point of abscissa $x$. Therefore, given a
central potential, the mass contained within a sphere of radius $r$ can be
measured simply by reading this $y$-intercept. In particular, for the
isochrone potentials the mass within a given sphere is also something that can
be geometrically read off the parabola, as depicted in Fig. 8.

Figure 8: The curve $\mathscr{C}$ is in solid black, and the rest of the
parabola in dashed black. The curve passes through the origin; therefore, the
mass $M(r)$ inside any sphere of radius $r$ is finite. It can be read off as
the $y$-intercept of the tangent at the point of abscissa $x$ (in red). Note
that this construction for the mass holds for any central potential.

####  III.2.3 New parameters

When working in the Hénon plane, with geometry and parabolae, the Latin
parameters $(a,b,c,d,e)$ are useful. In order to work with simpler expressions
when dealing with the potentials, and prepare for the next steps, we rewrite
the equations of the isochrone parabola $(\mathscr{P}_{i})$ with more adapted,
Greek parameters $(\varepsilon,\lambda,\omega,\mu,\beta)$. We will follow the
notations and definitions introduced in Simon-Petit  _et al._ (2018).

We first consider the top-oriented parabolae given by Eq. (32) with $d<0$. We
may combine the constants in the first two terms and define
$\varepsilon:=-c/d\in\mathbb{R}$, $\lambda:=-e/d\in\mathbb{R}$. Moreover,
since $d<0$ in this case, we may always write

$\omega^{2}:=-\frac{16a^{2}}{d}>0\,.$ (40)

Inserting the new parameters $(\varepsilon,\lambda,\omega)$ in Eq. (32) and
using $\psi(r)=Y(2r^{2})/2r^{2}$, we obtain for the first family of potentials
$\psi_{1}$ and/or parabolae $\mathscr{P}_{1}$, associated with top-oriented
parabolae,

• the Harmonic family:

$\psi_{1}(r)=\varepsilon+\frac{\lambda}{2r^{2}}+\frac{1}{8}\omega^{2}r^{2}\quad\leftrightarrow\quad\mathscr{P}_{1}:y=\varepsilon
x+\lambda+\frac{1}{16}\omega^{2}x^{2}\,.$ (41)

The name Harmonic comes from the fact that $\psi_{1}$ is a harmonic potential,
up to a constant $\varepsilon$ and a centrifugal-like term $\lambda/2r^{2}$.
The normalizing factor $1/16$ is chosen such that the radial period $T$
coincides exactly with the angular frequency $\omega$; i.e.,. $T=2\pi/\omega$,
as we shall find later on.

Now consider the $b\neq 0$ case, i.e., the convex branch $\mathscr{P}_{-}$ of
Eq. (34). Once again, we may define $\varepsilon:=-a/b\in\mathbb{R}$. With a
bit of rewriting, we can also introduce the $\lambda$ parameter of Eq. (36)
and move the square root of the rightmost term down, using the usual conjugate
trick. Furthermore, independently of the sign of $b\neq 0$, we may always
define

$\mu:=\sqrt{\frac{\delta|b|}{2b^{4}}}>0\,\quad\text{and}\quad\beta:=\sqrt{\frac{d^{2}-4b^{2}e}{8\delta|b|}}\geq
0\,.$ (42)

When $\beta\neq 0$, the parabola crosses the $y$-axis twice. Then either $b>0$
(right-oriented parabola) or $b<0$ (left-oriented parabola), as depicted in
Fig. 7. Inserting the new parameters $(\varepsilon,\lambda,\mu,\beta)$ in Eq.
(34) and using $\psi(r)=Y(2r^{2})/2r^{2}$, we obtain the second family of
potentials $\psi_{2}$ and/or parabolae $\mathscr{P}_{2}$, associated with
left-oriented parabolae,

• the Bounded family:

$\psi_{2}(r)=\varepsilon+\frac{\lambda}{2r^{2}}+\frac{\mu}{\beta+\sqrt{\beta^{2}-r^{2}}}\quad\leftrightarrow\quad\mathscr{P}_{2}:y=\varepsilon
x+\lambda+2\mu(\beta-\sqrt{\beta^{2}-x/2})\,.$ (43)

The name Bounded comes from the fact that $\psi_{2}$ is defined only on the
bounded interval $[0,\beta]$. Finally, the same parameters can be used to
define the third family of potentials $\psi_{3}$ and/or parabolae
$\mathscr{P}_{3}$, associated with right-oriented parabolae crossing the
$y$-axis twice, namely

• the Hénon family:

$\psi_{3}(r)=\varepsilon+\frac{\lambda}{2r^{2}}-\frac{\mu}{\beta+\sqrt{\beta^{2}+r^{2}}}\quad\leftrightarrow\quad\mathscr{P}_{3}:y=\varepsilon
x+\lambda+2\mu(\beta-\sqrt{\beta^{2}+x/2})\,.$ (44)

The name Hénon comes from the fact that $\psi_{3}$ is, up to a constant and a
centrifugal-like term, the potential found by Michel Hénon Hénon (1959a).
Finally, the case $\beta=0$ makes up for the fourth and last family of
potentials $\psi_{4}$ and/or parabolae $\mathscr{P}_{4}$, associated with
right-oriented parabolae crossing the $y$-axis once, namely

• the Kepler family:

$\psi_{4}(r)=\varepsilon+\frac{\lambda}{2r^{2}}-\frac{\mu}{r}\quad\leftrightarrow\quad\mathscr{P}_{4}:y=\varepsilon
x+\lambda-\mu\sqrt{2x}\,.$ (45)

The name Kepler comes from the fact that $\psi_{4}$ is, up to a constant and a
centrifugal-like term $\lambda/2r^{2}$, the usual Kepler potential.

With the new parameters, the result of this section can be summarized easily:
If a potential $\psi$ is isochrone, then it must be equal to one of the
$\psi_{i}$ with $i=1,2,3$ or $4$. In other words, if $\psi$ is isochrone, then
there exists some constants $(\varepsilon,\lambda)$ and $(\mu,\beta,\omega)$
such that $\psi(r)=\psi_{i}(r)$ for some $i=1,2,3,4$. We stress that by
definition, $\omega\neq 0$ , $\mu>0$ and $\beta\geq 0$. However, $\varepsilon$
and $\lambda$ are defined in such a way that they can take any real value, a
priori.

Lastly, let us make contact with Simon-Petit  _et al._ (2018) once again and
introduce some qualifiers for the different potentials. Notice that the
parameter $\lambda$ in Eqs. (41)-(45) is precisely the $y$-intercept of the
parabola, i.e., $Y(0)=\lambda$. According to our findings in Sec. III.2.2, any
potential with $\lambda\neq 0$ will be associated with an infinite mass at the
origin. Such potentials are coined gauged potentials in Simon-Petit  _et al._
(2018), as the term $\lambda/2r^{2}$ that makes them special looks like a
gauged angular momentum $\Lambda^{2}\rightarrow\Lambda^{2}+\lambda$ in the
effective potential formalism. Gauged potential are opposed to so-called
physical potentials that have $\lambda=0$, and that are associated with a
finite mass at the center.

###  III.3 Complete set of isochrone parabolae

In the last section, we have isolated four families of potentials $(\psi_{i})$
and their associated parabolae $(\mathscr{P}_{i})$, and discussed some of
their properties. We have shown that if a potential is isochrone, then its
curve $\mathscr{C}:y=Y(x)$ is the portion of a parabola $\mathscr{P}_{i}$ that
is convex and lies on the right half plane $x>0$.

What remains to be shown is the reciprocal of this statement, i.e., that any
such $\mathscr{C}$ is the curve of an isochrone potential. To this end, we
just need to show that $\mathscr{C}$ can always be intersected twice by some
line $\mathscr{L}:y=\xi x-\Lambda^{2}$. We shall do this by finding explicitly
which lines $\mathscr{L}$ can intersect $\mathscr{C}$. In so doing, we will
find two important results:

  * •

the set $(\psi_{i})$ is complete, i.e., it contains all and only the isochrone
potentials, and

  * •

necessary and sufficient conditions on $(\xi,\Lambda)$ such that the
particle’s orbit is bounded.

In what follows, we consider a curve $\mathscr{C}$ and a line
$\mathscr{L}:y=\xi x-\Lambda^{2}$. By assumption, $\mathscr{C}$ is on the
right half plane and on the convex portion of a parabola $\mathscr{P}_{i}$
given by Eqs. (41)-(45). When they exist, we denote by $x_{P}$ and
$x_{A}>x_{P}$ the abscissae of $P$ and $A$, the two intersections of
$\mathscr{L}$ with $\mathscr{C}$.

####  III.3.1 Top-oriented parabolae $\mathscr{P}_{1}$

Let us start with the family $\mathscr{P}_{1}$ given in Eq. (41). We fix the
parameters $(\varepsilon,\lambda,\omega)$ and look for the conditions on
$(\xi,\Lambda)$ under which the line $\mathscr{L}:y=\xi x-\Lambda^{2}$
intersects $\mathscr{P}_{1}$ twice. By definition, $P,A$ belong to both
$\mathscr{L}$ and $\mathscr{P}_{1}$; therefore, $x_{P},x_{A}$ are solutions to
$\xi x-\Lambda^{2}=\varepsilon x+\lambda+\omega^{2}x^{2}/16$. We may
equivalently write this equation in the following evocative form

$x^{2}-sx+p=0\,,\quad\quad\text{where}\quad\,\begin{cases}s:=16(\xi-\varepsilon)/\omega^{2}\,,\\\
p:=16(\Lambda^{2}+\lambda)/\omega^{2}\,,\end{cases}$ (46)

with the sum $s=x_{P}+x_{A}$ and product $p=x_{P}x_{A}$ of the two roots. Note
that $x_{P},x_{A}$ and therefore $s$ and $p$ are all functions of $\xi$ and
$\Lambda^{2}$. For a generic quadratic equation such as Eq. (46), two
solutions exist if and only if $\Delta:=s^{2}-4p>0$, and are given by
$(s\pm\sqrt{\Delta})/2$, with a minus sign for $x_{P}$ and a plus sign for
$x_{A}$. We want these solutions to lie on the convex branch of the parabola,
in order for an orbit to actually exist. This is always satisfied since the
parabola $\mathscr{P}_{1}$ is everywhere convex. Furthermore, we want them to
be strictly positive, in order for $P$ and $A$ to be in the right half plane
$x\geq 0$. It is sufficient to require $x_{P}>0$, because then
$x_{A}>x_{P}>0$.

The parameters of the potential are fixed; therefore, the condition
$x_{P}(\xi,\Lambda^{2})>0$, along with $\Lambda^{2}\geq 0$, defines a region
in the $(\xi,\Lambda^{2})$ plane that contains every pair $(\xi,\Lambda^{2})$
such that the orbit is periodic. Using the formula for $x_{P}$ in terms of $s$
and $p$, this domain is explicitly delimitated by the two inequalities

$\Lambda^{2}\geq 0\quad\text{and}\quad x_{P}(\xi,\Lambda^{2})>0\,,$ (47)

with $x_{P}=(s-\sqrt{s^{2}-4p})/2$ and $s,p$ given by Eq. (46). Outside this
region, there may be collision orbits (the particle avoids the origin and goes
to infinity), or no orbit at all (for instance in the $\Lambda^{2}<0$ region).
This is depicted in Fig. 9.

Figure 9: Bifurcation diagram for a harmonic potential $\psi_{1}$, with
$\lambda\geq 0$ (left) and $\lambda\leq 0$ (right). The axes are
$\xi=\varepsilon$ and $\Lambda^{2}=0$. The light grey region defined by the
inequalities (47) contains the $(\xi,\Lambda^{2})$ associated with bounded
motion. Following the convention of Arnold (Fig. (2.3) of Arnol’d (1995)), the
region of possible motion in the physical space is depicted as a light-red
region in the orbital plane. For a generic orbit, $0<r_{P}<r_{A}$ and the
motion takes place in an annulus. The black boundaries correspond to
degeneracies: The top one to circular motion $(r_{P}\rightarrow r_{A})$ and
the bottom one to trajectories spiraling toward the center $(r_{P}\rightarrow
0)$.

####  III.3.2 Left-oriented parabolae $\mathscr{P}_{2}$

We proceed similarly for the left-oriented parabolae $\mathscr{P}_{2}$,
associated with a Bounded potential. In particular, we fix the parameters
$(\varepsilon,\lambda,\mu,\beta)$ and look for a domain in the
$(\xi,\Lambda^{2})$ plane that contains all and only the periodic orbits. The
condition $P,A\in\mathscr{L}\cap\mathscr{P}_{2}$ translates algebraically into

$\xi x-\Lambda^{2}=\varepsilon
x+\lambda+2\mu\beta-2\mu\sqrt{\beta^{2}-x/2}\,.$ (48)

As for $\mathscr{P}_{1}$, with a bit of algebra we may write Eq. (48) as
$x^{2}-sx+p=0$ where $s:=x_{P}+x_{A}$ and $p:=x_{P}x_{A}$. In terms of
$(\xi,\Lambda^{2})$, we have explicitly

$s:=2\,\frac{(\Lambda^{2}+\lambda+2\mu\beta)(\xi-\varepsilon)-\mu^{2}}{(\xi-\varepsilon)^{2}}\quad\text{and}\quad
p:=\frac{(\Lambda^{2}+\lambda+4\mu\beta)(\Lambda^{2}+\lambda)}{(\xi-\varepsilon)^{2}}\,.$
(49)

As before, the solutions $x_{P},x_{A}$ must be strictly positive and this is
ensured by the condition $x_{P}>0$. At this point, choosing
$(\xi,\Lambda^{2})$ in the region $x_{P}(\xi,\Lambda^{2})>0$ ensures that
there are two intersections between $\mathscr{P}_{2}$ and $\mathscr{L}$.
Adding the condition $\Lambda^{2}\geq 0$ ensures that $x_{P}$ is on the convex
branch. However, a third condition is needed, namely that $A$ belongs to the
convex branch. To this end, notice that in Eq. (48), the minus sign in front
of $2\mu$ came from selecting the convex branch of the parabola. Therefore, to
ensure that $A$ is on this branch, it is sufficient to impose that $\xi
x_{A}-\Lambda^{2}\leq\varepsilon x_{A}+\lambda+2\mu\beta$. All in all, three
conditions are sufficient to draw the bifurcation diagram. They read
explicitly

$\Lambda^{2}\geq 0\,,\quad
x_{P}(\xi,\Lambda^{2})>0\quad\text{and}\quad(\xi-\varepsilon)\,x_{A}(\xi,\Lambda^{2})\leq\Lambda^{2}+\lambda+2\mu\beta\,.$
(50)

Once we express $x_{P}$ and $x_{A}$ in terms of $s,p$, and thus in terms of
$(\xi,\Lambda^{2})$ via Eq. (49), the three inequalities (50) delimit a region
with all and only the periodic orbits. This region is depicted for a typical
bounded potential $\psi_{2}$ in Fig. 10.

Figure 10: Bifurcation diagram for a bounded potential $\psi_{2}$, with
$\lambda\geq 0$ (left) and $\lambda\leq 0$ (right). The axes are
$\xi=\varepsilon+\mu/2\beta$ and $\Lambda^{2}=0$. All information is encoded
the same way as in Fig. 9.

####  III.3.3 Right-oriented parabolae $\mathscr{P}_{3}$ and $\mathscr{P}_{4}$

For the right-oriented parabolae, we proceed the same way. Noticing that a
parabola $\mathscr{P}_{3}$ can be obtained as the $\beta\rightarrow 0$ limit
of a parabola $\mathscr{P}_{4}$, we may focus on the latter. As before, the
condition $P,A\in\mathscr{P}_{4}$ translates into $x^{2}-sx+p=0$, where $s$
and $p$ are have the same expression as in Eq. (49), albeit with a plus sign
in front of $\mu^{2}$ for the former.

The conditions to be imposed to have a well-defined orbit are as before,
$x_{P}>0$, $\Lambda^{2}\geq 0$. These two ensure that $P$ is in the right half
plane and on the convex branch. However, this does not imply that $A$ is on
the convex branch, so we must, again, add a third condition. Therefore, all
three requirements are the same as in the $\mathscr{P}_{2}$ case, and the
bifurcation diagram can be depicted using the inequalities Eq. (50) (again,
with a plus sign in front of $\mu^{2}$). For the Hénon family, $\beta\neq 0$
and the bifurcation diagram is depicted in Fig. 11. The bifurcation diagram
for the Kepler family $\mathscr{P}_{4}$ is the $\beta\rightarrow 0$ limit of
Fig. 11 and coincides precisely with Fig. (2.3) of Arnol’d (1995).

Figure 11: Bifurcation diagram for a Hénon potential $\psi_{3}$, with
$\lambda\geq 0$ (left) and $\lambda\leq 0$ (right). The axes are
$\xi=\varepsilon-\mu/2\beta$ and $\Lambda^{2}=0$. All information is encoded
the same way as in Fig. 9.

##  IV Dynamics in isochrone potentials

Johannes Kepler published in his Astronomia Nova a set of three laws that
nowadays rightfully bear his name. The first law states that planets follow
elliptical orbits around the Sun. The second law states that an orbiting
planet always sweeps equal areas in equal times, and this holds for any
central potential, hence for isochrone ones. Last but not least, the third law
is arguably one of the most celebrated and useful equations in astronomy and
astrophysics. In modern notation, this law reads

$T^{2}=\frac{\pi^{2}}{2}\frac{\mu^{2}}{(-\xi)^{3}}$ (51)

with $T$ the period of motion171717Around a point mass, a test particle orbits
on a closed ellipse whose semimajor axis $a$ is such that $a\propto\xi^{-1}$
Arnol’d (1995). Consequently, Kepler’s third law is commonly written as
$T^{2}\propto a^{3}$. Note that here the radial period coincides with that of
the motion, i.e., $\vec{r}(t+T)=\vec{r}(t)$. of a test particle of energy
$\xi<0$ orbiting a point mass $\mu=GM$.

This fourth and last section is mainly dedicated to a generalization of
Kepler’s first and third laws. Regarding the third law, we will show that it
is actually inherent to isochrony, in the sense that any periodic orbit in any
isochrone potential satisfies a strikingly similar law. We shall interpret
this law in various geometrical contexts and will also point out a similar and
unified law for the apsidal angle $\Theta$. For the first law, we will provide
an explicit formula for isochrone orbits in polar coordinates, and show that
all isochrone orbits can be parameterized by a Keplerian ellipse. Finally, we
shall use these results to exhibit and classify isochrone orbits, i.e., orbits
of test particles in an isochrone potential.

###  IV.1 Isochrone Kepler’s laws for $T$ and $\Theta$

This subsection is divided into three parts. In Sec. IV.1.1 we derive the
generalized Kepler’s third law for the period $T$ in any isochrone potential,
by solving quadratic equations. Then in Sec. IV.1.2 we use the circular orbit
trick to get a similar law for $\Theta$. Lastly in Sec. IV.1.3 we provide an
alternative formulation of these laws, in terms of purely geometrical
quantities that can all be inferred solely from the parabola $\mathscr{P}$ and
the line $\mathscr{L}$.

####  IV.1.1 Laws for the radial period $T$

Let us consider a generic isochrone potential $\psi$ and a particle
$(\xi,\Lambda)$ that orbits periodically within it. As we did many times
above, in the Hénon plane $\psi$ is associated with a parabola $\mathscr{P}$,
and the particle is associated with a line $\mathscr{L}$ that intersects
$\mathscr{P}$. The equations for $\mathscr{P}$ and $\mathscr{L}$ are of the
form

$\mathscr{P}:(ax+by)^{2}+cx+dy+e=0\quad\text{and}\quad\mathscr{L}:y=\xi
x-\Lambda^{2}\,.$ (52)

Keeping the parabola $\mathscr{P}$ fixed, we take a line $(\xi,\Lambda)$ with
two intersections $P$ and $A$, both functions of $(\xi,\Lambda)$. Now since
$A$ and $P$ belong to both $\mathscr{P}$ and $\mathscr{L}$, we can eliminate
$y$ from the two equations in (52) and get an equation on $x$ whose solutions
are $x_{P}$ and $x_{A}$, the abscissa of $P$ and $A$. Re-arranging the result
gives the following quadratic equation:

$(a+b\xi)^{2}x^{2}+[(c+d\xi)-2b\Lambda^{2}(a+b\xi)]x+[b^{2}\Lambda^{4}+e-d\Lambda^{2}]=0\,,$
(53)

whose discriminant $\Delta$ is given by

$\Delta(\xi,\Lambda):=(c+d\xi)^{2}-4b\Lambda^{2}(c+d\xi)(a+b\xi)-4(a+b\xi)^{2}(e-d\Lambda^{2})\,.$
(54)

We will now compute the period $T$ with the help of Hénon’s formula for $T$
given in Eq. (21) proved in Sec. IV.1. First, we need the difference between
$x_{A}$ and $x_{P}$. This is simply a matter of writing the solutions to a
quadratic equation. We obtain easily

$(x_{A}-x_{P})^{2}=\frac{\Delta(\xi,\Lambda)}{(a+b\xi)^{4}}\,.$ (55)

Second we need a formula for $\Lambda_{C}$, the angular momentum of the
circular orbit with energy $\xi$. To find it, we notice that when we keep
$\xi$ fixed, the discriminant $\Delta(\xi,\Lambda)$ is strictly positive when
there are two intersections, and by definition vanishes for some
$\Lambda_{C}$, when there is only one intersection. This corresponds to a
circular orbit, obtained by translating $\mathscr{L}$ downward until $P$ and
$A$ degenerate into a single point $C$. Therefore, we have
$\Delta(\xi,\Lambda_{C})=0$, which may be solved for $\Lambda_{C}$. With the
help of Eq. (53) and some elementary algebra, we obtain

$\Lambda_{C}^{2}=\frac{4e(a+b\xi)^{2}-(c+d\xi)^{2}}{4\delta(a+b\xi)}\,,$ (56)

where $\delta$ is the discriminant of the parabola, cf. (31). Now we can
combine Hénon’s formula (21) with Eqs. (55) and (56). Once again, after some
easy algebra we obtain the following generalization of Kepler’s third law

$T^{2}=-\frac{\pi^{2}}{4}\frac{\delta}{(a+\xi b)^{3}}.$ (57)

A few remarks are in order here. First of all, we stress that Eq. (57) is
valid for any particle $(\xi,\Lambda)$ orbiting periodically in any isochrone
potential. In particular, this law is valid even for the potentials that were
discarded earlier, i.e., these that are decreasing around the origin, these
that contain infinite mass at the center and even the hollow ones, undefined
around the origin. As long as there is a periodic orbit in a isochrone
potential, physical or not, there is an associated parabola $\mathscr{P}$
given by Eq. (52) and its radial period $T$ verifies Eq. (57).

Second, we see that it involves in the numerator $\delta$ which is strictly
positive. Therefore, Eq. (57) implies that the denominator is strictly
negative and thus that $a+b\xi<0$. This is a general property that can be
traced back to the very existence of solutions to the quadratic equation (53).
We shall use this result later in Sec. IV.2 to find a parameterization of
isochrone orbits. Moreover, speaking of the parameters, we recover the two
well-known cases: When $a=0$ the parabola has horizontal symmetry and we have
$T^{2}\propto\xi^{-3}$, as in the Kepler potential. Similarly, when $b=0$, the
parabola has vertical symmetry and we have $T=\text{cst}$, i.e., $T$ is
independent of the properties of the particle, as for the harmonic potential.

####  IV.1.2 Laws for the apsidal angle $\Theta$

All the results presented in the last paragraphs regarding $T$ are also true
for the apsidal angle $\Theta$. In particular, we can use the Hénon formula
(22) in order to write the apsidal angle for any orbit solely in terms of
$\Lambda$ and the parameters $(a,b,c,d,e)$. To this end, we start, as usual,
with some geometry.

Consider a line $\mathscr{L}$ intersecting a generic isochrone parabola
$\mathscr{P}$, both given by Eq. (52). Since $\Theta$ is independent of $\xi$,
we may choose a value of $\xi$ such that the orbit is circular. This can be
done as follows. Keeping $\Lambda$ fixed, decreasing $\xi$ defines other lines
with the same $\Lambda$ and thus the same apsidal angle $\Theta(\Lambda)$ for
the associated orbits. In particular, $\xi$ can reach a critical value
$\xi_{C}$ such that the line $\mathscr{L}$ becomes tangent to $\mathscr{P}$,
at some point of abscissa $x_{C}$. It is important to notice that $\xi_{C}$
and $x_{C}$ are function of $\Lambda$ only.

Let us focus on this very line $\mathscr{L}_{C}:y=\xi_{C}x-\Lambda^{2}$ and
the associated circular orbit. Its orbital radius is $r_{C}$, such that
$2r_{C}^{2}=x_{C}$. The period $T(\xi_{C})$ of this orbit is given by Eq.
(57). Now by definition of the angular momentum, we have, for this circular
orbit $\Lambda=r_{C}^{2}\dot{\theta}$. Since $x_{C}=2r_{C}^{2}$, this can be
turned into a differential equality
$2\Lambda\mathrm{d}t=x_{C}\mathrm{d}\theta$. Now, by definition of $\Theta$,
integrating the latter over a period $T(\xi_{C})$ readily gives

$\Theta=2\Lambda\frac{T(\xi_{C})}{x_{C}}\,.$ (58)

Let us stress again that both $\xi_{C}$ and $x_{C}$ are functions of
$\Lambda$, thus we just need a formula for these in terms of $(a,b,c,d,e)$ and
$\Lambda$. Now we can apply the results of the last section, regarding the
intersections of $\mathscr{P}$ and $\mathscr{L}$, below Eq. (52). In
particular, in the present context $\xi_{C}$ is such that
$\Delta(\xi_{C},\Lambda)=0$, with $\Delta$ given by Eq. (54). After some
algebra, the solution for $\xi_{C}$ is easily found to be

$\xi_{C}(\Lambda)=-\frac{a\Xi_{C}-c}{b\Xi_{C}-d}\quad\text{where}\quad\Xi_{C}:=2b\Lambda\pm
2\sqrt{b^{2}\Lambda^{4}-d\Lambda^{2}+e}\,,$ (59)

and the $+/-$ sign should be used for the left-/right-oriented parabolae,
respectively. Regarding the quantity $x_{C}$, it can be found by writing the
solution to Eq. (53) when $\Delta=0$. We obtain easily

$x_{C}(\Lambda)=\frac{b\Lambda^{2}}{a+b\xi_{C}}-\frac{c+d\xi_{C}}{2(a+b\xi_{C})^{2}}\,,$
(60)

where $\xi_{C}$ is given in terms of $\Lambda$ by Eq. (59).

Now it is just a matter of inserting Eqs. (60) and (59) into Eq. (58) and do
some algebra to obtain a formula for $\Theta$. After a rather lengthy but
simple computation, we obtain the following law, valid for any particle
orbiting in any isochrone potential

$\frac{\Theta^{2}}{\pi^{2}\Lambda^{2}}=\frac{2b^{2}\Lambda^{2}-d}{b^{2}\Lambda^{4}-d\Lambda^{2}+e}+\frac{2b}{\sqrt{b^{2}\Lambda^{4}-d\Lambda^{2}+e}}\,.$
(61)

As for Eq.(57), we stress that Eq. (61) is valid for any orbit in any
isochrone potential, even the gauged and hollow ones (discarded at the end of
Sec. III.2 due to unusual physical properties. As a corollary of this general
formula, one may insert the Greek parameters
$(\varepsilon,\lambda,\omega,\mu,\beta)$ introduced earlier, and find
agreement with the results of Simon-Petit  _et al._ (2018).

####  IV.1.3 geometrical reading of the third laws

The computation of the period $T$ and apsidal angle $\Theta$ via Eqs. (57) and
(61) involves the parameters $a,b,c,d,e$, and can thus be made only if we know
the algebraic equation of the parabola. Here, we show that it is also possible
to express $T$ and $\Theta$ entirely in terms of geometrical quantities, i.e.,
compute them solely with Euclidean geometry, once a parabola $\mathscr{P}$ and
a line $\mathscr{L}$ is drawn in the Hénon plane. For the period $T$, we need
to define three geometrical quantities

  * •

$\vec{\xi}:=(1,\xi)$, the natural tangent vector to the line $\mathscr{L}$,

  * •

$\vec{N}=(a,b)$, the natural tangent vector to the symmetry axis of
$\mathscr{P}$ (which controls its orientation), and

  * •

and $R=\delta/2|\vec{N}|^{3}$, the radius of curvature of $\mathscr{P}$ at its
apex (which controls its aperture).

The expressions for these quantities can be easily derived with the help of
Sec. III.1. We can now rewrite Eq. (57) without the $(a,b,c,d,e)$ parameters,
using the unit vector $\vec{n}=\vec{N}/|\vec{N}|$, simply as

$T^{2}=\frac{\pi^{2}}{2}\frac{R}{|\vec{n}\cdot{\vec{\xi}}\,|^{3}}\,.$ (62)

This formula should be compared to Kepler’s third law for the radial period as
given by Eq. (51). In a similar fashion, we can make a geometrical
construction for the law of the apsidal angle $\Theta$. In particular, let
$\ell_{\pm}$ be the ordinate of the intersection points between the branch
$\mathscr{P}_{\pm}$ and the $y$-axis. As we have seen already in Sec. III.1,
we have

$\ell_{\pm}=-\frac{d\pm\sqrt{d^{2}-4b^{2}e}}{2b^{2}}\,$ (63)

In terms of the Greek parameters, we can show easily that the quantity
$\ell_{+}$ is nothing but $\lambda$, and that $\ell_{-}$ is
$\lambda+4\mu\beta$. Now we can easily turn Eq. (63) into
$b^{2}\Lambda^{4}-d\Lambda^{2}+e=b^{2}(\ell_{+}+\Lambda^{2})(\ell_{-}+\Lambda^{2})$
and notice that the left-hand side appears precisely in the denominator in Eq.
(61). Therefore, we insert this result in Eq. (61), make a partial fraction
decomposition for the first term and obtain

$\Theta=\frac{\pi}{\sqrt{1+\ell_{+}/\Lambda^{2}}}+\frac{\iota\pi}{\sqrt{1+\ell_{-}/\Lambda^{2}}}\,,$
(64)

where $\iota\in\\{-1,0,1\\}$ is simply the sign of $b$ and determines the
orientation (resp. left, top, bottom) of the parabola. As for the period $T$,
we see that the apsidal angle $\Theta$ can be found with only geometrical
quantities that can be read off the parabola. In particular,
$\ell_{\pm}/\Lambda^{2}$ is simply the ratio of the vertical distances between
the $y$-intercept of $\mathscr{L}$ and the branches181818For the Kepler
family, the two intersections degenerate into one and $\ell_{+}=\ell_{-}$. For
the Harmonic family, $\ell_{+}$ goes to $+\infty$ (think of a $\pi/2$-rotation
turning $y=-\sqrt{x}$ into $y=x^{2}$). $\mathscr{P}_{\pm}$, as depicted in
Fig. 12.

Figure 12: geometrical quantities involved in the geometrical laws for $T$
(Eq. (62)) and for $\Theta$ (Eq. (64)). In this figure, $\vec{n}$ is a unit
vector and gives the asymptotic direction of the parabola, $R$ is the
curvature radius at the apex, and $\ell_{\pm}$ are the vertical distances
between the $y$-intercept of $\mathscr{L}$ and the $y$-intercept of
$\mathscr{P}_{\pm}$.

Finally, at the end of App. B we provide yet another form for the third law
$T(\xi)$ and $\Theta(\Lambda)$. For $T$, this alternative form involves the
slope and curvature of the parabola at the circular point $C$ uniquely
associated with the orbit. For $\Theta$, this alternative involves the
curvature of the effective potential, when looked at in the Binet variable
$u=1/r$. The interested reader may find these results useful in order to go
further into the geometrical properties of isochrony, e.g., for academic
purposes. Along these lines, we stress that what led to the mathematical
equalities (57), (61) and other third laws in App. B can be seen as a
geometrical method to compute the rather complex-looking integrals (14) and
(22), with $Y(x)$ given by Eq. (32) or Eq. (34). In particular, the fact that
these rather complex-looking integrals do not depend on $\Lambda$ for $T$, and
on $\xi$ for $\Theta$, is quite remarkable.

###  IV.2 Isochrone orbits transformations

In this section, we provide a geometrical analysis that leads naturally to an
explicit and analytic parameterization of any isochrone orbit in polar
coordinates. The essential idea is the following: An isochrone orbit is
associated with an arc of parabola in the Hénon plane. There is one isochrone
orbit for which we know an analytic expression: The Keplerian ellipse. Using
linear transformations in the Hénon plane, we show how to map any arc of
parabola to a Keplerian one, and therefore establish a one-to-one
correspondence between any isochrone orbit and a Keplerian ellipse, the latter
being used to parameterize the former.

####  IV.2.1 Reduced orbit

As we have seen many times before, an arc of parabola $\mathscr{A}$ in the
Hénon plane $(x,y)$ is associated with an isochrone orbit in the physical
space that will be denoted by $\mathscr{O}$. By conservation of angular
momentum, the particle orbits within a plane, equipped with the usual polar
coordinates $(r,\theta)$. We shall always choose the angle $\theta$ such that
$\theta=0$ at periapsis $r=r_{P}$.

When the particle moves on an isochrone orbit $\mathscr{O}$, its radius $r$
changes periodically and can be mapped to a point $M$ that travels back and
forth on the arc $\mathscr{A}$. However, the converse is not true: A point
$M\in\mathscr{A}$ of abscissa $x=2r^{2}$ corresponds to an infinite number of
points on $\mathscr{O}$, namely the points
$(r,\theta+k\Theta)_{k\in\mathbb{Z}}$, precisely because of the radial
periodicity. To get a one-to-one correspondence, we can quotient the full
orbit $\mathscr{O}$ by reflexions and rotations, to get the reduced orbit
$\mathscr{O}_{o}$, as depicted in Fig. 13. The full orbit $\mathscr{O}$ can be
constructed from $\mathscr{O}_{o}$, which acts as a generator of the orbit and
which, contrary to the full orbit $\mathscr{O}$, is in a one-to-one
correspondence with the arc $\mathscr{A}$: a point
$(r,\theta)\in\mathscr{O}_{o}$ is uniquely linked to a point
$(x,y)\in\mathscr{A}$ via $x=2r^{2}$.

Figure 13: The construction of a full orbit $\mathscr{O}$ (right) from a a
reduced orbit $\mathscr{O}_{o}$ (left). All points of $\mathscr{O}_{o}$ have a
different radius $r$ varying increasingly between periapsis $r=r_{P}$ and
apoapsis $r=r_{A}$. By reflexion with respect to $\theta=0$, one obtains the
portion travelled in one radial period $T$, and the opening angle is therefore
$\Theta$. By successive rotations of angle $\Theta$, the full orbit
$\mathscr{O}$ can be constructed.

####  IV.2.2 Kepler parabola

We will need in this section a few results about the Kepler parabola
$y^{2}=2\mu^{2}x$. It is associated with the usual Kepler potential
$\psi(r)=-\mu/r$. The bifurcation diagram in terms of $(\xi,\Lambda)$ for the
Kepler potential is given in Fig. 11 with $\beta=\varepsilon=\lambda=0$, or
equivalently in Fig. (2.3) of Arnol’d (1995). Requiring that the values of
$(\xi,\Lambda)$ generate a periodic orbit (i.e., that they are in the grey
region of Fig. 11) is equivalent to the following algebraic inequalities:

$\Lambda^{2}\geq 0\quad\text{and}\quad
0>\xi\geq-\frac{\mu^{2}}{2\Lambda^{2}}\,.$ (65)

In the Kepler potential, any bounded orbit is an ellipse whose focus is at the
center of polar coordinates. An elliptic orbit $\mathscr{O}$ is made of one
periapsis and one apoapsis, joined by two symmetrical portions, as can be seen
in the bottom left of Fig. 14. We will take the reduced orbit
$\mathscr{O}_{o}$ to be the upper portion. From the classical solution to the
Kepler problem (see, e.g., Arnol’d (1995)), the reduced orbit
$\mathscr{O}_{o}$ can be given the following parametric
representation191919Note that if the periapsis is at $(r,\theta)=(r_{P},0)$,
the reduced orbit $\mathscr{O}_{o}$ is simply the upper half of $\mathscr{O}$.

$\begin{cases}r(s)=p\,(1+\varepsilon\cos s)^{-1}\\\
\theta(s)=s\end{cases}\\!\\!\\!\\!\\!\\!\\!,\quad s\in[0,\pi]\,.$ (66)

In these equations, $\varepsilon$ is the eccentricity of the orbit, and $p$ is
its semi-latus rectum Arnol’d (1995). They depend explicitly on the energy
$\xi$ and angular momentum $\Lambda$ of the particle, as well as the central
mass $\mu:=GM$ of the Keplerian potential. They are given by

$\varepsilon:=\sqrt{1+\frac{2\Lambda^{2}\xi}{\mu^{2}}}\quad\text{and}\quad
p:=\frac{\Lambda^{2}}{\mu}\,,$ (67)

and one can see that the conditions (65) are actually equivalent to
$\varepsilon\in[0,1[$ and $p\geq 0$.

####  IV.2.3 Linear transformation

We consider a reduced isochrone orbit $\mathscr{O}_{o}$ in a central isochrone
potential with finite mass at the origin. Recall that $\mathscr{O}_{o}$ is in
a one-to-one correspondence with an arc of parabola $\mathscr{A}$ in the Hénon
plane that passes through the origin (as explained in Sec. III.2.2). This arc
$\mathscr{A}$ is defined as the portion of a parabola $\mathscr{P}$ that lies
below a given line $\mathscr{L}$, both given by Eq. (52) with $e=0$

$\mathscr{P}:(ax+by)^{2}+cx+dy=0\quad\text{and}\quad\mathscr{L}:y=\xi
x-\Lambda^{2}\,.$ (68)

Now let us apply the following linear transformation $F$ to the Hénon plane

$F:(x,y)\mapsto(-cx-dy,ax+by)=:(\bar{x},\bar{y})\,.$ (69)

Following Eq. (69), any quantity $X$ that has been mapped by $F$ will be
denoted with a bar as $\bar{X}:=F(X)$. For instance, a point $M$ of
coordinates $(x,y)$ on $\mathscr{A}$ will be mapped to the point $\bar{M}$
with coordinates $(\bar{x},{\bar{y}})=F(x,y)$ given by Eq. (69). Since the set
of parabolae and the set of lines are stable under affine transformations (and
thus linear ones), $\bar{\mathscr{P}}=F(\mathscr{P})$ is still a parabola and
$\bar{\mathscr{L}}=F(\mathscr{L})$ still a line. The parameters $(a,b,c,d)$ in
Eq. (69) are precisely these of the parabola $\mathscr{P}$ in Eq. (68) and are
not chosen randomly. It is straightforward to find its implicit equation which
reads

$\bar{\mathscr{P}}:\bar{y}^{2}=\bar{x}\,.$ (70)

In view of the previous subsection, it is clear from Eq. (70) that $F$ maps
$\mathscr{P}$ to a Keplerian parabola with mass parameter $\mu=1/\sqrt{2}$.
Regarding the image $\bar{\mathscr{L}}$ of the line $\mathscr{L}$, a quick
computation gives the following equation

$\bar{\mathscr{L}}:\bar{y}=\bar{\xi}\bar{x}-\bar{\Lambda}^{2}\,,\quad\text{with}\quad\bar{\xi}=-\frac{a+b\xi}{c+d\xi}\quad\text{and}\quad\bar{\Lambda}^{2}=-\frac{\delta\Lambda^{2}}{c+d\xi}\,.$
(71)

The image arc $\bar{\mathscr{A}}$ is a portion of $\bar{\mathscr{P}}$,
although we do not know yet if it lies below $\bar{\mathscr{L}}$. If it does,
then $\bar{\mathscr{A}}$ is a Keplerian arc and the associated orbit is an
ellipse. Let us first ensure that $\bar{\mathscr{A}}$ indeed corresponds to a
well-defined elliptic orbit. According to the inequalities (65) with
$\mu=1/\sqrt{2}$, a Keplerian orbit is periodic provided that
$\bar{\Lambda}^{2}\geq 0$, $\bar{\xi}<0$ and
$\bar{\xi}>-1/4\bar{\Lambda}^{2}$. We now argue that these three conditions
are always satisfied, in the three following steps:

  * •

From Kepler’s generalized third law (57) and $\delta>0$, we have $a+b\xi<0$.
Consequently, by Eq. (71), the two conditions $\bar{\xi}<0$ and
$\bar{\Lambda}^{2}\geq 0$ hold if and only if $c+d\xi<0$.

  * •

The condition $c+d\xi<0$ is a geometrical consequence of all hypotheses
$(H_{i})$ that are, by assumption, verified since the initial orbit
$\mathscr{O}$ is isochrone. (The proof is easy but not central here; it can be
found in App. E.) At this stage, we thus have $\bar{\xi}<0$ and
$\bar{\Lambda}^{2}\geq 0$.

  * •

Linear transformations preserve the existence of intersection points;
therefore, $\bar{\mathscr{L}}$ intersects the Kepler parabola
$\bar{\mathscr{P}}$ twice. Along with $\bar{\xi}<0$ and $\bar{\Lambda}^{2}\geq
0$, we can check easily that these intersections are necessarily on the convex
branch. Consequently, the orbit is an ellipse with eccentricity
$\bar{\varepsilon}=\sqrt{1+4\bar{\Lambda}^{2}\bar{\xi}}\in[0,1[$, and
therefore, $\bar{\xi}>-1/4\bar{\Lambda}^{2}$ holds.

To summarize, we can map any isochrone arc $\mathscr{A}$ to a Keplerian one
$\bar{\mathscr{A}}$ with mass $\mu=1/\sqrt{2}$ using an appropriate linear
transformation $F$ given by Eq. (69). According to Keplerian dynamics, the
orbit $\bar{\mathscr{O}}$ associated with $\bar{\mathscr{A}}$ is an ellipse
whose polar equation is

$\bar{r}(\bar{\theta})=\frac{\bar{p}}{1+\bar{\varepsilon}\cos\bar{\theta}}\quad\text{with}\quad\bar{\varepsilon}=\sqrt{1+4\bar{\Lambda}^{2}\bar{\xi}}\quad\text{and}\quad\bar{p}=\sqrt{2}\bar{\Lambda}^{2}\,.$
(72)

We thus have a mapping between the generic isochrone arc $\mathscr{A}$ and the
Keplerian one $\bar{\mathscr{A}}$, i.e., we have established the upper part of
Fig. 14. The next step is to extend this to the lower part of Fig. 14, i.e.,
link the polar coordinates of each orbit.

Figure 14: Any reduced isochrone orbit $\mathscr{O}_{o}=(r,\theta)$ (red,
bottom right) can be put in correspondence with a reduced Keplerian ellipse
$\bar{\mathscr{O}}_{o}=(\bar{r},\bar{\theta})$ (red, bottom left). The mapping
$r(\bar{r}),\theta(\bar{\theta})$ can be obtained by going into the Hénon
plane and making a linear transformation $F$ to relate the arcs of parabolae
associated with $\mathscr{O}_{o}$ (red, top right) and $\bar{\mathscr{O}}_{o}$
(red, top left) (see details in the text).

####  IV.2.4 General isochrone orbit

Consider a point $N=(r,\theta)$ on a reduced isochrone orbit
$\mathscr{O}_{o}$. It is in a one-to-one correspondence with the point
$M\in\mathscr{A}$ of abscissa $x=2r^{2}$. The point $M=(x,y)$ is in turn
associated with a unique point $\bar{M}=(\bar{x},\bar{y})$ on the Keplerian
parabola (70) and thus with a unique point $N=(\bar{r},\bar{\theta})$ on the
reduced elliptic orbit $\bar{O}_{o}$. This is all depicted in Fig. 14. The
goal now is to express $(r,\theta)$ of the generic isochrone orbit in terms of
$(\bar{r},\bar{\theta})$.

We start with the radius. By inverting Eq. (69) we write
$x=\delta^{-1}(b\bar{x}+d\bar{y})$. Now $\bar{M}$ is on the convex branch of
the Keplerian parabola (70); therefore, $\bar{y}=-\sqrt{\bar{x}}$. Combining
these two equations readily gives a relation between the abscissa of the two
points $M$ and $\bar{M}$, namely

$x=\delta^{-1}(b\bar{x}-d\sqrt{\bar{x}})\quad\Rightarrow\quad
r=\sqrt{\frac{\bar{r}}{\delta}\biggl{(}b\bar{r}-\frac{d}{\sqrt{2}}\biggr{)}}\,,$
(73)

where we simply used the definition of Hénon variables for each orbit, i.e.,
$\bar{x}=2\bar{r}^{2}$ and $x=2r^{2}$ to get the second equation, and
$\delta=ad-bc\neq 0$ (see Sec. III.1). Now if we take a point
$(\bar{r},\bar{\theta})$ on the Keplerian reduced ellipse, then while
$\bar{\theta}$ varies in $[0,\pi]$, $\bar{r}$ changes according to Eq. (72),
and $r$ changes as well according to Eq. (73). Therefore, we can use
$\bar{\theta}$ as a parameter, denoted $s$, to track the radius $r$ on
$\mathscr{O}_{o}$. To this end, we insert Eq. (72) into Eq. (73) and perform
some algebraic manipulations to find that the radius $r$ of the particle on a
generic reduced isochrone orbit $\mathscr{O}_{o}$ can be parameterized by
$r=\varrho(s)$, $s\in[0,\pi]$, where

$\varrho(s):=\frac{\sqrt{c_{1}+c_{2}\cos s}}{1+c_{3}\cos s}\,,$ (74)

for some constants $c_{1},c_{2}$ and $c_{3}$ that depend algebraically on
$(a,b,c,d)$ and $(\xi,\Lambda)$ given by

$c_{1}=\frac{2b\delta\Lambda^{4}}{(c+d\xi)^{2}}+\frac{c_{2}}{c_{3}}\,,\quad
c_{2}=\frac{\Lambda^{2}d}{c+d\xi}c_{3}\quad\text{and}\quad
c_{3}=\sqrt{1+\frac{4\delta\Lambda^{2}(a+b\xi)}{(c+d\xi)^{2}}}\,.$ (75)

We note incidentaly that $c_{3}$ is actually the eccentricity
$\bar{\varepsilon}$ of the parameterizing ellipse (72). Now we seek to find a
similar parameterization for the angle $\theta$. Since $\bar{\theta}=s$ is our
parameter, all we need is an expression of $\theta$ in terms of
$\bar{\theta}$. To this end, we start by writing the following chain rule

$\frac{\mathrm{d}\theta}{\mathrm{d}\bar{\theta}}=\frac{\mathrm{d}\theta}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}\bar{x}}\frac{\mathrm{d}\bar{x}}{\mathrm{d}\bar{t}}\frac{\mathrm{d}\bar{t}}{\mathrm{d}\bar{\theta}}\,.$
(76)

In this equation, the time $\bar{t}$ is the one associated with the dynamics
of the Keplerian orbit $\bar{\mathscr{O}}$, i.e., such that energy and angular
momentum conservation read

$\frac{1}{16}\biggl{(}\frac{\mathrm{d}\bar{x}}{\mathrm{d}\bar{t}}\biggr{)}^{2}=\bar{\xi}\bar{x}-\bar{\Lambda}^{2}-\bar{y}\quad\,\,\text{and}\quad\,\,\bar{\Lambda}=\frac{\bar{x}}{2}\frac{\mathrm{d}\bar{\theta}}{\mathrm{d}\bar{t}}\,.$
(77)

We now express each factor on the right-hand side of Eq. (76) in terms of
$\bar{\theta}$, in order to integrate a first order ODE. For the first and
last terms, we use angular momentum conservation in each orbit: Eq. (13) for
$\mathscr{O}$ and Eq. (77) for $\bar{\mathscr{O}}$. Similarly, in the second
and second-to-last terms we use energy conservation: Eq. (12) for
$\mathscr{O}$ and Eq. (77) for $\bar{\mathscr{O}}$. Inserting these results in
Eq. (76) readily gives

$\frac{\mathrm{d}\theta}{\mathrm{d}\bar{\theta}}=\frac{\bar{x}}{x}\frac{\mathrm{d}x}{\mathrm{d}\bar{x}}\frac{\Lambda}{\bar{\Lambda}}\biggl{(}\frac{\bar{\xi}\bar{x}-\bar{\Lambda}^{2}-\bar{y}}{\xi
x-\Lambda^{2}-y}\biggr{)}^{1/2}\,.$ (78)

To simplify this equation, we express $(\bar{x},\bar{y})$ in terms of $(x,y)$
using Eq. (69) and $(\bar{\xi},\bar{\Lambda})$ in terms of $(\xi,\Lambda)$
using Eq. (71). When doing so, the last two terms on the right-hand side
compensate each other exactly. The only terms contributing on the right-hand
side of Eq. (78) are the first two, and they can be simplified with the help
of Eq. (73). In the end, we find

$\frac{\mathrm{d}\theta}{\mathrm{d}\bar{\theta}}=\frac{1}{2}+\frac{1}{2}\frac{b}{b-d/\sqrt{2}\bar{r}}\,.$
(79)

The final step is to insert Eq. (72) into Eq. (79). We then obtain a first-
order ODE that can then be integrated using the usual change of variables
$u=\tan\bar{\theta}/2$. Once this integration is done and the initial
condition is chosen,202020We choose $\theta$ such that $\theta=0$ at initial
$r=r_{P}$. Since $\bar{r}_{P}$ is sent to $r_{P}$, we require $\theta=0$ when
$\bar{\theta}=0.$, we find that the angle $\theta$ of the particle on a
generic reduced isochrone orbit $\mathscr{O}_{o}$ can be parameterized by
$\theta=\vartheta(s)$, with $s\in[0,\pi]$, where

$\vartheta(s):=\frac{s}{2}+c_{4}\arctan\biggl{(}c_{5}\tan\frac{s}{2}\biggr{)}\,,$
(80)

for some constants $c_{4}$ and $c_{5}$ that depend algebraically on
$(a,b,c,d)$ and $(\xi,\Lambda)$; given by

$c_{4}=\frac{b\Lambda}{\sqrt{b^{2}\Lambda^{2}-d}}\quad\text{and}\quad
c_{5}=\sqrt{1-\frac{2d(c+d\xi)c_{3}}{2b\delta\Lambda^{2}+d(c+d\xi)(1+c_{3})}}\,.$
(81)

####  IV.2.5 Summary and remarks

To summarize, we have found a parameterization for any reduced orbit
$\mathscr{O}_{o}$ in an isochrone potential $\psi$ with finite central mass.
To get the full orbit $\mathscr{O}$ from $\mathscr{O}_{o}$, we follow Fig. 13.
In particular, we add to $\mathscr{O}_{o}$ its symmetric with respect to the
$x$-axis, by extending the range of the parameter $s$ from $[0,\pi]$ to
$[-\pi,\pi].$212121Indeed, from Eqs. (74) and (80), two points
$(\varrho(s),\vartheta(s))$ and $(\varrho(s),\vartheta(-s))$ are symmetric
with respect to $\theta=0$ for $s\in[0,\pi]$ simply because $\cos$ is even and
id, $\tan$ and $\arctan$ are odd.. We then obtain a piece $\mathscr{O}_{T}$ of
the orbit that spans a full radial period $T$, or equivalently a full apsidal
angle $\Theta$. The full orbit $\mathscr{O}$ is then obtained by copying and
pasting the piece $\mathscr{O}_{T}$, albeit rotated anti-clockwise by an angle
$n\Theta$, for all $n\in\mathbb{Z}$. In particular, any orbit $\mathscr{O}$ in
a potential with finite central mass can be parameterized by

$\mathscr{O}=\bigcup_{n\in\mathbb{Z}}\mathscr{O}_{n}\quad\text{where}\quad\mathscr{O}_{n}:=\\{(r,\theta)=(\varrho(s),\vartheta(s)+n\Theta)\,,s\in[-\pi,\pi]\\}\,,$
(82)

with $\varrho(s)$ and $\vartheta(s)$ given by Eqs. (74) and (80),
respectively. By construction, when $s=0$, the particle is at periapsis
$(r_{P},0)$ and when $s=\pi$ it is at apoapsis $(r_{A},\Theta/2)$. The latter
implies that $\Theta=\pi(1+c_{4})$, a result that can be checked by comparing
Eqs. (61) and (81). The special case of the Keplerian ellipse of eccentricity
$\varepsilon$ and semi-latus rectum $p$ corresponds to
$(c_{1},c_{2},c_{3},c_{4},c_{5})=(p^{2},0,\varepsilon,1,1)$.

This parameterization covers any isochrone orbit in a potential associated
with a finite mass at the center. However, it can be extended easily to orbits
in gauged potentials (with $\lambda\neq 0$, i.e., infinite central mass) and
to hollow potentials (with $x_{v}>0$, i.e., undefined around the origin), by
considering affine transformations instead of only linear ones. Indeed,
starting from the appropriate parabola crossing the origin, with a vertical
(resp horizontal) translation, one can reach any orbit in a gauged (resp
hollow) potential. In particular, one can follow the previous method and send
any parabola $\mathscr{P}:(ax+by)^{2}+cx+dy+e=0$ to the Keplerian parabola
$\bar{\mathscr{P}}:\bar{y}^{2}=\bar{x}$ by applying to $\mathscr{P}$ the
affine transformation $F\circ G$, composed of the linear map $F$ given by Eq.
(69) and the translation $G:x\mapsto x-e$. The computation can be done to find
an analytic parameterization, with a little more work in the integration of
the ODE expressing $\theta$ in terms of $\bar{\theta}$, cf.. Eq. (78). We
leave this as an interseting exercise to the interested reader.

Speaking of Eq. (78), we have seen that the last two terms on the right-hand
side cancel each other. Notice that it would also have been the case if the
following assumptions had been made: $\xi x-y=\bar{\xi}\bar{x}-\bar{y}$ and
$\Lambda=\bar{\Lambda}$. In Simon-Petit  _et al._ (2018), the authors
precisely make these assumptions and the consequence was twofold. On the one
hand, not all orbits can be reached from the Keplerian one (only the ones
associated with arcs that verify theses two geometrical constraints).
Therefore, the so-called bolst transformations cannot bridge between any two
isochrone orbits. On the other hand, these bolsts form a subgroup of the
linear transformations, whose additive representation exhibits similarities
with the Lorentz group (to some extent), allowing for some analogies with
special relativity, for a particular subclass of bolsts (the so-called
‘$\bm{i}-$bolst). However, as we have seen, the integration of Eq. (78) is
tractable without any additional assumption, and considering linear
transformations is the only way to describe all isochrone orbits. To
summarize, although we believe that the special relativistic analogies
presented in Simon-Petit  _et al._ (2018) may be of pedagogical interest, the
fundamental group associated with isochrony is that of parabola arcs equipped
with affine transformation. Any other subgroup will necessarily miss the
description of some isochrone orbits.

###  IV.3 Classification of isochrone orbits

Now that we have an analytic expression for any isochrone orbit, we will
classify each of them according to the isochrone potential in which it exists.
In classical textbooks, the two academic orbits turn out to be ellipses: In
the Kepler problem, these ellipses have the origin at one of their foci, and
in the harmonic problem, the ellipse is centered on the origin. We will of
course recover these results here, and try to exhibit the plethora of orbits
arising from all four families of isochrone potentials, one by one.

####  IV.3.1 General properties of isochrone orbits

It is well known that in gravitational mechanics, a periodic orbit in a
generic radial potential consists in a rosette Arnol’d (1995). Although no
clear definition of a rosette exists, all have in common a generally not-
closed flower-shaped plane curve that may wrap numerous times around the
origin while oscillating between an perapsis and apoapsis. Of course, we will
recover all these results here. In Figs. 15 through 17, we depict the orbits
of particles in each of the four families of isochrone potentials, such as
defined in Sec. III.2.3. However, all isochrone orbits exhibit similar
properties, due to the following fact: Any isochrone potential $\psi(r)$ can
be written as

$\psi(r)=\varepsilon+\frac{\lambda}{2r^{2}}+f(r)\,,$ (83)

where $f$ is an increasing function or $r$, as can be checked on the
definitions of the potentials $\psi_{i}$ in Sec. III.2.3.

In the case $\lambda\leq 0$, it is immediate from Eq. (83) that $\psi$ will be
increasing and therefore be associated with a gravitational potential, i.e.,
with positive mass density $\rho$. Consequently, the orbiting particle will
feel an ever-attracting force and its orbit will be some kind of rosette
Arnol’d (1995), as is well known in classical mechanics. In particular, the
apsidal angle in such a case will always verify $\Theta\geq\pi$ since the
particle, when approaching the origin, misses it as its angular velocity
$v_{\theta}=\Lambda/r$ increases, while $r\rightarrow 0$. Given that the
harmonic and Kepler potentials are isochrone, it is no surprise that all
isochrone orbits with $\lambda\leq 0$ will be some kind of precessing
ellipses, as we shall see in the following sections.

We mention that in the $\lambda\leq 0$ case, any particle $(\xi,\Lambda)$ with
$\Lambda^{2}=-\lambda$ will fall toward the center without stopping, and will
take an infinite amount of time to reach it, as on the innermost orbit
depicted in Fig. 15. This is because the $\lambda/2r^{2}$ term will balance
the centrifugal term $\Lambda^{2}/2r^{2}$ and the particle will not feel that
centrifugal wall anymore. In terms of the parabola, this happens when the line
$\mathscr{L}$ intersects the parabola $\mathscr{P}$ once, on the $y$-axis.
This is a generalization of the well-known radial orbits Arnol’d (1995), i.e.,
these with $\Lambda=0$. When $\lambda=0$, only $\Lambda=0$ travel on radial
orbits. As their angular velocity $v_{\theta}=\Lambda/r$ vanishes they go in a
straight line to the center. When $\lambda<0$, although the centrifugal wall
is not here anymore, they still have $v_{\theta}\neq 0$ and will thus inspiral
toward the center. These orbits are all depicted as the innermost ones in Fig.
15 through Fig. 17.

In the case $\lambda>0$, Eq. (83) shows that $\psi$ will be decreasing around
the origin $r=0$, and therefore be associated with a repulsive force in this
region, associated with a negative mass density. The particle will therefore
be repelled at periapsis. Far from the origin, however, the potential is
always decreasing, and the particle will be attracted at apoapsis. This
situation is closer to electrostatics than to gravitational dynamics, and
shows that isochrony is not unique to gravitational systems, and can be found
in the motion of charged, test particles in central electrostatic potentials.
This special property of $\lambda>0$ potentials will imply that $\Theta$ can
take value in $[0,\pi]$, and isochrone orbits will be drastically different.

We now turn to the analysis of orbits in each of the four families of
isochrone potentials. We stress that the general shape of the orbits can be
classified only by the value of the $\lambda$ parameter. In particular, we
will set $\varepsilon=0$ for each potential as it just amounts to re-scaling
the potential energy, and with a good choice of units for time and space, we
may always set $\omega=1$ for the harmonic family and $\mu=\beta=1$ in the
Bounded, Hénon and Kepler families. These choices do not change the general
characteristics of the orbit. A dynamical system formulation of the problem
(detailed in App. A) has been integrated numerically and used to check (and
found perfect agreement with) all the isochrone formulae: (57) for the radial
period $T$, (61) for the apsidal angle $\Theta$ and the parameterization (82)
for the shape of the orbit.

####  IV.3.2 Harmonic family

According to Sec. III.2.3, a potential $\psi$ in the Harmonic family is given
by

$\psi(r)=\varepsilon+\frac{\lambda}{2r^{2}}+\frac{1}{8}\omega^{2}r^{2}\,,$
(84)

for some $\omega>0$ and $(\varepsilon,\lambda)\in\mathbb{R}^{2}$. Given the
potential, the values of $(\xi,\Lambda)$ that yield periodic orbits are given
by the inequalities (47). With the help of Sec. III.2.3, we may insert the
Greek parameters in place of the Latin ones into Eq. (57) and (61) to find the
period $T(\xi)$ and apsidal angle $\Theta(\Lambda)$ in terms of
$\varepsilon,\lambda,\omega$

$T=\frac{2\pi}{\omega}\,,\quad\text{and}\quad\Theta=\frac{\pi\Lambda}{\sqrt{\Lambda^{2}+\lambda}}\,.$
(85)

• In the case $\lambda=0$, we have $\Theta=\pi$ for all orbits. Up to the
additive constant $\varepsilon$, $\psi_{1}$ is the well-known harmonic (or
Hooke) potential. The dynamics can be solved analytically, and the shape of
the orbit is an ellipse centered on the origin (bottom left of Fig. 15).

• In the case $\lambda<0$, from Eq. (85) we have $\Theta>\pi$. When
$\pi<\Theta<2\pi,$ the particle makes less than one turn in one radial period
$T$. When $\Theta>2\pi$ the orbit winds up at least once around the origin,
and the winding number can become arbitrarily large with $\lambda$. One such
orbit is depicted on the bottom-right of Fig. 15.

• In the case $\lambda>0$, we have $\Theta\in]0,\pi[$. In this peculiar case,
the orbits need many periods in order to make a complete turn around the
origin. This is because particles are repelled when reaching their periapsis.
Such orbits are depicted at the top of Fig. 15. When the orbit is close to a
circular one this gives rather odd shapes, such as the top-right of Fig. 15.

Figure 15: Five isochrone orbits, generated by the same particle
$(\xi,\Lambda)$ orbiting in five different potentials of the harmonic family
$\psi_{1}$, with varying $\lambda$. In the corners, the four orbits are
depicted over several periods $T$. The curves in the middle, aside each orbit
in the corner, is the highlighted, first period $[0,T]$ of each orbit,
allowing for an easier comparison. Each of these orbits has the same initial
position $(r,\theta)=(r_{0},0)$ (black dot on the right) and same initial
velocity. The innermost orbit (in red) spirals toward the origin, and the four
outer ones are found at $r=r_{0}$ again (black dots on the dashed circle
$r=r_{0}$) after one radial period $T$, by definition. The spiraling one
corresponds to the $\lambda=-\Lambda^{2}$ case, discussed in the text.

####  IV.3.3 Bounded family

According to Sec. III.2.3, a potential $\psi$ in the Bounded family is given
by

$\psi(r)=\varepsilon+\frac{\lambda}{2r^{2}}+\frac{\mu}{\beta+\sqrt{\beta^{2}-r^{2}}}\,,$
(86)

for some $\mu>0,\beta>0$ and $(\varepsilon,\lambda)\in\mathbb{R}^{2}$. Given
the potential, the values of $(\xi,\Lambda)$ that yield periodic orbits are
given by the inequalities (50). As established in Sec. IV.1, the period
$T(\xi)$ and apsidal angle $\Theta(\Lambda)$ of the orbit are given by

$T=\frac{\pi}{\sqrt{2}}\frac{\mu}{(\varepsilon-\xi)^{3/2}}\quad\text{and}\quad\Theta=\frac{\pi\Lambda}{\sqrt{\Lambda^{2}+\lambda}}-\frac{\pi\Lambda}{\sqrt{\Lambda^{2}+\lambda+4\mu\beta}}\,.$
(87)

The most striking feature of orbits in Bounded potentials is the angular,
almost non-differentiable, turning point at the apoapsis, as depicted in Fig.
16. In fact, these orbits are smooth and we provide some insight as to why
they seem pointy in App. F. Regarding the classification of orbits, it will be
very similar to that of the harmonic family, by examining the function
$\Theta(\Lambda)$ given by Eq. (87), the properties of which can be found in
App. D. We set $\varepsilon=0$ and $\mu=\beta=1$ in Eq. (86) by a good choice
of units, and the shape of the orbits depends on the sign of $\lambda$.

• In the case $\lambda=0$, $\Theta$ decreases with $\Lambda$, but varies in
$]0;\pi]$. The particle needs many periods to make a full rotation around the
center. If the apoapsis are peaked, then this can lead to peculiar, star-
shaped orbits, such as the bottom left one in Fig. 16. It is even possible to
tune $\Lambda$ so that $\Theta$ is commensurable with $\pi$ in order to obtain
any regular polygon whose vertices are the apsides of the orbit.

• In the case $\lambda<0$, $\Theta$ decreases with $\Lambda$ and can take
arbitrary values in $]0,+\infty[$. As we said for the harmonic family, the
orbit may wrap around the origin numerous times in one period, as depicted at
the bottom of Fig. 16.

• In the case $\lambda>0$, $\Theta$ is not monotonous with respect to
$\Lambda$. It is increasing from $0$ to some maximum value
$\Theta_{\text{max}}<\pi$ when $\Lambda$ equals some critical value
$\Lambda_{o}$, and then, it decreases to zero for $\Lambda_{o}$. In
particular, all orbits have a maximum apsidal angle that is less than $\pi$.
However, we are again in the case where the particle is repelled at periapsis,
giving the orbits a different look than the $\lambda=0$ case. Two exemples are
depicted at the top of Fig. 16.

Figure 16: Some orbits in various potentials of the Bounded family $\psi_{2}$,
with varying $\lambda$. See Fig. 15 for the explanation of the figure.

####  IV.3.4 Hénon family

According to Sec. III.2.3, a potential $\psi$ in the Hénon family is given by

$\psi(r)=\varepsilon+\frac{\lambda}{2r^{2}}-\frac{\mu}{\beta+\sqrt{\beta^{2}+r^{2}}}\,,$
(88)

for some $\mu>0,\beta\geq 0$ and $(\varepsilon,\lambda)\in\mathbb{R}^{2}$.
Given the potential, the values of $(\xi,\Lambda)$ that yield periodic orbits
are given by the inequalities (50) (with a minus sign in front of $\mu^{2}$).
As established in Sec. IV.1, the period $T(\xi)$ and apsidal angle
$\Theta(\Lambda)$ of the orbit are given by

$T=\frac{\pi}{\sqrt{2}}\frac{\mu}{(\xi-\varepsilon)^{3/2}}\quad\text{and}\quad\Theta=\frac{\pi\Lambda}{\sqrt{\Lambda^{2}+\lambda}}+\frac{\pi\Lambda}{\sqrt{\Lambda^{2}+\lambda+4\mu\beta}}\,.$
(89)

Regarding the classification of orbits, we apply the same method that we used
for the two other families. In particular, the variations of the function
$\Theta(\Lambda)$ by Eq. (89) are given in App. D and we set $\varepsilon=0$
and $\mu=\beta=1$ in Eq. (86) by a good choice of units. The shape of the
orbits depends on the sign of $\lambda$.

• In the case $\lambda=0$, $\Theta$ increases with $\Lambda$ and varies in
$[\pi;2\pi[$. The particle needs at least two periods to make a full rotation
around the center. The Kepler potential belongs to the Hénon family with
$\lambda=\beta=0$, and has $\Theta=2\pi$, recovering the elliptic orbit. It is
thus not a surprise that most orbits in the Hénon family resemble precessing
ellipses. One such orbit is depicted at the bottom of Fig. 17.

• In the case $\lambda<0$, $\Theta$ is, in general, not monotonous with
respect to $\Lambda$. The precise shape of the function $\Theta(\Lambda)$ can
be found in App. C, but generally speaking, $\Theta$ is decreasing from
$+\infty$ to some minimum value $\Theta_{\text{min}}<\pi$ when $\Lambda$
equals some critical value $\Lambda_{o}$, and it increases to reach $2\pi$ for
$\Lambda>\Lambda_{o}$. For some values of $\lambda$, the critical angular
momentum $\Lambda_{o}$ goes to $+\infty$, and $\Theta(\Lambda)$ is then
strictly decreasing, varying between $2\pi$ and $+\infty$. In either case,
$\Theta>\pi$ and the periapsis can be at an arbitrarily large radius, leading
to an orbit with numerous windings around the center, as depicted on the
bottom-right of Fig. 17.

• In the case $\lambda>0$, the apsidal angle $\Theta$ is strictly increasing
between $0$ and $2\pi$. This case is peculiar because the shape of the orbit
will depend on the location of the periapsis. Indeed, note that since
$\lambda>0$, the potential is always decreasing in some region surrounding the
origin. If the periapsis is in this region, then the particle will be
repelled, and we will have necessarily $0<\Theta<\pi$, as usual for repelled
orbits. On the contrary, if the periapsis is outside the region where the
potential decreases, the particle is always attracted and $\pi<\Theta<2\pi$.

Figure 17: Some orbits in various potentials of the Hénon family $\psi_{3}$,
with varying $\lambda$. See Fig. 15 for the explanation of the figure. The
Hénon family contains the usual Kepler potential when $(\beta,\lambda)=(0,0)$,
with closed ellipses. Consequently, for $(\beta,\lambda)\neq(0,0)$ the orbits
are similar to precessing ellipses.

## Conclusion

In this paper, we have tried to answer as thoroughly as possible the following
mathematical physics question: What are the characteristics of orbits in
isochrone potentials? To this end we have extended some results of Simon-Petit
_et al._ (2018) and managed to fill in some gaps along the way, privileging a
geometrical approach as much as possible, in the spirit of Michel Hénon’s
work. We often made contact with the results of Simon-Petit  _et al._ (2018),
and argued that this paper should be seen as the second part of a work toward
the completion of Michel Hénon’s endeavor, namely the systematic study of
isochrone potentials and orbits.

After some reminders about orbital dynamics in central potentials in the first
section, in the second one we have presented and argued how the Hénon
variables $(x,Y)$ are well suited for the study of a test particle in an
arbitrary central potential. The reason is that, in these variables, the
particle is associated with a line $\mathscr{L}$ whose two degrees of freedom
(slope and $y$-intercept) are in a one-to-one correspondence with the
properties of the particle (energy and angular momentum). We have used this
Hénon formalism to give a geometrical proof of the central result of
isochrony: A potential $\psi$ is isochrone if and only if its curve
$\mathscr{C}$ in Hénon’s variables is an arc of parabola, using a beautiful
result of Euclidean geometry tracing back to Archimedes. We have given insight
as to why these special curves are parabolae and not any other type of curve
in Sec. II.2.4.

To establish that the set of isochrone potentials is really in a one-to-one
correspondence with parabolae in the Hénon plane, we needed to find the
explicit bifurcation diagrams for each potential. In particular, we built on
the Keplerian picture exposed in Arnol’d (1995) and provided the set of
particles $(\xi,\Lambda)$ that have bounded motion in any isochrone potential,
as depicted in Fig. 9 through 11. By a careful analysis of the set of
parabolae summarized in Fig. 7, we have derived a classification of isochrone
potentials. We have found the one exposed in Hénon (1959a) and Simon-Petit
_et al._ (2018), as well as the hollow potentials, with non-usual physical
properties, but nonetheless bounded isochrone orbits, and therefore necessary