Maximally Superintegrable Systems in a Magnetic Field with Higher Order
Integrals

An Infinite Family of Maximally Superintegrable
Systems in a Magnetic Field with Higher Order
Integrals

Antonella MARCHESIELLO † and Libor ŠNOBL ‡

A. Marchesiello and L. Šnobl

† Czech Technical University in Prague, Faculty of Information Technology,
† Department of Applied Mathematics, Thákurova 9, 160 00 Prague 6, Czech
Republic<EMAIL_ADDRESS>
‡ Czech Technical University in Prague, Faculty of Nuclear Sciences and
Physical Engineering,
‡ Department of Physics, Břehová 7, 115 19 Prague 1, Czech Republic
<EMAIL_ADDRESS>
Received April 10, 2018, in final form August 24, 2018; Published online
August 31, 2018

We construct an additional independent integral of motion for a class of three
dimensional minimally superintegrable systems with constant magnetic field.
This class was introduced in [J. Phys. A: Math. Theor. 50 (2017), 245202, 24
pages] and it is known to possess periodic closed orbits. In the present paper
we demonstrate that it is maximally superintegrable. Depending on the values
of the parameters of the system, the newly found integral can be of
arbitrarily high polynomial order in momenta.

integrability; superintegrability; higher order integrals; magnetic field

37J35; 78A25

##  1 Introduction

In our recent paper [13] we found a class of minimally superintegrable systems
in three spatial dimensions with constant magnetic field which possesses
closed bounded periodic trajectories for a particular choice of parameters.
Namely, a quantity $\kappa$ constructed out of them (cf. equation (2.7) below)
must be rational, $\kappa=\frac{m}{n}$. For three particular choices of
$\kappa$ this system is known to be maximally superintegrable with integrals
of at most second order in momenta [13]. Thus a natural question arises asking
whether also for the remaining values of the parameters satisfying the
rationality constraint a missing independent integral can be constructed.

In this paper we describe how the considered system can be reduced to the two
dimensional anisotropic harmonic oscillator and how the known integrals of the
anisotropic oscillator give rise to a new integral for the system with the
magnetic field. Assuming we have $\kappa=\frac{m}{n}$ where $m$ and $n$ are
incommensurable, the additional integral is of order $m+n-1$ in momenta. Its
leading order terms involve angular momenta linearly.

For the sake of clearness, let us recall that in classical mechanics (with or
without magnetic field) integrability means that there exist $N$ integrals of
motion $X_{j}$ including the Hamiltonian that Poisson commute pairwise, are
well defined functions on the phase space and are functionally independent.
The system is superintegrable if it allows $k$ further independent integrals
$Y_{a}$ that Poisson commute with the Hamiltonian, but not necessarily with
each other, nor with the integrals $X_{j}$. The integer $k$ satisfies $1\leq
k\leq N-1$ where $k=1$ and $k=N-1$ correspond to minimal and maximal
superintegrability, respectively. Similarly, in quantum mechanics we assume
that the integrals are well-defined commuting self-adjoint operators,
polynomial in the operators $\hat{x}_{j}$ and $\hat{p}_{j}$ representing the
coordinates and the momenta, or more general objects, such as convergent
series in these operators. Requirement of independence in this case means that
no nontrivial fully symmetrized polynomial in the integrals of motion should
vanish.

Maximally superintegrable systems are of special interest in classical physics
because all finite trajectories in these systems are closed in configuration
space and the motion is periodic. In quantum mechanics the energy levels are
degenerate and it has been conjectured that maximally superintegrable systems
are exactly solvable [24].

Most of the recent research on superintegrability focused on “natural”
Hamiltonians with scalar potentials. For early systematic study in three
dimensions see [6, 7, 12, 27]. This paper belongs to a series of papers [13,
14, 15] studying superintegrability of three dimensional systems with magnetic
fields. We refer the reader to the papers [14, 15] for discussion of a general
motivation for our research and to [13] for details concerning the
introduction of the system considered in this paper. For a general discussion
of integrability and superintegrability see a recent review [18], for
superintegrability in the presence of vector potentials, e.g., [2, 4, 11, 16,
17, 20, 21, 28]. The influence of constant magnetic field on the motion in a
given potential was studied, e.g., in [5, 26] where the two body Coulomb
problem was studied. Despite the similarities the reader should notice the
differences between the two approaches: in [5] additional, so called
particular, integrals conserved only on special “superintegrable” trajectories
were constructed whereas we consider the appearance of additional global
integrals when a particular relation among the parameters of the system holds.

The structure of the paper is as follows: In Section 2 we describe our system,
its integrals and its trajectories as presented in [13]. In Section 3 we
reduce its dynamics to the well-known case of two dimensional anisotropic
harmonic oscillator (cf. [10, 19, 22]). In Section 4 we use the known
integrals of the anisotropic oscillator to construct previously unknown
integrals for the three dimensional problem under investigation. In Section 5
we present an explicit example. In the last section we conclude with a summary
of our results and comment on the superintegrability of the quantum analogue
of our system.

##  2 The system

We consider the Hamiltonian system on the phase space with coordinates
$(\vec{x},\vec{p})$ where $\vec{x}=(x,y,z)$ and $\vec{p}=(p_{1},p_{2},p_{3})$.
We assume that its magnetic field and effective potential are given by

$\displaystyle\vec{B}(\vec{x})=(-\Omega_{1},\Omega_{2},0),\qquad
W(\vec{x})=\frac{\Omega_{1}\Omega_{2}}{2S}(Sx-y)^{2},$ (2.1)

where $\Omega_{1}$, $\Omega_{2}$, $S$ are real constants such that $S\neq 0$
and $\Omega_{1}^{2}+\Omega_{2}^{2}\neq 0$. Its Hamiltonian can be written as

$\displaystyle H=\frac{1}{2}\big{(}\vec{p}^{A}\big{)}^{2}+W(\vec{x}),$ (2.2)

where

$\displaystyle p_{j}^{A}=p_{j}+A_{j}(\vec{x})$

are gauge covariant expressions for the momenta (for our choice of mass equal
to 1 they coincide with the velocities). We notice that in the usual
formulation of Hamiltonian dynamics which we shall use here for computational
reasons, the equations of motion on the phase space and thus also the
integrals of motion expressed in terms of the canonical variables $x_{i}$ and
$p_{j}$ depend on the chosen gauge. However, the existence of the integrals as
well as their expression in terms of the coordinates $x_{i}$ and the
covariantized momenta $p_{j}^{A}$ does not depend on the choice of gauge. This
is best seen from the fact that the (non-canonical) Poisson brackets among the
coordinates $x_{i}$ and $p_{j}^{A}$

$\displaystyle\\{x_{i},x_{j}\\}_{\rm
P.B.}=0,\qquad\big{\\{}x_{i},p_{j}^{A}\big{\\}}_{\rm
P.B.}=\delta_{ij},\qquad\big{\\{}p_{i}^{A},p_{j}^{A}\big{\\}}_{\rm
P.B.}=-\epsilon_{ijk}B_{k}$

as well as Poisson bracket of any functions expressed in terms of them are
explicitly gauge invariant (see, e.g., [25, Remark 5.1.10(6), p. 217]). Thus
the notions of integrability and superintegrability are gauge-invariant.

The system (2.1) is already known to be minimally superintegrable [13]. It has
three Cartesian type integrals

$\displaystyle X_{0}=p_{3}^{A}+\Omega_{2}x+\Omega_{1}y,$ $\displaystyle
X_{1}=\big{(}p_{1}^{A}\big{)}^{2}-2\Omega_{2}xp_{3}^{A}-\Omega_{2}^{2}x^{2}+\Omega_{1}\Omega_{2}x(Sx-2y),$
$\displaystyle
X_{2}=\big{(}p_{2}^{A}\big{)}^{2}-2\Omega_{1}yp_{3}^{A}-\Omega_{1}^{2}y^{2}+\frac{\Omega_{1}\Omega_{2}}{S}y(y-2Sx)$
(2.3)

on which the Hamiltonian is polynomially dependent,
$H=\frac{1}{2}\big{(}X_{0}^{2}+X_{1}+X_{2}\big{)}$, and an additional first
order integral

$\displaystyle X_{3}=p_{1}^{A}+Sp_{2}^{A}-(S\Omega_{1}+\Omega_{2})z.$ (2.4)

The classical trajectories of the system (2.1) are known, cf. [13]

$\displaystyle
x(t)=\frac{1}{\omega_{1}^{2}}\big{(}\big{(}\omega_{1}^{2}x_{0}-\Omega_{2}p_{30}\big{)}\cos(\omega_{1}t)+\omega_{1}p_{10}\sin(\omega_{1}t)+\Omega_{2}p_{30}\big{)},$
$\displaystyle
y(t)=\frac{1}{\omega_{2}^{2}}\big{(}\big{(}\omega_{2}^{2}y_{0}-\Omega_{1}p_{30}\big{)}\cos(\omega_{2}t)+\omega_{2}p_{20}\sin(\omega_{2}t)+\Omega_{1}p_{30}\big{)},$
$\displaystyle
z(t)=\frac{1}{\Omega_{1}S+\Omega_{2}}\bigg{(}p_{10}(\cos(\omega_{1}t)-1)+Sp_{20}(\cos(\omega_{2}t)-1)$
$\displaystyle\hphantom{z(t)=}{}+\frac{\Omega_{2}p_{30}-\omega_{1}^{2}x_{0}}{\omega_{1}}\sin(\omega_{1}t)+\frac{\Omega_{1}p_{30}-\omega_{2}^{2}y_{0}}{\omega_{2}}\sin(\omega_{2}t)\bigg{)}+z_{0},$
(2.5)

where we introduced the constants

$\displaystyle\omega_{1}=\sqrt{\Omega_{2}(\Omega_{1}S+\Omega_{2})},\qquad\omega_{2}=\sqrt{\frac{\Omega_{1}}{S}(\Omega_{1}S+\Omega_{2})}=\sqrt{\frac{\Omega_{1}}{S\Omega_{2}}}\omega_{1}$
(2.6)

in order to shorten the terms in (2.5).

In [13] it was also proven that in the special cases

$\displaystyle S=\frac{\Omega_{1}}{\Omega_{2}},\qquad
S=4\frac{\Omega_{1}}{\Omega_{2}}\qquad\text{and}\qquad
S=\frac{\Omega_{1}}{4\Omega_{2}}$

the system (2.1) is maximally superintegrable, with the additional integral of
order $1$ and $2$, respectively. In the following we prove that the system
(2.1) is maximally superintegrable whenever the trajectories (2.5) are
periodic (or, equivalently, closed), i.e., for

$\displaystyle
S=\frac{\Omega_{1}}{\Omega_{2}}\kappa^{2},\qquad\text{where}\quad\kappa=\frac{m}{n},\quad
m,n\in\mathbb{N}\ \text{are}\ \text{incommensurable},$ (2.7)

with the fifth integral of order $m+n-1$ in the momenta $p_{1}$, $p_{2}$,
$p_{3}$. We notice that systems with the parameters $\Omega_{1}$,
$\Omega_{2}$, $\kappa$ and $\Omega_{2}$, $\Omega_{1}$, $\frac{1}{\kappa}$ are
equivalent, differ just by a choice of Cartesian coordinates, cf. [13].

We present the system as expressed in (2.1) since this is the form in which
its mathematical structure is most easy to analyze. However, this is not the
best point of view for its physical interpretation. Through a rotation the
system can be brought to a form where either the magnetic field $\vec{B}$ or
the harmonic potential $W$ is aligned with one of the coordinate axis. In [13]
we demonstrated its form when the magnetic field is aligned with the $x$-axis;
however it may be more illuminating to rotate it so that the potential $W$
acts along a coordinate axis. Thus we rotate our coordinate system about the
$z$-axis by an angle $\alpha$ such that
$\tan\alpha=\frac{1}{S}=\frac{\Omega_{2}}{\kappa^{2}\Omega_{1}}$. The system
takes the form (2.2) with the harmonic oscillator potential acting along one
coordinate axis,

$\displaystyle\tilde{W}(\tilde{x},\tilde{y},\tilde{z})=\frac{1}{2}\left(\kappa^{2}\Omega_{1}^{2}+\frac{\Omega_{2}^{2}}{\kappa^{2}}\right)\tilde{x}^{2}\equiv\frac{1}{2}(\hat{\omega})^{2}\tilde{x}^{2}$
(2.8)

in the constant magnetic field

$\displaystyle\vec{\hat{B}}=\frac{1}{\sqrt{\Omega_{2}^{2}+\kappa^{4}\Omega_{1}^{2}}}\big{(}{-}\kappa^{2}\Omega_{1}^{2}-\Omega_{2}^{2},\Omega_{1}\Omega_{2}\big{(}\kappa^{2}-1\big{)},0\big{)}\equiv\big{(}\hat{B}\cos\beta,\hat{B}\sin\beta,0\big{)}.$

Vice versa, we may, at least on some open neighborhood, express $\Omega_{1}$,
$\Omega_{2}$ and $\kappa$ in terms of the three new parameters $\hat{\omega}$,
$\hat{B}$ and $\beta$ by solving algebraic equations. Such three dimensional
unidirectional harmonic oscillator inserted in a constant magnetic field is
minimally superintegrable with two first order integrals $\tilde{p}_{2}^{A}$
and $\tilde{p}_{3}^{A}-\hat{B}y\cos\beta+\hat{B}x\sin\beta$ and one
independent second order integral. We see that the integral (2.4) reflects the
invariance of the system under translation along the $\tilde{y}$ direction.
The solutions of the equations of motion exhibit oscillatory behaviour with
two independent frequencies

$\displaystyle\omega_{1}=\frac{1}{\sqrt{2}}\sqrt{\hat{B}^{2}+\hat{\omega}^{2}+\sqrt{\big{(}\hat{B}^{2}+\hat{\omega}^{2}\big{)}^{2}-4\hat{B}^{2}\hat{\omega}^{2}\cos(\beta)^{2}}},$
$\displaystyle\omega_{2}=\frac{1}{\sqrt{2}}\sqrt{\hat{B}^{2}+\hat{\omega}^{2}-\sqrt{\big{(}\hat{B}^{2}+\hat{\omega}^{2}\big{)}^{2}-4\hat{B}^{2}\hat{\omega}^{2}\cos(\beta)^{2}}}$

(they coincide with the frequencies $\omega_{1,2}$ in (2.6)). When these
frequencies become resonant,

$\displaystyle\frac{\omega_{1}}{\omega_{2}}=\kappa=\frac{m}{n},$

a new higher order integral to be constructed below appears.

##  3 Reduction to the anisotropic oscillator

Oscillator potentials and constant magnetic fields share some similarities, as
noticed, e.g., in [3]. On the other hand, the first order integrals $X_{0}$
and $X_{3}$ in (2.3) and (2.4) show that the system can be regarded as an
oscillator in the direction of $\tilde{x}$, cf. (2.8) plus a constant magnetic
field. In the following we see how, by using the known first order integrals,
the system (2.1) can indeed be reduced to a two dimensional anisotropic
oscillator for the parameter $S$ satisfying (2.7). With gauge chosen as

$\displaystyle\vec{A}=(0,0,-\Omega_{2}x-\Omega_{1}y),$ (3.1)

the Hamilton equations read

$\displaystyle\dot{x}=p_{1},\qquad\dot{y}=p_{2},\qquad\dot{z}=-\Omega_{1}y-\Omega_{2}x+p_{3},\qquad\dot{p}_{3}=0,$
$\displaystyle\dot{p}_{1}=-\big{(}\Omega_{2}^{2}+\kappa^{2}\Omega_{1}^{2}\big{)}x+\Omega_{2}p_{3},\qquad\dot{p}_{2}=-\left(\Omega_{1}^{2}+\frac{\Omega_{2}^{2}}{\kappa^{2}}\right)y+\Omega_{1}p_{3}.$

By substituting $p_{3}\equiv p_{30}$ and by the shift

$\displaystyle
x=X+\frac{\Omega_{2}p_{30}}{\Omega_{2}^{2}+\Omega_{1}^{2}\kappa^{2}},\qquad
y=Y+\frac{\Omega_{1}p_{30}\kappa^{2}}{\Omega_{2}^{2}+\Omega_{1}^{2}\kappa^{2}},$
(3.2)

the equations simplify to

$\displaystyle\dot{P}_{1}=-\left(\Omega_{2}^{2}+\kappa^{2}\Omega_{1}^{2}\right)X,\qquad\dot{P}_{2}=-\left(\Omega_{1}^{2}+\frac{\Omega_{2}^{2}}{\kappa^{2}}\right)Y,\qquad\dot{X}=P_{1},\qquad\dot{Y}=P_{2},$
(3.3) $\displaystyle\dot{z}=-\Omega_{1}Y-\Omega_{2}X+p_{30},$ (3.4)

where $P_{1}$, $P_{2}$ are the momenta conjugated to the new space coordinates
$X$, $Y$ (once evaluated they are equal to the original $p_{1}$, $p_{2}$). By
solving the first two equations in (3.3) with respect to $X$ and $Y$ and
substituting into (3.4) we find

$\displaystyle\Omega_{2}\dot{P}_{1}+\Omega_{1}\kappa^{2}\dot{P}_{2}-\big{(}\Omega_{2}^{2}+\Omega_{1}^{2}\kappa^{2}\big{)}\dot{z}=0$

corresponding to the integral (2.4). The dynamics are thus reduced to the
dynamics of an anisotropic oscillator, whose frequency ratio is $\kappa$ and
canonical coordinates are $(X,Y,P_{1},P_{2})$. It is known [22] that if
$\kappa$ satisfies (2.7), such an oscillator is superintegrable. Let us
henceforth restrict to this case and set

$\displaystyle\omega^{2}=\frac{\Omega_{1}^{2}}{n^{2}}+\frac{\Omega_{2}^{2}}{m^{2}}.$

The Hamiltonian of the two degree of freedom (from now on abbreviated to
d.o.f.) oscillator (3.3) is obtained by substituting (3.2) into the
Hamiltonian (2.2) and reads

$\displaystyle
H_{2}=\frac{1}{2}\big{(}P_{1}^{2}+P_{2}^{2}\big{)}+\frac{1}{2}\omega^{2}\big{(}m^{2}X^{2}+n^{2}Y^{2}\big{)}.$
(3.5)

By introducing complex coordinates $z_{1}=iP_{1}+m\omega X$,
$z_{2}=iP_{2}+n\omega Y$, the ring of the invariants of the oscillator (3.5)
is generated by [10, 22]

$\displaystyle I_{1}=z_{1}\bar{z}_{1},\qquad I_{2}=z_{2}\bar{z}_{2},\qquad
I_{3}=\operatorname{Re}\big{(}z_{1}^{n}\bar{z}_{2}^{m}\big{)},\qquad
I_{4}=\operatorname{Im}\big{(}z_{1}^{n}\bar{z}_{2}^{m}\big{)}.$ (3.6)

The invariants (3.6) are clearly not independent; they satisfy the relation

$\displaystyle I_{3}^{2}+I_{4}^{2}=I_{1}^{n}I_{2}^{m}.$

Equivalently, the integrals $I_{3}$, $I_{4}$ can be expressed in terms of
Chebyshev polynomials as [8]

$\displaystyle
I_{3}=|z_{1}|^{n-1}|z_{2}|^{m-1}\left(|z_{1}||z_{2}|T_{n}\left(\frac{\operatorname{Re}{z_{1}}}{|z_{1}|}\right)T_{m}\left(\frac{\operatorname{Re}{z_{2}}}{|z_{2}|}\right)\right.$
$\displaystyle\left.\hphantom{I_{3}=}{}+\operatorname{Im}{z_{1}}\operatorname{Im}{z_{2}}U_{n-1}\left(\frac{\operatorname{Re}{z_{1}}}{|z_{1}|}\right)U_{m-1}\left(\frac{\operatorname{Re}z_{2}}{|z_{2}|}\right)\right),$
$\displaystyle
I_{4}=|z_{1}|^{n-1}|z_{2}|^{m-1}\left(|z_{2}|\operatorname{Im}{z_{1}}U_{n-1}\left(\frac{\operatorname{Re}z_{1}}{|z_{1}|}\right)T_{m}\left(\frac{\operatorname{Re}{z_{2}}}{|z_{2}|}\right)-\right.$
$\displaystyle\left.\hphantom{I_{4}=}{}-|z_{1}|\operatorname{Im}z_{2}T_{n}\left(\frac{\operatorname{Re}z_{1}}{|z_{1}|}\right)U_{m-1}\left(\frac{\operatorname{Re}{z_{2}}}{|z_{2}|}\right)\right),$
(3.7)

where $T_{n}$, $U_{n}$ denote the Chebyshev polynomial of degree $n$ of the
first and second type, respectively.

As we will show in the next section, the integrals $I_{1}$ and $I_{2}$
correspond to the Cartesian type integrals $X_{1}$ and $X_{2}$ of the original
system while $I_{3}$ (or $I_{4}$) gives a new independent integral for the
system (2.1). We find it interesting to note that, under the assumption (2.7),
the system reduces to the Landau problem (i.e., a particle moving in a
constant magnetic field without any potential force) in the limit
$\Omega_{1}\rightarrow 0$ if $S$ is kept constant, i.e.,
$\kappa\rightarrow+\infty$, $\Omega_{1}\approx\frac{1}{\kappa^{2}}$. This
reflects the fact that one of the frequencies in the oscillator (3.5) goes to
zero, and the polynomial integrals (3.7) are becoming of increasing order
until they are lost in the limit.

##  4 The fifth integral

Let us first invert the shift (3.2). The integrals $I_{1}$ and $I_{2}$ give,
after neglecting terms proportional to $p_{30}^{2}$,

$\displaystyle\tilde{I}_{1}=p_{1}^{2}-2\Omega_{2}xp_{30}+\frac{\Omega_{1}^{2}m^{2}+\Omega_{2}^{2}n^{2}}{n^{2}}x^{2},$
(4.1)
$\displaystyle\tilde{I}_{2}=p_{2}^{2}-2\Omega_{1}yp_{30}+\frac{\Omega_{1}^{2}m^{2}+\Omega_{2}^{2}n^{2}}{m^{2}}y^{2}.$
(4.2)

By substituting back $p_{3}=p_{30}$ into (4.1)–(4.2) we see they correspond to
the Cartesian type integrals $X_{1}$ and $X_{2}$ of (2.1).

Similarly, we can find the expressions of the integrals $I_{3}$ and $I_{4}$ in
the original phase space coordinates. We find it convenient to work with the
following series expressions for Chebyshev polynomials

$\displaystyle
T_{n}(a)=\sum_{k=0}^{[\frac{n}{2}]}\binom{n}{2k}a^{n-2k}\big{(}a^{2}-1\big{)}^{k},\qquad
U_{n}(a)=\sum_{k=0}^{[\frac{n}{2}]}\binom{n+1}{2k+1}a^{n-2k}\big{(}a^{2}-1\big{)}^{k},$

so that we can explicitly write the two integrals as polynomials in the
momenta. Namely, after inverting the shift (3.2) and substituting
$p_{30}=p_{3}$ we have, in gauge covariant form,

$\displaystyle
X_{4}=\sum_{k=0}^{[\frac{n-1}{2}]}(-1)^{k}\binom{n}{2k+1}\big{(}m\omega\tilde{X}^{A}\big{)}^{n-2k-1}\big{(}p_{1}^{A}\big{)}^{2k+1}$
$\displaystyle\hphantom{X_{4}=}{}\times\sum_{k=0}^{[\frac{m-1}{2}]}(-1)^{k}\binom{m}{2k+1}\big{(}n\omega\tilde{Y}^{A}\big{)}^{m-2k-1}\big{(}p_{2}^{A}\big{)}^{2k+1}$
$\displaystyle\hphantom{X_{4}=}{}+\sum_{k=0}^{[\frac{n}{2}]}(-1)^{k}\binom{n}{2k}\big{(}m\omega\tilde{X}^{A}\big{)}^{n-2k}\big{(}p_{1}^{A}\big{)}^{2k}\sum_{k=0}^{[\frac{m}{2}]}(-1)^{k}\binom{m}{2k}\big{(}n\omega\tilde{Y}^{A}\big{)}^{m-2k}\big{(}p_{2}^{A}\big{)}^{2k}$
(4.3)

and

$\displaystyle
X_{5}=\sum_{k=0}^{[\frac{n-1}{2}]}(-1)^{k}\binom{n}{2k+1}\big{(}m\omega\tilde{X}^{A}\big{)}^{n-2k-1}\big{(}p_{1}^{A}\big{)}^{2k+1}\sum_{k=0}^{[\frac{m}{2}]}(-1)^{k}\binom{m}{2k}\big{(}n\omega\tilde{Y}^{A}\big{)}^{m-2k}\big{(}p_{2}^{A}\big{)}^{2k}$
$\displaystyle\hphantom{X_{5}=}{}-\sum_{k=0}^{[\frac{n}{2}]}(-1)^{k}\binom{n}{2k}\big{(}m\omega\tilde{X}^{A}\big{)}^{n-2k}\big{(}p_{2}^{A}\big{)}^{2k}$
$\displaystyle\hphantom{X_{5}=}{}\times\sum_{k=0}^{[\frac{m-1}{2}]}(-1)^{k}\binom{m}{2k+1}\big{(}n\omega\tilde{Y}^{A}\big{)}^{m-2k-1}\big{(}p_{1}^{A}\big{)}^{2k+1},$
(4.4)

where

$\displaystyle\tilde{X}^{A}=x-\frac{n^{2}\Omega_{2}\big{(}p_{3}^{A}+\Omega_{2}x+\Omega_{1}y\big{)}}{n^{2}\Omega_{2}^{2}+m^{2}\Omega_{1}^{2}},\qquad\tilde{Y}^{A}=y-\frac{m^{2}\Omega_{1}\big{(}p_{3}^{A}+\Omega_{2}x+\Omega_{1}y\big{)}}{n^{2}\Omega_{2}^{2}+m^{2}\Omega_{1}^{2}}.$

Since $I_{1}$, $I_{2}$, $I_{3}$ are independent integrals for the oscillator
system, by applying the chain rule we see that the integrals $X_{0}$, $X_{1}$,
$X_{2}$, $X_{4}$ form a set of independent integrals for the original system,
where $X_{0}=p_{3}$. Moreover, if the gauge is chosen as in (3.1), none of
them depends on the $z$ variable, while $X_{3}$ does. Therefore the five
integrals $X_{0}$, $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$ are independent. This
implies the maximally superintegrability of the system (2.1). Notice that the
same argument also applies to $X_{0}$, $X_{1}$, $X_{2}$, $X_{3}$, $X_{5}$.

Let us now discuss the order of the new integrals $X_{4}$, $X_{5}$. From the
expressions (4.3) and (4.4), it is clear that their order is at most $m+n$.
However from (4.3) we see that the terms of order $m+n$ in $X_{4}$ are only of
the form

$\displaystyle\alpha_{k}\gamma_{j}p_{1}^{2k}p_{2}^{2j}p_{3}^{n+m-2(k+j)},\qquad
k=0,\ldots,\left[\frac{n}{2}\right],\qquad
j=0,\ldots,\left[\frac{m}{2}\right]$ (4.5)

or

$\displaystyle\beta_{k}\delta_{j}p_{1}^{2k+1}p_{2}^{2j+1}p_{3}^{n+m-2(k+j+1)},\qquad
k=0,\ldots,\left[\frac{n-1}{2}\right],\qquad
j=0,\ldots,\left[\frac{m-1}{2}\right],$ (4.6)

where $\alpha_{j}$, $\beta_{j}$, $\gamma_{j}$, $\delta_{j}$ are coefficients
whose explicit expression can be found through (4.3). We can eliminate all the
terms of the form (4.5) by subtracting the integrals

$\displaystyle\alpha_{k}\gamma_{j}X_{0}^{n+m-2(k+j)}X_{1}^{k}X_{2}^{j},\qquad
k=0,\ldots,\left[\frac{n}{2}\right],\qquad
j=0,\ldots,\left[\frac{m}{2}\right].$ (4.7)

Similarly, we can subtract

$\displaystyle\frac{\beta_{k}\delta_{j}}{2}X_{0}^{n+m-2(k+j+1)}X_{1}^{k}X_{2}^{j}\left(\frac{\Omega_{2}}{\kappa^{2}\Omega_{1}}\big{(}X_{3}^{2}-X_{1}\big{)}-\kappa^{2}\frac{\Omega_{1}}{\Omega_{2}}X_{2}\right),$
(4.8)

where $k=0,\ldots,\big{[}\frac{n-1}{2}\big{]}$,
$j=0,\ldots,\big{[}\frac{m-1}{2}\big{]}$, and eliminate all the terms (4.6).
Therefore the order of the integral $X_{4}$ can be reduced to $m+n-1$. By
construction the terms of order $m+n-1$ of the reduced integral
$\tilde{X}_{4}$ take the form of products of $m+n-1$ linear momenta and one
coordinate. Since the highest order terms of any integral must belong to the
enveloping algebra of the Euclidean algebra [14, 18], we deduce that each of
the highest order terms of $\tilde{X}_{4}$ is a product of $m+n-2$ linear
momenta and one angular momentum. Since the leading order terms of all the
other independent integrals $X_{0}$, $X_{1}$, $X_{2}$, $X_{3}$ contain only
linear momenta, it is not possible to further reduce the order of the integral
$\tilde{X}_{4}$ by polynomial combinations of the other integrals.

Concerning the order of $X_{5}$, we notice that this integral contains the
highest order terms of the type

$\displaystyle\alpha_{k}\gamma_{j}p_{2}^{2k}p_{1}^{2j+1}p_{3}^{n-2(k+j)-1},\qquad
k=0,\ldots,\left[\frac{m}{2}\right],\qquad
j=0,\ldots,\left[\frac{n-1}{2}\right]$

or

$\displaystyle\beta_{k}\delta_{j}p_{1}^{2k+1}p_{2}^{2j}p_{3}^{n-2(k+j)-1},\qquad
k=0,\ldots,\left[\frac{m-1}{2}\right],\qquad
j=0,\ldots,\left[\frac{n}{2}\right],$

whose order as polynomials in $p_{1}$ and $p_{2}$ is odd. Therefore, it is not
possible to eliminate them by polynomial combinations of the other integrals.
As far as we can see, the order of $X_{5}$ cannot be reduced and it is $m+n$.

##  5 Example: $\boldsymbol{n=2}$, $\boldsymbol{m=3}$

In order to illustrate the concepts and general results introduced above let
us consider a particular nontrivial example. We choose the constants $n=2$ and
$m=3$, i.e., $\kappa=\frac{3}{2}$. Thus the Hamiltonian of the 2 d.o.f.
oscillator (3.5) reads

$\displaystyle
H_{2}=\frac{1}{2}\big{(}P_{1}^{2}+P_{2}^{2}\big{)}+\frac{1}{2}\omega^{2}\big{(}9X^{2}+4Y^{2}\big{)},\qquad\omega^{2}=\frac{1}{4}\Omega_{1}^{2}+\frac{1}{9}\Omega_{2}^{2}.$

The integral $X_{4}$ is of order $n+m-1=4$. Thus its leading order terms are
fourth order terms in the enveloping algebra of the Euclidean algebra, linear
in angular momenta $l_{j}$ and cubic in linear momenta $p_{j}$. Explicitly,
they are as follows

$\displaystyle
X_{4}^{\mathrm{(h.o.)}}=\frac{1}{\sqrt{9\Omega_{1}^{2}+4\Omega_{2}^{2}}}\bigg{(}\left(\frac{16\Omega_{2}^{3}}{9\Omega_{1}}+4\Omega_{1}\Omega_{2}\right)l_{2}p_{2}^{2}p_{3}-4\Omega_{1}\Omega_{2}(3l_{2}p_{3}+8l_{3}p_{2})p_{3}^{2}$
$\displaystyle\hphantom{X_{4}^{\mathrm{(h.o.)}}=}{}-\big{(}4\Omega_{2}^{2}+9\Omega_{1}^{2}\big{)}(l_{1}p_{3}+l_{3}p_{1})p_{2}^{2}+27\Omega_{1}^{2}(l_{1}p_{3}+l_{3}p_{1}t)p_{3}^{2}\bigg{)}.$
(5.1)

The highest order terms take the same form also when $X_{4}$ is expressed in a
gauge covariant way, using $p_{j}^{A}=p_{j}+A_{j}(\vec{x})$ and
$l_{j}^{A}=\sum_{k,l}\epsilon_{jkl}x_{k}p_{l}^{A}$ instead of $p_{j}$,
$l_{j}$.

For the sake of completeness let us also write down rather unwieldy expression
for the lower order terms (for our fixed gauge (3.1) since the gauge covariant
expression is even more cumbersome)

$\displaystyle X_{4}-X_{4}^{\mathrm{(h.o.)}}=2\Omega_{1}\tau
y^{2}p_{1}^{2}p_{3}-2\tau\left(3\Omega_{1}x+\frac{8}{9}\Omega_{2}y\right)yp_{1}p_{2}p_{3}-\frac{8\Omega_{2}\tau}{9}yzp_{1}p_{3}^{2}$
$\displaystyle\hphantom{X_{4}-X_{4}^{\mathrm{(h.o.)}}=}{}+\tau\left(\frac{\Omega_{1}}{2}\big{(}9x^{2}+y^{2}-z^{2}\big{)}+2\Omega_{2}xy+\frac{2}{9}\frac{\Omega_{2}^{2}}{\Omega_{1}}\big{(}x^{2}-z^{2}\big{)}\right)p_{2}^{2}p_{3}$
$\displaystyle\hphantom{X_{4}-X_{4}^{\mathrm{(h.o.)}}=}{}-\frac{1}{2\tau}\left(27\left(x^{2}-\frac{1}{3}y^{2}-z^{2}\right)\Omega_{1}^{3}-36\Omega_{1}^{2}\Omega_{2}xy\right.$
$\displaystyle\left.\hphantom{X_{4}-X_{4}^{\mathrm{(h.o.)}}=}{}+4\Omega_{2}^{2}\Omega_{1}\big{(}3x^{2}+4y^{2}-3z^{2}\big{)}-\frac{64\Omega_{2}^{3}}{9}xy\right)p_{3}^{3}$
$\displaystyle\hphantom{X_{4}-X_{4}^{\mathrm{(h.o.)}}=}{}-2\Omega_{1}\tau
yzp_{2}p_{3}^{2}-\frac{\tau^{3}}{27}y^{3}p_{1}^{2}+\frac{\tau^{3}}{3}xy^{2}p_{1}p_{2}+\frac{4\Omega_{2}\tau^{3}}{81\Omega_{1}}y^{2}zp_{1}p_{3}$
$\displaystyle\hphantom{X_{4}-X_{4}^{\mathrm{(h.o.)}}=}{}-\frac{\tau^{3}}{4}x^{2}yp_{2}^{2}+\frac{\tau^{3}}{9}y^{2}zp_{2}p_{3}$
$\displaystyle\hphantom{X_{4}-X_{4}^{\mathrm{(h.o.)}}=}{}-\tau\left(\Omega_{1}^{2}\left(9\frac{x^{2}}{4}+2y^{2}-z^{2}\right)+\frac{4\Omega_{2}^{2}}{9}\left(x^{2}-\frac{1}{3}y^{2}-z^{2}\right)+\frac{16\Omega_{2}^{3}}{81\Omega_{1}}xy\right)yp_{3}^{2}$
$\displaystyle\hphantom{X_{4}-X_{4}^{\mathrm{(h.o.)}}=}{}+\frac{1}{18\Omega_{1}}\tau^{3}\left(\left(\Omega_{1}y-\frac{2}{3}\Omega_{2}x\right)^{2}-\left(\Omega_{1}^{2}+\frac{4}{9}\Omega_{2}^{2}\right)z^{2}\right)y^{2}p_{3}+\frac{\tau^{5}}{108}y^{3}x^{2},$

where $\tau=\sqrt{9\Omega_{1}^{2}+4\Omega_{2}^{2}}=6\omega$.

Sample trajectories for two different choices of the frequencies
$\Omega_{1,2}$ are shown in Fig. 1.

Figure 1: Sample trajectories for $n=2$, $m=3$, $\vec{x}_{0}=(1,0,0)$,
$\vec{p}_{0}=\big{(}0,1,\frac{1}{2}\big{)}$ and $\Omega_{1}=1$,
$\Omega_{2}=\frac{3}{2}$ (solid line) versus $\Omega_{1}=1$,
$\Omega_{2}=\frac{1}{2}$ (dashed line).

##  6 Conclusions

We have demonstrated in a constructive way that the classical system given by
the potentials (2.1) is maximally superintegrable whenever the parameters
satisfy the rationality constraint (2.7). The constructed fifth independent
integral is polynomial in the momenta and coordinates and it is of order
$m+n-1$ where $m$ and $n$ are incommensurable integers such that
$S=\frac{m^{2}\Omega_{1}}{n^{2}\Omega_{2}}$. Its leading order terms contain
angular momenta, in contrast with all the other, previously known integrals
for the system (2.1).

The explicit form of the integral $X_{4}$ given as the expression (4.3) minus
terms of the form (4.7) and (4.8) is unfortunately rather complicated, cf.
(5.1). We were not yet able to obtain any better insight into the structure of
the monomials in (5.1). In particular we would like to be able to predict the
monomials appearing in the highest order terms for arbitrary $m$, $n$,
together with relations between their coefficients. This understanding should
be postponed to future work.

Up to this point our analysis was purely classical. Thus a natural question
arises whether its results can be taken over into the quantum case. We notice
that the quantum analogues of the expressions $z_{1}$ and $z_{2}$

$\displaystyle\hat{z}_{1}=i\hat{P}_{1}+m\omega\hat{X},\qquad\hat{z}_{2}=i\hat{P}_{2}+n\omega\hat{Y}$

satisfy $[\hat{z}_{1},\hat{z}_{2}]=0$. Thus the hermitean expressions

$\displaystyle\hat{I}_{3}=\frac{1}{2}\big{(}\hat{z}_{1}^{n}\big{(}\hat{z}_{2}^{\dagger}\big{)}^{m}+\hat{z}_{2}^{m}\big{(}\hat{z}_{1}^{\dagger}\big{)}^{n}\big{)},\qquad\hat{I}_{4}=\frac{1}{2i}\big{(}\hat{z}_{1}^{n}\big{(}\hat{z}_{2}^{\dagger}\big{)}^{m}-\hat{z}_{2}^{m}\big{(}\hat{z}_{1}^{\dagger}\big{)}^{n}\big{)}$
(6.1)

are again integrals of motion,

$\displaystyle\big{[}\hat{H},\hat{I_{3}}\big{]}=0,\qquad\big{[}\hat{H},\hat{I_{4}}\big{]}=0$

(this claim can be also verified directly through a simple commutator
evaluation, see also [10]). Thus the integrals of the 2 d.o.f. anisotropic
oscillator are preserved by the quantization although their explicit
expression as polynomials in $\hat{X}$, $\hat{Y}$ and $\hat{P}_{1,2}$ needs to
be symmetrized due to presence of terms involving the same component of both
the coordinate and the momentum, e.g., $\hat{X}$ and $\hat{P_{1}}$ in
$\hat{z}_{1}$.

In order to return to the system (2.1) we assume that the gauge is fixed as in
(3.1). We notice that the Hamiltonian as well as the integrals (2.3) and (2.4)
contain only commuting terms in each of their monomials, thus can be taken
into quantum mechanics without any need for symmetrization. In the
substitution

$\displaystyle\widehat{X}=\hat{x}-\frac{n^{2}\Omega_{2}\hat{p_{3}}}{n^{2}\Omega_{2}^{2}+m^{2}\Omega_{1}^{2}},\qquad\widehat{Y}=\hat{y}-\frac{m^{2}\Omega_{1}\hat{p_{3}}}{n^{2}\Omega_{2}^{2}+m^{2}\Omega_{1}^{2}}$
(6.2)

we have only commuting variables $\hat{x}$, $\hat{y}$ and $\hat{p_{3}}$. The
momentum $\hat{p_{3}}$ also commutes with $\hat{p}_{1}$ and $\hat{p}_{2}$.
Thus substituting (6.2) into the expressions (6.1) for $\hat{I}_{3}$ and
$\hat{I}_{4}$ one can directly obtain the quantum integrals

$\displaystyle\hat{X}_{4}=\frac{1}{2}\left(\left(i\hat{p}_{1}-\frac{\Omega_{2}\hat{p_{3}}}{m\omega}+m\omega\hat{x}\right)^{n}\left(-i\hat{p}_{2}-\frac{\Omega_{1}\hat{p_{3}}}{n\omega}+n\omega\hat{y}\right)^{m}+\text{h.c.}\right)$
(6.3)

and

$\displaystyle\hat{X}_{5}=\frac{1}{2i}\left(\left(i\hat{p}_{1}-\frac{\Omega_{2}\hat{p_{3}}}{m\omega}+m\omega\hat{x}\right)^{n}\left(-i\hat{p}_{2}-\frac{\Omega_{1}\hat{p_{3}}}{n\omega}+n\omega\hat{y}\right)^{m}-\text{h.c.}\right),$
(6.4)

where “h.c.” stands for hermitean conjugate. Expanding the powers in (6.3) and
(6.4) one obtains quantum analogues of equations (4.3) and (4.4) as their
properly symmetrized versions.

Also the argument concerning the lowering of the order of the integral $X_{4}$
remains the same in the quantum case, thus the integral $\hat{X}_{4}$ makes
the quantum system maximally superintegrable of order $(m+n-1)$.

Let us notice that in accordance with [1] and [23] both the Hamilton–Jacobi
and the Schrödinger equations separate in Cartesian coordinates. E.g., the
Hamilton’s principal function $S(\vec{x},t)$ can be written as

$\displaystyle S(\vec{x},t)=-Et+K_{3}z+S_{1}(x)+S_{2}(y),$

where the functions $S_{1,2}$ are solutions of

$\displaystyle
S_{1}^{\prime}(x)=\pm\sqrt{-\big{(}\kappa^{2}\Omega_{1}^{2}+\Omega_{2}^{2}\big{)}x^{2}+2K_{3}x\Omega_{2}+2K_{1}},$
$\displaystyle
S_{2}^{\prime}(y)=\pm\sqrt{-\left(\Omega_{1}^{2}+\frac{\Omega_{2}^{2}}{\kappa^{2}}\right)y^{2}+2K_{3}y\Omega_{1}-K_{3}^{2}-2K_{1}-2E},$

expressible in terms of square roots and inverse trigonometric functions.
Whether it separates also in some other coordinate system, i.e., whether the
maximally superintegrable system (2.1) is multiseparable, remains to our
knowledge an open question.

Our considerations are by construction nonrelativistic. We mention that some
of the nonrelativistic superintegrable systems with magnetic fields give rise
also to their superintegrable relativistic versions, as was observed in [9],
e.g., in the case of helical undulator. Due to the complicated structure of
the integral (4.3) we are presently unable to construct its relativistic
version and thus we do not know whether the relativistic analogue of the
system (2.1) is also superintegrable.

### Acknowledgments

This research was supported by the Grant Agency of the Czech Republic, project
17-11805S. The authors thank Pavel Winternitz for discussions on the subject
of this paper.

## References

  * [1] Benenti S., Chanu C., Rastelli G., Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds, J. Math. Phys. 42 (2001), 2065–2091.
  * [2] Bérubé J., Winternitz P., Integrable and superintegrable quantum systems in a magnetic field, J. Math. Phys. 45 (2004), 1959–1973, math-ph/0311051.
  * [3] Dehin D., Hussin V., A simple connection between the motion in a constant magnetic field and the harmonic oscillator, Helv. Phys. Acta 60 (1987), 552–559.
  * [4] Dorizzi B., Grammaticos B., Ramani A., Winternitz P., Integrable Hamiltonian systems with velocity-dependent potentials, J. Math. Phys. 26 (1985), 3070–3079.
  * [5] Escobar-Ruiz M.A., Turbiner A.V., Two charges on a plane in a magnetic field: special trajectories, J. Math. Phys. 54 (2013), 022901, 15 pages, arXiv:1208.2995.
  * [6] Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666–5676.
  * [7] Evans N.W., Group theory of the Smorodinsky–Winternitz system, J. Math. Phys. 32 (1991), 3369–3375.
  * [8] Evans N.W., Verrier P.E., Superintegrability of the caged anisotropic oscillator, J. Math. Phys. 49 (2008), 092902, 10 pages, arXiv:0808.2146.
  * [9] Heinzl T., Ilderton A., Superintegrable relativistic systems in spacetime-dependent background fields, J. Phys. A: Math. Theor. 50 (2017), 345204, 14 pages.
  * [10] Jauch J.M., Hill E.L., On the problem of degeneracy in quantum mechanics, Phys. Rev. 57 (1940), 641–645.
  * [11] Labelle S., Mayrand M., Vinet L., Symmetries and degeneracies of a charged oscillator in the field of a magnetic monopole, J. Math. Phys. 32 (1991), 1516–1521.
  * [12] Makarov A.A., Smorodinsky J.A., Valiev K., Winternitz P., A systematic search for nonrelativistic systems with dynamical symmetries, Nuovo Cimento A 52 (1967), 1061–1084.
  * [13] Marchesiello A., Šnobl L., Superintegrable 3D systems in a magnetic field corresponding to Cartesian separation of variables, J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages.
  * [14] Marchesiello A., Šnobl L., Winternitz P., Three-dimensional superintegrable systems in a static electromagnetic field, J. Phys. A: Math. Theor. 48 (2015), 395206, 24 pages, arXiv:1507.04632.
  * [15] Marchesiello A., Šnobl L., Winternitz P., Spherical type integrable classical systems in a magnetic field, J. Phys. A: Math. Theor. 51 (2018), 135205, 24 pages.
  * [16] McIntosh H.V., Cisneros A., Degeneracy in the presence of a magnetic monopole, J. Math. Phys. 11 (1970), 896–916.
  * [17] McSween E., Winternitz P., Integrable and superintegrable Hamiltonian systems in magnetic fields, J. Math. Phys. 41 (2000), 2957–2967.
  * [18] Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
  * [19] Nakagawa K., Yoshida H., A list of all integrable two-dimensional homogeneous polynomial potentials with a polynomial integral of order at most four in the momenta, J. Phys. A: Math. Gen. 34 (2001), 8611–8630, nlin.SI/0108051.
  * [20] Pucacco G., On integrable Hamiltonians with velocity dependent potentials, Celestial Mech. Dynam. Astronom. 90 (2004), 111–125.
  * [21] Pucacco G., Rosquist K., Integrable Hamiltonian systems with vector potentials, J. Math. Phys. 46 (2005), 012701, 25 pages, nlin.SI/0405065.
  * [22] Rodríguez M.A., Tempesta P., Winternitz P., Reduction of superintegrable systems: the anisotropic harmonic oscillator, Phys. Rev. E 78 (2008), 046608, 6 pages, arXiv:0807.1047.
  * [23] Shapovalov V.N., Bagrov V.G., Meshkov A.G., Separation of variables in the stationary Schrödinger equation, Sov. Phys. J. 15 (1972), 1115–1119.
  * [24] Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248–4257, hep-th/0011209.
  * [25] Thirring W., Classical mathematical physics. Dynamical systems and field theories, 3rd ed., Springer-Verlag, New York, 1997.
  * [26] Turbiner A.V., Escobar-Ruiz M.A., Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions, J. Phys. A: Math. Theor. 46 (2013), 295204, 15 pages, arXiv:1303.2345.
  * [27] Verrier P.E., Evans N.W., A new superintegrable Hamiltonian, J. Math. Phys. 49 (2008), 022902, 8 pages, arXiv:0712.3677.
  * [28] Zhalij A., Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case, J. Phys. Conf. Ser. 621 (2015), 012019, 8 pages, arXiv:0812.2682.