# Various disguises of the Pais-Uhlenbeck oscillator

Mahmut Elbistan1111mailto<EMAIL_ADDRESS>and Krzysztof
Andrzejewski2222mailto<EMAIL_ADDRESS>
###### Abstract

Beginning with a simple set of planar equations, we discuss novel realizations
of the Pais-Uhlenbeck oscillator in various contexts. First, due to the bi-
Hamiltonian character of this model, we develop a Hamiltonian approach for the
Eisenhart-Duval lift of the related dynamics. We apply this approach to the
previously worked example of a circularly polarized periodic gravitational
wave. Then, we present our further results. Firstly, we show that the
transverse dynamics of the Lukash plane wave and a complete gravitational wave
pulse can also lead to the Pais-Uhlenbeck oscillator. We express the related
Carroll Killing vectors in terms of the Pais-Uhlenbeck frequencies and derive
extra integrals of motion from the conformal Newton-Hooke symmetry. In
addition, we find that the $3+1$ dimensional Penning trap can be canonically
mapped to the 6th order Pais-Uhlenbeck oscillator. We also carry the problem
to the non-commutative plane. Lastly, we discuss other examples like the
motion of a charged particle under electromagnetic field created with double
copy.

1) Physics Department, Boğaziçi University, 34342 Bebek / Istanbul, Turkey

2) Faculty of Physics and Applied Informatics, University of Lodz, Pomorska
149/153, 90-236, Lodz, Poland

###### Contents

  1. 1 Introduction
  2. 2 Trapping and the PU oscillator
    1. 2.1 Setting of the problem
    2. 2.2 Explicit link to PU oscillator
  3. 3 Plane waves, ED lift and Hamiltonian formalism
    1. 3.1 ED lift and symmetries
    2. 3.2 Hamiltonian approach
    3. 3.3 BJR coordinates and Hamiltonian formalism
  4. 4 Circularly polarized periodic gravitational wave
    1. 4.1 Relation to PU oscillator
    2. 4.2 Integrals of motion in rotating Brinkmann coordinates
  5. 5 Lukash plane wave and PU oscillator
  6. 6 Complete gravitational wave pulse and PU model
  7. 7 Ideal Penning trap and $n=3$ PU oscillator
  8. 8 Non-commuting coordinates and the PU oscillator
  9. 9 Other examples
    1. 9.1 Double copy and electromagnetic configurations for the PU oscillator
    2. 9.2 Hill’s equations
    3. 9.3 Lagrange points and Rydberg atoms
    4. 9.4 Gravitational trapping
  10. 10 Discussion

##  1 Introduction

It has been quite a time since Pais and Uhlenbeck [1] invented their
$1+1$-dimensional non-relativistic (NR) mechanical model, now called the Pais-
Uhlenbeck (PU) oscillator, as a prototype for higher order theories (see e.g.
[2] for a review). Namely, it is defined by the following Lagrangian

$L=-\frac{1}{2}{x}\prod_{k=1}^{n}\left(\frac{d^{2}}{dt^{2}}+\lambda^{2}_{k}\right){x}.$
(1.1)

One of the crucial observations [1] is that the Hamiltonian of the $2n$th
order PU oscillator can be written as a combination of $n$ decoupled harmonic
oscillators with an alternating sign

$H=\frac{1}{2}\sum_{k=1}^{n}(-1)^{k+1}(p_{k}^{2}+\lambda_{k}^{2}x_{k}^{2}).$
(1.2)

As a feature of higher derivative theories, the energy (1.2) is not positive
definite. Ostrogradski’s Hamiltonian for the PU oscillator necessarily has a
linear term in one of the momenta333See eqn. (2.19) for $n=2$. and the
associated Lagrangian begins with a square of $n$th derivative of the
position.

Since it is an interesting topic, the PU oscillator has been the subject of
further studies. For instance, its quantization was discussed [3],
interactions were added [4, 5, 6], odd order generalization was made [7] and
lastly damped case was worked out [8]. Its relation to curl forces and to
gyroscopic systems were also noted recently in [9] and [10], respectively.

The goal of this paper is to reveal another aspect of the PU oscillator,
namely its realization in various and seemingly different contexts like
gravitational waves and NR ion traps. Moreover, following [11, 12] we give a
formulation of the PU oscillator in non-commuting coordinates. We further
relate the motion of a charged particle in the electromagnetic background
created via double copy [13] to the $4$th order PU oscillator. So, in spite of
being a higher derivative theory, PU oscillator appears in the dynamics of
several interesting physical systems with second order equations of motion.

For our aim, an important clue comes from the symmetries of the PU oscillator.
Let us note that the symmetries of the $4$th order PU oscillator span 2 copies
of the Heisenberg algebra for generic $\lambda_{i}$s [14]. In the case of odd
frequencies the algebra becomes richer: It is the $\ell=\frac{2n-1}{2}$
conformal Newton-Hooke (NH) algebra [15] which is equivalent to the Galilei
algebra up to a redefinition. There has been a renewed interest for this group
in the field of fluid dynamics [16, 17, 18].

Recently in [19] it was shown that the $1+1$-dimensional 4th order PU
oscillator can be related to a $1+3$ dimensional circularly polarized periodic
gravitational wave (CPP GW) via the Eisenhart-Duval (ED) lift a.k.a. Bargmann
framework [20, 21, 22, 23]. The ED lift provides a correspondence between a
certain class of Riemannian manifolds called the Bargmann manifolds and lower
dimensional NR systems. Basically, the transverse part of the geodesic
equations of the former coincide with the Newtonian equations of the latter.
The correspondence shown in [19] is surprising because it is generically
assumed that for a $1+3$ dimensional plane wave, the associated NR theory is
$2+1$ dimensional. There is no contradiction though as it is the phase space
dimension which counts and the phase space of a $1+1$ dimensional PU
oscillator is $4$ dimensional just as a $2+1$ dimensional NR theory.

In the case of the plane gravitational waves, the transverse part of the
geodesic equations define anisotropic, time-dependent and coupled oscillators
in the form of Sturm-Liouville (SL) problem [24] which is very difficult to
solve. However, the situation changes if there exists an extra symmetry. In
the former example of the CPP GW, apart from the generic $5$-parameter Carroll
symmetry of the plane wave [25] and the homothety [26], there exists an extra
screw isometry that led us to new coordinates where equations become time-
independent and easy to solve. Then, the Carroll symmetries were projected and
became NH charges for the corresponding PU oscillator [19].

Apart from the CPP GW, there exists a second vacuum plane wave solution called
the Lukash plane wave [26] which is endowed with an extra isometry. Its
geodesic motion was studied in [27]. There is also a geodesically complete
vacuum plane wave profile studied in [13, 28] which has an extra conformal
symmetry. In both examples, using those extra symmetries, transverse part of
geodesic motion can be cast into the form

$\displaystyle x^{\prime\prime}-\omega y^{\prime}-\Omega_{+}^{2}x$
$\displaystyle=0,$ (1.3a) $\displaystyle y^{\prime\prime}+\omega
x^{\prime}-\Omega_{-}^{2}y$ $\displaystyle=0.$ (1.3b)

Here, for the first time, we will relate those transverse motions in the
Lukash and the complete plane waves to a $1+1$ dimensional $4$th order PU
oscillator.

After ending our discussion about plane waves, next we consider lower
dimensional NR physics. In that context, our first example will be the
$3+1$-dimensional Penning trap [29], an experimental device to store molecular
ions. The ED lift of the Penning trap and the related geodesic motion was
previously studied in [30]. Here, we will show that the planar dynamics of the
Penning trap is described by (1.3) and with the addition of the third
direction it can be surprisingly mapped into a $1+1$ dimensional but $6$th
order PU oscillator. The physical parameters of the Penning trap, namely
modified cyclotron, magnetron and axial frequencies turn out be frequencies
$\lambda_{1},\lambda_{2},\lambda_{3}$ of the PU oscillator (1.2) in a nice
way.

It is known that in $2+1$ dimensions the Galilei group admits a second central
extension (see [31] and references therein). This situation can be realized in
condensed matter systems where the momentum dependent Berry phases come into
play [32]. We work out equations (1.3) in the non-commutative plane by simply
postulating a non-commutativity parameter $\theta$, see (8.1). As a result, NH
symmetry is enhanced with a second central extension and we show that even
this exotic model can be mapped to a $4$th order $1+1$ dimensional PU
oscillator with $\theta$-dependent frequencies. This new result may yield
potential applications of the PU oscillator in condensed matter theory.

Lastly, we discuss the interrelation between the PU oscillator and the motion
of a charged particle in electromagnetic fields created by double copy [33,
13], Hill’s equations [34], Lagrange points [35] and gravitational trapping
[36].

The outline of the paper is as follows: At Section 2, we begin with our
generic planar trapping problem and link it to the PU oscillator. At Section
3, we discuss the related ED lifted plane wave metrics and their symmetries.
We also develop a Hamiltonian approach to avoid the ambiguity related to the
bi-Hamiltonian aspect of the PU model. As a bonus, we find the canonical
transformation between the underlying NR systems of the Brinkmann and Baldwin,
Jeffery and Rosen (BJR) metrics. At Section 4, we revisit the CPP GW case and
write explicitly its Carroll Killing vectors and additional symmetries
originating from the NH symmetry of the PU oscillator. Sections 5 and 6 are
devoted to the Lukash plane wave and the complete pulse example and their
relation to the PU oscillator, respectively. At section 7, we show that the
Penning trap can be mapped to a $n=3$ PU oscillator. At section 8, the
coordinates are promoted to be non-commuting, then the associated exotic
problem and its symmetries are discussed. We mention other related examples at
Section 9. Finally, at Section 10 we summarize our results and comment on
future problems.

##  2 Trapping and the PU oscillator

###  2.1 Setting of the problem

Let us consider the Lagrangian

$L=\frac{1}{2}M(x^{\prime 2}+y^{\prime
2})-\frac{M\omega}{2}(yx^{\prime}-xy^{\prime})\mp\frac{M}{2}(\Omega_{+}^{2}x^{2}+\Omega_{-}^{2}y^{2}),$
(2.1)

where $M$ is a non-zero parameter. The corresponding Hamiltonian is of the
form

$H=\frac{p_{x}^{2}+p_{y}^{2}}{2M}+\frac{\omega}{2}(yp_{x}-xp_{y})+\frac{M\omega^{2}}{8}(x^{2}+y^{2})\pm\frac{M}{2}(\Omega_{+}^{2}x^{2}+\Omega_{-}^{2}y^{2}).$
(2.2)

Note that by the replacement $H\rightarrow-H,\omega\rightarrow-\omega$ we can
restrict to $M>0$; moreover using the coordinate change $x\leftrightarrow y$
we may choose $\omega>0$. The Hamiltonian (2.2) for $+$ sign describes a
charged particle in a static homogeneous magnetic field (in the symmetric
gauge) and an anisotropic harmonic potential. Because of the loss of symmetry
caused by the anisotropic term we cannot reduce it to a one-dimensional
problem in the standard way. Despite this fact, there is a canonical
transformation leading to the sum of two independent harmonic oscillators (in
particular, the energy is positive-definite), see [37, 38, 39, 40].

However, for our further purposes more interesting will be the second case,
i.e. with the inverted anisotropic potential (minus sign in (2.2)). To get
some insight into this choice of sign, first, let us analyse the corresponding
equations of motion (1.3) where derivation $\\{^{\prime}\\}$ is w.r.t. to NR
time. We assume no prior relations between $\omega$ and $\Omega_{\pm}$ so that
our findings can be applied to various cases. Note that without the Coriolis
force i.e., $\omega=0$, the motions will be uncoupled and they will be
described by hyperbolic functions444It is not excluded that those motions can
be described by trigonometric functions when $\Omega_{\pm}^{2}<0$. In that
case one should be careful about reality/hermiticity of the related
Hamiltonian. Here we consider the case $\Omega_{\pm}>0$.. It is this Coriolis
force or magnetic field that makes trapping possible (within a suitable
parameter range for $\omega,\Omega_{\pm}$). Similar logic applies to a Penning
trap [29] where motions are stabilized via a constant magnetic field.
Alternatively, one can think of equations (1.3) as a generalized version of
planar Hill’s equations for the Earth-Moon-Sun system [34].

Eqs. (1.3) can be transformed into a set of two decoupled linear oscillators;
so the corresponding Hamiltonian should be a combination of two oscillator
Hamiltonians. Since we started with inverted anisotropic oscillators, the
Hamiltonian is not positively defined. Thus, instead of the sum, for specific
values of parameters we may obtain the difference of two oscillators and
consequently the PU model (1.2). Before we perform a detailed analysis at the
Hamiltonian level confirming this conjecture; first, let us have a look at
this problem more directly. Deriving and substituting equations in each other
we get a decoupled but the 4th order equation either for $x$ or $y$ as

$x^{\prime\prime\prime\prime}+(\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2})x^{\prime\prime}+\Omega_{+}^{2}\Omega_{-}^{2}x=0.$
(2.3)

Comparing eq. (2.3) with the equation of motion following from (1.1) (for
$n=2$)

$x^{\prime\prime\prime\prime}+(\lambda_{1}^{2}+\lambda_{2}^{2})x^{\prime\prime}+\lambda_{1}^{2}\lambda_{2}^{2}x=0,$
(2.4)

we conclude that to obtain the PU model $\lambda_{1,2}$ should satisfy the
following conditions

$\lambda_{1}^{2}\lambda_{2}^{2}=\Omega_{+}^{2}\Omega_{-}^{2},\quad\lambda_{1}^{2}+\lambda_{2}^{2}=\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2}.$
(2.5)

In the particular case of $\omega^{2}=2(\Omega_{+}^{2}+\Omega_{-}^{2})$ the
equation (2.3) automatically takes the PU form [19]. These equations are
symmetric under $\lambda_{1}\leftrightarrow\lambda_{2}$; for $\lambda_{2}^{2}$
we found the algebraic relation

$\lambda_{2}^{4}-(\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2})\lambda_{2}^{2}+\Omega_{+}^{2}\Omega_{-}^{2}=0,$
(2.6)

that yields two roots

$\lambda_{2}^{2}=\frac{(\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2})\pm\sqrt{\Delta}}{2},\quad\quad\Delta=(\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2})^{2}-4\Omega_{+}^{2}\Omega_{-}^{2}.$
(2.7)

We make the choice

$\lambda_{1}^{2}=\frac{(\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2})+\sqrt{\Delta}}{2},\qquad\lambda_{2}^{2}=\frac{(\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2})-\sqrt{\Delta}}{2}.$
(2.8)

However, we should consider only real and positive $\lambda^{2}$’s, thus some
additional relations on $\Omega_{\pm}^{2}$ and $\omega$ are needed. First,
$|\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2}|>2\Omega_{+}\Omega_{-}$; however,
for $\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2}<0$ we have $\lambda_{2}^{2}<0$
thus we assume $\omega^{2}-\Omega_{+}^{2}-\Omega_{-}^{2}>0$. In view of this
for $\omega>\Omega_{+}+\Omega_{-}$ we obtain real and positive
${\lambda^{2}}^{\prime}s$.

Although the above intuitive approach is useful to obtain necessary conditions
for the PU model, it may be misleading. Had we chosen the first case with plus
sign in (2.2), again we would have ended up with a $4$th order equation
analogous to (2.3). However, in this case the resulting theory is the sum of
two harmonic oscillators [12, 39, 40]. Such a situation is related to the bi-
Hamiltonian property of the PU model [41]. Therefore, a more rigorous and
explicit link to PU oscillator at the Hamiltonian level will be presented in
the next section.

Based on the analysis made in [4, 41], we note that there might be other cases
where our results can be straightforwardly extended. For instance, when one of
the frequencies, say $\Omega_{-}$, is vanishing, the solution (2.8) again
works with $\lambda_{2}=0$ and $\lambda_{1}^{2}=\omega^{2}-\Omega_{+}^{2}$.
This case will be separately handled at section 9.2 when we discuss Hill’s
equations [34]. Let us also mention that when $\Delta<0$, $\lambda_{1,2}^{2}$
are complex conjugate to each other.

###  2.2 Explicit link to PU oscillator

The relation between second order coupled equations (1.3) and the PU model has
been recently discussed in various ways and contexts, including curl forces
and optics [9], Coriolis force and CCP GW [19], gyroscopic systems and dark
energy [10]. Since, for our further considerations the Hamiltonian (2.2) with
the minus sign will be the starting point, we need to generalize the approach
in [19]. To this end let us introduce the kinematical momenta
$\Pi_{i}=(\Pi_{x},\Pi_{y})$ as

$\Pi_{i}=p_{i}+\frac{M\omega}{2}\epsilon_{ij}x^{j},$ (2.9)

and write the associated symplectic structure in terms of that non-canonical
coordinates $(x,y,\Pi_{x},\Pi_{y})$

$\displaystyle H$
$\displaystyle=\frac{\Pi_{x}^{2}}{2M}+\frac{\Pi_{y}^{2}}{2M}-\frac{M}{2}(\Omega_{+}^{2}x^{2}+\Omega_{-}^{2}y^{2}),$
(2.10a) $\displaystyle\sigma$ $\displaystyle=d\Pi_{i}\wedge dx^{i}+M\omega
dx\wedge dy.$ (2.10b)

The symplectic structure (2.10) directly yields equations (1.3).

Our goal is to apply the method of chiral decomposition [37, 38] and split the
coupled theory (2.10) above. This amounts to trade the phase space coordinates
$(x,y,\Pi_{x},\Pi_{y})$ with the new ones $(X_{+}^{1,2},X_{-}^{1,2})$

$\displaystyle\Pi_{x}$
$\displaystyle=\alpha_{+}X_{+}^{2}+\alpha_{-}X_{-}^{2},\quad\Pi_{y}=-\beta_{+}X_{+}^{1}-\beta_{-}X_{-}^{1},$
(2.11a) $\displaystyle x$ $\displaystyle=X_{+}^{1}+X_{-}^{1},\quad\quad\quad
y=X_{+}^{2}+X_{-}^{2},$ (2.11b)

and solve for the following equations

$\displaystyle\beta_{+}\beta_{-}$
$\displaystyle=M^{2}\Omega_{+}^{2},\quad\quad\alpha_{+}\alpha_{-}=M^{2}\Omega_{-}^{2},$
(2.12a) $\displaystyle\alpha_{+}+\beta_{-}$
$\displaystyle=M\omega,\quad\quad\alpha_{-}+\beta_{+}=M\omega.$ (2.12b)

Omitting details, below we present our solutions for the constants
$\beta_{\pm},\alpha_{\pm}$ in terms of diagonalization frequencies (the PU
frequencies) $\lambda_{1,2}$ (2.8) as

$\displaystyle\beta_{-}$
$\displaystyle=\frac{M}{\omega}(\lambda_{1}^{2}+\Omega_{+}^{2}),\quad\quad\beta_{+}=\frac{M\omega\Omega_{+}^{2}}{\lambda_{1}^{2}+\Omega_{+}^{2}},$
(2.13a) $\displaystyle\alpha_{+}$
$\displaystyle=\frac{M}{\omega}(\lambda_{2}^{2}+\Omega_{-}^{2}),\quad\quad\alpha_{-}=\frac{M\omega\Omega_{-}^{2}}{\lambda_{2}^{2}+\Omega_{-}^{2}}.$
(2.13b)

Substituting them into eqs. (2.10), we obtain a symplectic structure which is
decomposed into $+$ and $-$ sectors

$\displaystyle H$
$\displaystyle=-\frac{M(\lambda_{1}^{2}-\lambda_{2}^{2})}{2}\left(\frac{\lambda_{2}^{2}}{(\lambda_{2}^{2}+\Omega_{-}^{2})}(X_{+}^{1})^{2}+\frac{\lambda_{2}^{2}+\Omega_{-}^{2}}{\omega^{2}}(X_{+}^{2})^{2}-\frac{\lambda_{1}^{2}}{(\lambda_{1}^{2}+\Omega_{+}^{2})}(X_{-}^{2})^{2}-\frac{\lambda_{1}^{2}+\Omega_{+}^{2}}{\omega^{2}}(X_{-}^{1})^{2}\right),$
(2.14a) $\displaystyle\sigma$
$\displaystyle=\frac{M(\lambda_{1}^{2}-\lambda_{2}^{2})}{\omega}(dX_{+}^{1}\wedge
dX_{+}^{2}-dX_{-}^{1}\wedge dX_{-}^{2}).$ (2.14b)

Let us remind that in the above formulae, we assume that $\lambda_{i}^{2}$s
(2.8) are real and positive (see the previous section). Although (2.14) looks
a bit complicated as it stands, the related Euler-Lagrange equations become
the ones for the PU oscillator with frequencies $\lambda_{1,2}$ (2.8) 555At
this point, one can easily find generic solutions for $x,\ y$ (2.11b) to show
that they both solve for the PU equation (2.3) and (1.3).

$(X_{+}^{1,2})^{\prime\prime}+\lambda_{2}^{2}X_{+}^{1,2}=0,\quad(X_{-}^{1,2})^{\prime\prime}+\lambda_{1}^{2}X_{-}^{1,2}=0.$
(2.15)

When we pass to Darboux coordinates by means of the following simple
redefinition

$\displaystyle\sqrt{\frac{M(\lambda_{1}^{2}-\lambda_{2}^{2})(\lambda_{1}^{2}+\Omega_{+}^{2})}{\omega^{2}}}X_{-}^{1}$
$\displaystyle=-p_{1},\quad\quad\sqrt{\frac{M(\lambda_{1}^{2}-\lambda_{2}^{2})}{\lambda_{1}^{2}+\Omega_{+}^{2}}}X_{-}^{2}=x^{1},$
(2.16a)
$\displaystyle\sqrt{\frac{M(\lambda_{1}^{2}-\lambda_{2}^{2})(\lambda_{2}^{2}+\Omega_{-}^{2})}{\omega^{2}}}X_{+}^{2}$
$\displaystyle=-p_{2},\quad\quad\sqrt{\frac{M(\lambda_{1}^{2}-\lambda_{2}^{2})}{\lambda_{2}^{2}+\Omega_{-}^{2}}}X_{+}^{1}=x^{2},$
(2.16b)

we see that the Hamiltonian consists of two harmonic oscillators, surprisingly
with a relative minus sign between them:

$H=H_{1}-H_{2},\qquad
H_{i}=\frac{1}{2}\left(p_{i}^{2}+\lambda_{i}^{2}(x^{i})^{2}\right),$ (2.17)

cf. (1.2). Finally, there exists a final canonical transformation [2]

$\displaystyle x^{1}$
$\displaystyle=\frac{p_{x}+\lambda_{1}^{2}v}{\lambda_{1}\sqrt{(\lambda_{1}^{2}-\lambda_{2}^{2})}},\quad
p_{1}=\frac{\lambda_{1}(p_{v}+\lambda_{2}^{2}x)}{\sqrt{(\lambda_{1}^{2}-\lambda_{2}^{2})}},$
(2.18a) $\displaystyle x^{2}$
$\displaystyle=\frac{p_{v}+\lambda_{1}^{2}x}{\sqrt{(\lambda_{1}^{2}-\lambda_{2}^{2})}},\quad
p_{2}=\frac{p_{x}+\lambda_{2}^{2}v}{\sqrt{(\lambda_{1}^{2}-\lambda_{2}^{2})}},$
(2.18b)

which maps (2.17) to Ostragadski’s Hamiltonian of the PU oscillator,

$H=H_{PU}=p_{x}v+\frac{p_{v}^{2}}{2}+\frac{(\lambda_{1}^{2}+\lambda_{2}^{2})v^{2}}{2}-\frac{\lambda_{1}^{2}\lambda_{2}^{2}x^{2}}{2}.$
(2.19)

To sum up, the Hamiltonian (2.2), for the case of minus sign, can lead to the
PU oscillator with $\lambda^{2}$s being positive. The question is in what
context such a peculiar choice of potential and, in consequence, a possible
realization of the PU dynamics appear. This is the main aim of our further
considerations. In particular, we will show that for some gravitational and
electromagnetic fields and for a Penning trap such a situation occurs.

##  3 Plane waves, ED lift and Hamiltonian formalism

In view of the above considerations the crucial question is whether we can
find some physically interesting models with the Hamiltonian (2.10). The hint
to this problem is the so-called ED lift. Namely, it turns out that for some
plane wave space-times the transverse part of the geodesic equations decouple
and take the form of the 2-dimensional (time-dependent) Newton’s equations.
Conversely, some classical equations (in particular eq. (1.3)) can be embedded
into higher dimensional relativistic geodesics. Such an approach is known as
the ED lift and has been studied in various places and contexts (see e.g.,
[42, 43] for a review). However, there is a subtle point related to the fact
the PU oscillator is a bi-Hamiltonian system [41] what, in turn, involves to
use the Hamiltonian formalism rather than equations of motion. We will address
to this issue at Section 3.2. Firstly, we will discuss the lift of (1.3) and
its symmetry aspects.

###  3.1 ED lift and symmetries

When the NR system (2.10) is ED lifted, we get the following plane-wave metric

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dx^{2}+dy^{2}+2dudv+\omega(xdy-
ydx)du+(\Omega_{+}^{2}x^{2}+\Omega_{-}^{2}y^{2})du^{2},$ (3.1)

in rotating Brinkmann coordinates $x^{\mu}=(x,y,u,v)$. Conversely, we can use
the null geodesic condition

$H_{geo}=\frac{g^{\mu\nu}p_{\mu}p_{\nu}}{2}\equiv 0,\qquad
p_{\mu}=(p_{i},p_{u},p_{v}),$ (3.2)

to recover the underlying NR dynamics (1.3) where $u$ plays the role of NR
time and its conjugate momentum $p_{u}$ becomes $-H$ (2.10). Coordinate $v$
turns out to be the associated NR action and the conserved $p_{v}$ is
identified with the NR mass $M$.

The plane waves have a 5-parameter isometry group which is isomorphic to the
$2+1$ dimensional Carroll group with broken rotations [25] (or 2 copies of
Heisenberg algebra). Finding Killing vectors amounts to deal with a matrix SL
equation and only a few exact solutions are known.

Having already solved the underlying NR motion, we computed the Killing
vectors of the metric (3.1) in terms of the related PU oscillator frequencies
$\lambda_{1,2}$ as666Note that we assume $\lambda_{1,2}^{2}>0$,
$\sqrt{\Delta}=\lambda_{1}^{2}-\lambda_{2}^{2}$ and $\Omega_{\pm}>0$ while
deriving (3.3). Situation changes for Hill’s equations where
$\Omega_{-}=\lambda_{2}=0$, see Section 9.2.

$\displaystyle Y_{1}^{c}$
$\displaystyle=\frac{1}{2(\lambda_{2}^{2}+\Omega_{-}^{2})}\Big{(}\omega(\lambda_{2}^{2}-\Omega_{-}^{2})y\cos\lambda_{2}u-\lambda_{2}(\Omega_{+}^{2}-\Omega_{-}^{2}+\sqrt{\Delta})x\sin\lambda_{2}u\Big{)}\partial_{v}+\cos\lambda_{2}u\partial_{x}-\frac{\omega\lambda_{2}}{\lambda_{2}^{2}+\Omega_{-}^{2}}\sin\lambda_{2}u\partial_{y},$
(3.3a) $\displaystyle Y_{2}^{c}$
$\displaystyle=\frac{1}{2(\lambda_{1}^{2}+\Omega_{+}^{2})}\Big{(}-\omega(\lambda_{1}^{2}-\Omega_{+}^{2})x\cos\lambda_{1}u+\lambda_{1}(\Omega_{+}^{2}-\Omega_{-}^{2}+\sqrt{\Delta})y\sin\lambda_{1}u\Big{)}\partial_{v}+\frac{\omega\lambda_{1}}{\lambda_{1}^{2}+\Omega_{+}^{2}}\sin\lambda_{1}u\partial_{x}+\cos\lambda_{1}u\partial_{y},$
(3.3b) $\displaystyle Y_{1}^{b}$
$\displaystyle=-\frac{\omega}{\sqrt{\Delta}}\left[\Big{(}\frac{(\Omega_{+}^{2}-\Omega_{-}^{2}+\sqrt{\Delta})}{2\omega}x\cos\lambda_{2}u+\frac{(\lambda_{2}^{2}-\Omega_{-}^{2})}{2\lambda_{2}}y\sin\lambda_{2}u\Big{)}\partial_{v}+\frac{(\lambda_{2}^{2}+\Omega_{-}^{2})}{\omega\lambda_{2}}\sin\lambda_{2}u\partial_{x}+\cos\lambda_{2}u\partial_{y}\right],$
(3.3c) $\displaystyle Y_{2}^{b}$
$\displaystyle=-\frac{\omega}{\sqrt{\Delta}}\left[\Big{(}\frac{(\Omega_{+}^{2}-\Omega_{-}^{2}+\sqrt{\Delta})}{2\omega}y\cos\lambda_{1}u+\frac{(\lambda_{1}^{2}-\Omega_{+}^{2})}{2\lambda_{1}}x\sin\lambda_{1}u\Big{)}\partial_{v}+\cos\lambda_{1}u\partial_{x}-\frac{(\lambda_{1}^{2}+\Omega_{+}^{2})}{\omega\lambda_{1}}\sin\lambda_{1}u\partial_{y}\right].$
(3.3d)

The Carroll translations $Y_{i}^{c}$ and Carroll boosts $Y_{i}^{b}$ are
completed with the $v-$translation $Y^{v}=\partial_{v}$ to satisfy the
aforementioned Carroll algebra

$[Y_{i}^{c},Y_{j}^{b}]=-\delta_{ij}\partial_{v}.$ (3.4)

Apart from the generic homothetic vector field
$Y_{h}=2v\partial_{v}+x^{i}\partial_{i}$, our metric (3.1) has an extra
isometry which is a $u$-translation $\partial_{u}$. Thus we come to the
conclusion that if there exists a plane wave whose underlying NR system can be
mapped to the $1+1$ dimensional PU oscillator, it should be endowed with an
additional symmetry777Extra isometry $\partial_{u}$ corresponds to the
conserved Hamiltonian of the PU oscillator (2.19) after null projection [19]..

Coming to possible cases, there are only 2 exact plane gravitational waves
endowed with an extra isometry [26]. First one is the CPP GW whose relation to
the PU oscillator through the ED lift was worked out in [19]. The second
vacuum example with an extra isometry is the Lukash plane wave and its
geodesic motion was studied in [27]. At Section 5, we will show that it can be
also related to the PU oscillator in a non-trivial way. Moreover, there is a
third example of polarized plane gravitational waves with an extra conformal
symmetry [28]. We will study this example at Section 6 and will show that its
underlying theory can be described by a 1-dimensional PU oscillator.

Killing vectors (3.3) will help us to spell out the symmetries of various
systems that we link to the PU oscillator in the following sections. For
instance, using them we write down the isometries of CPP GW (4.8) that become
the NH symmetry of the PU oscillator when projected. They also allow us to
derive conserved charges of the Penning trap (7.11) which will be linked to a
$1+1$ dimensional but the $6$th order PU oscillator.

Finally, let us make the following change

${\bm{x}}=R^{-1}(U){\bm{X}},\quad u=U,\quad v=V,\qquad\qquad
R(U)=\left(\begin{array}[]{cc}\cos(\omega U/2)&-\sin(\omega U/2)\\\
\sin(\omega U/2)&\cos(\omega U/2)\end{array}\right),$ (3.5)

and express (3.1) in the standard Brinkmann coordinates
$X^{\mu}=(X^{1},X^{2},U,V)$

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
d\bm{X}^{2}+2dUdV+K_{ij}(U)X^{i}X^{j}dU^{2},$ (3.6) $\displaystyle K_{11}$
$\displaystyle=$ $\displaystyle\Omega_{+}^{2}\cos^{2}(\omega
U/2)+\Omega_{-}^{2}\sin^{2}(\omega U/2)-\frac{\omega^{2}}{4},$ $\displaystyle
K_{22}$ $\displaystyle=$ $\displaystyle\Omega_{+}^{2}\sin^{2}(\omega
U/2)+\Omega_{-}^{2}\cos^{2}(\omega U/2)-\frac{\omega^{2}}{4},$ $\displaystyle
K_{12}$ $\displaystyle=$ $\displaystyle\sin(\omega U/2)\cos(\omega
U/2)(\Omega_{+}^{2}-\Omega_{-}^{2}).$

The trace of the $2\times 2$ $K$ matrix is
$\mathrm{Tr}{(K)}=\Omega_{+}^{2}+\Omega_{-}^{2}-\frac{\omega^{2}}{2}$. When
$\mathrm{Tr}{(K)}=0$, metric (3.6) is a vacuum solution of Einstein’s
equations and it is called as an exact plane gravitational wave. CPP and
Lukash plane waves belong to this class. So, depending on the parameters
$\omega,\Omega_{\pm}$ our metric at hand can be a vacuum solution or not.

In the standard Brinkmann coordinates (3.6), the extra symmetry becomes the
screw isometry

$Y_{s}=\partial_{U}+\frac{\omega}{2}\epsilon_{ij}X^{i}\partial_{j},$ (3.7)

which is a combination of a $U$ translation and a half-angle rotation on the
plane. Moreover, in that coordinate system the metric is in the Kerr-Schild
form and this will allow us to discuss double copy at Section 9.1.

###  3.2 Hamiltonian approach

In this section we present an alternative approach to the ED lift with a
special emphasis put on the Hamiltonian formalism. This is motivated by the
fact that the PU model is a bi-Hamiltonian system; thus we cannot consider
only equations of motion (in particular the geodesic ones); they can be
obtained from different Hamiltonian formalisms. To this end let us consider
two approaches.

First, we start with a pp-wave metric888We can eliminate the term with
$\omega$ by a redefinition of $K$; however, such a form will be more useful
for us. in Brinkmann coordinates $(X^{i},U,V)$

$g=K({\bm{X}},U)dU^{2}+2dUdV+d{\bm{X}}^{2}+\omega{\bm{X}}^{T}\epsilon
d{\bm{X}}dU,$ (3.8)

which includes both (3.1) and (3.6) as special cases ($\epsilon$ is the
antisymmetric matrix with off-diagonal entries $\pm 1$). The Lagrangian of a
relativistic particle in an electromagnetic potential $A_{\mu}$ is of the form

$L=-m\sqrt{-g_{\mu\nu}{{\dot{X}}^{\mu}}{\dot{X}}^{\nu}}+eA_{\mu}{\dot{X}}^{\mu},$
(3.9)

where dot refers to the derivative w.r.t. an arbitrary parameter $\tau$ and we
adopt the conventions $(-,+,+,+)$. In our further considerations we will
restrict to the form ${\bf A}=0$, $A=A_{U}(U,{\bf X})$ and $A_{V}=0$ where
$U=(X^{3}-X^{0})/\sqrt{2}$ is the “light-cone” coordinate. Then we have

$L=-m\sqrt{-2\dot{U}\dot{V}-\dot{\bf
X}^{2}-K\dot{U}^{2}-\omega{\bm{X}}^{T}\epsilon\dot{\bm{X}}\dot{U}}+eA\dot{U}.$
(3.10)

Since $L$ is homogeneous of the first degree in velocities the Hamiltonian is
zero. To describe the evolution one has to fix the gauge by choosing a time
parameter $\tau=\tau(X^{\mu})$. Let us fix the gauge demanding
$\tau=\tau(U)=U$. At the Lagrangian level this leads to

$L=-m\sqrt{-2V^{\prime}-{({\bf
X}^{\prime})}^{2}-K-\omega{\bm{X}}^{T}\epsilon{\bm{X}}^{\prime}}+eA.$ (3.11)

where prime stands for the derivative w.r.t. $U$. Then one of e.q.m. gives
$-2V^{\prime}-({\bf
X}^{\prime})^{2}-K-\omega{\bm{X}}^{T}\epsilon{\bm{X}}^{\prime}=const=m^{2}/P_{V}^{2}$
where $P_{V}$ is the canonical momentum conjugated to $V$ coordinate;
moreover, we see that $\tau=U$ becomes the affine parametrization. Now, we can
form the Hamiltonian formalism. The phase space obtained is six-dimensional,
$(V,P_{V})$ and $({\bf X},{\bf P})$ with the canonical Poisson brackets, and
the Hamiltonian, for positive $P_{V}$, is of the following form

$H=\frac{({\bf P}+{\omega
P_{V}}\epsilon{\bm{X}}/2)^{2}+m^{2}}{2P_{V}}-eA-\frac{P_{V}}{2}K.$ (3.12)

The above Hamiltonian does not depend on the $V$ coordinate and thus
$P_{V}=const$. In consequence for the particle with fixed momentum $P_{V}$ and
$m$ the dynamics of the transversal coordinates $({\bf X},{\bf P})$ is
governed by the Hamiltonian of the form (3.12).

An alternative and more rigorous approach is based on the Dirac method for the
time-dependent constraints (see e.g. [44]). To this end let us note that the
canonical momenta for the Lagrangian (3.10)

$P_{U}=\frac{m\dot{V}+mK\dot{U}+m\omega{\bm{X}}^{T}\epsilon\dot{\bm{X}}/2}{\sqrt{-g_{\mu\nu}{\dot{X}^{\mu}}{\dot{X}^{\nu}}}}+eA,\quad
P_{V}=\frac{m\dot{U}}{\sqrt{-g_{\mu\nu}{\dot{X}^{\mu}}{\dot{X}^{\nu}}}},\quad{\bf
P}=\frac{m\dot{\bm{X}}-m{\dot{U}}\omega\epsilon{\bm{X}}/2}{\sqrt{-g_{\mu\nu}{\dot{X}^{\mu}}{\dot{X}^{\nu}}}},$
(3.13)

satisfy the condition $2(-P_{U}+eA+\frac{K}{2}P_{V})P_{V}-({\bf
P}+P_{V}\omega\epsilon{\bm{X}}/2)^{2}=m^{2}$; moreover,
$\textrm{sign}(P_{V})=\textrm{sign}(\dot{U})$ . In consequence, we arrive at
the constraint

$\phi=P_{U}-eA-\frac{K}{2}P_{V}+\frac{({\bf
P}+P_{V}\omega\epsilon{\bm{X}}/2)^{2}+m^{2}}{2P_{V}}.$ (3.14)

Due to the above constraint we can only express the velocities
$V,\dot{\bm{X}}$ in terms of momenta $P_{V},{\bf P}$ (there remains not
specified function $\dot{U}$). Namely, we obtain

$\dot{\bm{X}}=\frac{\dot{U}}{P_{V}}({\bf
P}+P_{V}\omega\epsilon{\bm{X}}/2),\quad\dot{V}=-\dot{U}\left(\frac{K}{2}+\frac{{\bf
P}^{2}+m^{2}}{2P_{V}^{2}}-\frac{1}{8}\omega^{2}{\bm{X}}^{2}\right),$ (3.15)

and $U$ is an arbitrary function of $\tau$. In view of the above the
Hamiltonian $H_{c}=P_{U}\dot{U}+P_{V}\dot{V}+{\bf P}\cdot\dot{\bm{X}}-L$ takes
the form

$H_{c}=\dot{U}\phi,$ (3.16)

thus $\phi$ is a first class constraint. We break the reparametrisation
symmetry by adding a new constraint,

$\psi=U-\tau.$ (3.17)

Then $\\{\psi,\phi\\}=1$ and thus $\psi$ and $\phi$ belong to the second class
constraints. Moreover, demanding

$\psi^{\prime}=\partial_{\tau}\psi+\\{\psi,H_{c}\\}=0,$ (3.18)

(i.e. no further constraints) we have $\dot{U}=1$ and $H_{c}=\phi$ (for
$\textrm{sign}(P_{V})=-1$ we should consider the constraint $\psi=U+\tau$,
then $H_{c}=-\phi$). Unfortunately the constraint $\psi$ depends on $\tau$
thus we cannot directly perform the Dirac procedure (it is valid for the time-
independent case). To get rid of this problem we perform the canonical
transformation changing only the one variable $\tilde{U}=U-\tau$
($\tilde{P}_{\mu}=P_{\mu}$, $\tilde{V}=V$, $\tilde{\bm{X}}={\bm{X}}$). The
corresponding generating function $W$ is of the form

$W(X^{\mu},\tilde{P}_{\nu},\tau)=X^{\mu}\tilde{P}_{\nu}-\tau\tilde{P}_{\tilde{U}}.$
(3.19)

Then $\psi=0$ is equivalent to $\tilde{\psi}=\tilde{U}=0$ ($\tau$ independent)
and the Hamiltonian takes the form

$\tilde{H}=H_{c}+\partial_{\tau}W=\phi-P_{U}=\frac{({\bf P}+{\omega
P_{V}}\epsilon{\bm{X}}/2)^{2}+m^{2}}{2P_{V}}-eA-\frac{P_{V}}{2}K$ (3.20)

(with $U$ replaced by $\tilde{U}+\tau$). Now, we are in the position to
perform the standard Dirac procedure i.e. restrict dynamics to the constraint
surface (eliminating $\tilde{U}$ and $\tilde{P}_{\tilde{U}}=P_{U}$ variables).
Then the Hamiltonian (3.20) coincides with (3.12); moreover, the Dirac bracket
between ${\bm{X}},V$ and the corresponding canonical momenta take the standard
form. In the case of negative $P_{V}$, after redefinition of
$\tau\rightarrow-\tau$, the Hamiltonian $\tilde{H}$ can be rewritten in the
form (3.12) with negative $P_{V}$. Finally, let us note that for the free
particle we have $P_{V}\neq 0$ (the same holds when the $A,K$ vanish at
infinites); in our convention the case $P_{V}<0$ describes particles
(antiparticles are with $P_{V}>0$). Moreover, $\tilde{H}=-P_{U}$ (not $P_{U}$,
despite the fact that $U$ takes the role of time). Finally, we can skip the
constant term $\frac{m^{2}}{2P_{V}}$ (using the canonical transformation
generated by $W=-\frac{m^{2}}{2P_{V}}U$). So we can restrict ourselves to the
massless case, $m=0$; however we should keep in mind that in this case we
exclude particles with $P_{V}=0$ (massless particles traveling in the same
direction that the wave).

In summary, the transversal part of relativistic dynamics is described by the
two-dimensional NR Hamiltonian, where $U$ plays the role of time. This is an
alternative description of the ED lift; namely, instead of geodesic equations
(equations of motion), we directly refer to the Hamiltonian formalism what, in
turn, is relevant for the PU model.

The above results will form the starting point for our further considerations.
Namely, we will show that for some plane GW (and/or their electromagnetic
counterparts) the Hamiltonian (3.20) can be related (by a suitable canonical
transformation) to the Hamiltonian (2.2) with minus sign (and the non-zero
parameter $M$).

###  3.3 BJR coordinates and Hamiltonian formalism

Before we go further, let us recall that for the gravitational background
defined by (3.8) with the quadratic form $K(U,{\bm{X}})=K_{ij}(U)X^{i}X^{j}$
($\omega=0$ and $A_{\mu}=0$), there is an alternative coordinate system called
the BJR coordinates in which the metric takes the form

$ds^{2}=a_{ij}(u)d\alpha^{i}d\alpha^{j}+2duds.$ (3.21)

These coordinates are more transparent for the symmetry analysis and weak
field approximation of the relativistic particle; however, in contrast to the
Brinkmann ones they are not complete. In this section we describe the
canonical transformation which leads to the Hamiltonian in the BJR coordinates
(also when $\omega\neq 0$). This will allow us to describe the PU model in the
BJR framework. At the level of metrics, the following coordinate
transformations

$U=u,\quad X^{i}=O^{i}_{j}(u)\alpha^{j},\quad
V=s-\frac{1}{4}\alpha^{i}\alpha^{j}\frac{da_{ij}}{du},\qquad
a_{ij}=(O^{T}O)_{ij},$ (3.22)

yield the BJR metric (3.21) with the modified SL problem for $2\times 2$
$O$-matrix

$O^{\prime\prime}_{ij}=\omega\epsilon_{ik}O^{\prime}_{kj}+K_{ik}O_{kj},\qquad\big{(}(O^{T})^{\prime}O\big{)}_{ij}=(O^{T}O^{\prime})_{ij}-\omega(O^{T}\epsilon
O)_{ij}.$ (3.23)

The BJR profile $a_{ij}$ is a $2\times 2$ symmetric matrix depending only on
the coordinate $u$. Then, Killing vectors of (3.21) can be found easily as

$Y^{v}=\frac{\partial}{\partial s},\quad
Y^{c}_{i}=\frac{\partial}{\partial\alpha^{i}},\quad
Y^{b}_{i}=S^{ij}(u)\frac{\partial}{\partial\alpha^{j}}-\alpha^{i}\frac{\partial}{\partial
s},$ (3.24)

cf. (3.3). In these coordinates, $s$ and $\alpha^{i}$-translations are
symmetries and related conserved quantities are just canonical momenta $p_{s}$
and $p_{i}$. $Y^{b}_{i}$ are called the Carroll boosts and
$S^{ij}(u)=\int^{u}a^{ij}d\tilde{u}$ is the Souriau’s matrix with
$a^{ij}a_{jk}=\delta^{i}_{k}$. The BJR metric (3.21) is also a Bargmann
manifold i.e., it is endowed with a covariantly constant null Killing vector
$\partial_{v}$. Therefore, by a null reduction the Hamiltonian in BJR
coordinates will be of the form

$H_{BJR}=\frac{a^{ij}k_{i}k_{j}}{2p_{s}},$ (3.25)

moreover, we have $p_{s}=P_{V}$. First, let us note that we should have $\
L_{BJR}=\frac{1}{2}p_{s}a_{ij}(u)(\alpha^{i})^{\prime}(\alpha^{j})^{\prime}$,
$(\alpha^{i})^{\prime}=\frac{d\alpha^{i}}{du}$ and

$L_{B}=L_{BJR}+\frac{dF}{du},$ (3.26)

where the generating function $F$ is
$F=\frac{p_{s}}{4}\alpha^{i}\alpha^{j}a_{ij}^{\prime}$; this can be confirmed
using (3.22). Rewriting (3.26) as

$P_{i}(X^{i})^{\prime}-H_{B}=k_{i}(\alpha^{i})^{\prime}-H_{BJR}+\frac{dF}{du},$
(3.27)

where $k_{i}=p_{s}a_{ij}(\alpha^{j})^{\prime}$ are the conserved canonical
momenta. Next choosing $W$ as $F=W(X^{i},k_{j},u)-k_{i}\alpha^{i}$, we obtain
the canonical transformation of the associated Hamiltonians

$H_{B}=H_{BJR}-\frac{\partial W(X^{i},k_{i},u)}{\partial u},$ (3.28)

augmented with the definitions

$P_{i}=\frac{\partial W}{\partial X^{i}},\quad\alpha^{i}=\frac{\partial
W}{\partial k_{i}},$ (3.29)

and the generating function

$W(X^{i},k_{j},u)=k_{i}O^{-1}_{ij}X^{j}+\frac{p_{s}}{4}a^{\prime}_{ij}O^{-1}_{ik}O^{-1}_{jl}X^{k}X^{l}.$
(3.30)

By direct calculations we check that it relates the Hamiltonian (3.12) (or
(3.20)) to the Hamiltonian (3.25). Thus, we conclude that for the plane wave
metric leading to the PU oscillator we can find, via a canonical
transformation (3.28), the description of the PU dynamics also in the BJR
coordinates.

##  4 Circularly polarized periodic gravitational wave

The simplest example whose underlying NR system is linked to the PU oscillator
via the ED lift is the CPP GW [19]. First, we will use it as a testing ground
for our Hamiltonian approach. We will also discuss the BJR metric and its
relation to the PU oscillator. Secondly, we will comment on the additional
integrals of motion coming from the $\ell=\frac{3}{2}$ conformal NH group.

###  4.1 Relation to PU oscillator

The Brinkmann profile (3.6) of CPP GW is

$K^{(1)}(U,{\bm{X}})=b{\bm{X}}^{T}\left(\begin{array}[]{cc}\cos(\omega
U)&\sin(\omega U)\\\ \sin(\omega U)&-\cos(\omega
U)\end{array}\right){\bm{X}};$ (4.1)

with amplitude $b>0$ and frequency $\omega$. It is a vacuum solution of the
Einstein equations. Let us consider the $U$-dependent canonical transformation
to the new coordinates ${\bf x},{\bf p}$ generated by the function:

$W({\bm{X}},{\bf p},U)={\bf p}^{T}R^{-1}(U){\bm{X}},$ (4.2)

where $R(U)$ is given by (3.5). Then the Hamiltonian (3.20) for the profile
$K=K^{(1)}$ transforms into the following one

$H({\bm{x}},{\bf p})=\frac{{\bf
p}^{2}}{2P_{V}}+\frac{bP_{V}}{2}((x^{2})^{2}-(x^{1})^{2})+\frac{\omega}{2}(x^{2}p_{1}-x^{1}p_{2})$
(4.3)

so it takes the form of (2.2) with minus sign after identification $M=P_{V}$
and

$\Omega_{+}^{2}=\frac{\omega^{2}}{4}+b=\lambda^{2}_{1},\qquad\Omega_{-}^{2}=\frac{\omega^{2}}{4}-b=\lambda_{2}^{2}.$
(4.4)

Thus for parameters $\omega^{2}>4b$ the frequencies $\lambda_{1,2}^{2}$ (2.8)
are positive and the underlying/transverse dynamics can be described by the PU
model as outlined at Section 2. In the other case, $\omega^{2}<4b$, we obtain
the sum of the ordinary and inverted harmonic oscillator.

Now, let us mention that we can pass to BJR coordinates via (3.22). In the CPP
GW case, $O$-matrix was found in [19] as

$O=\begin{pmatrix}\cos(\Omega_{-}u)&\frac{\omega}{2\Omega_{+}}\sin(\Omega_{+}u)\\\
-\frac{\omega}{2\Omega_{-}}\sin(\Omega_{-}u)&\cos(\Omega_{+}u)\end{pmatrix},$
(4.5)

and the Souriau matrix is

$S=\frac{2}{(\Omega_{+}^{2}-\Omega_{-}^{2})\det{O}}\begin{pmatrix}\scriptstyle\frac{\omega^{2}}{4\Omega_{+}}\cos(\Omega_{-}u)\sin(\Omega_{+}u)-\Omega_{-}\sin(\Omega_{-}u)\cos(\Omega_{+}u)&-\frac{\omega}{2}\\\
\scriptstyle-\frac{\omega}{2}&\scriptstyle\Omega_{+}\cos(\Omega_{-}u)\sin(\Omega_{+}u)-\frac{\omega^{2}}{4\Omega_{-}}\sin(\Omega_{-}u)\cos(\Omega_{+}u)\end{pmatrix}.$
(4.6)

One can explicitly write the symmetry generators (3.24) and the associated
conserved charges now. The BJR profile $a_{ij}(u)$ and/or the time dependent
metric of the underlying NR system is

$\displaystyle a_{11}$
$\displaystyle=\cos^{2}\Omega_{-}u+\frac{\omega^{2}}{4\Omega_{-}^{2}\sin^{2}\Omega_{-}u},$
(4.7a) $\displaystyle a_{12}$
$\displaystyle=\frac{\omega}{2\Omega_{+}}\sin\Omega_{+}u\cos\Omega_{-}u-\frac{\omega}{2\Omega_{-}u}\sin\Omega_{-}u\cos\Omega_{+}u=a_{21},$
(4.7b) $\displaystyle a_{22}$
$\displaystyle=\cos^{2}\Omega_{+}u+\frac{\omega^{2}}{4\Omega_{+}^{2}}\sin^{2}\Omega_{+}u.$
(4.7c)

Via (3.28), BJR Hamiltonian can be mapped to that of the rotating Brinkmann
coordinates (2.2) which is already related to the PU oscillator (2.19).

###  4.2 Integrals of motion in rotating Brinkmann coordinates

As we have seen above in the rotating coordinates $(u,x,y,v)$ the transversal
Hamiltonian is $u$-independent thus the additional isometry is obviously
$u-$translation. This is of course true for the CPP GW in which case the
metric takes the form (3.1) with $\Omega_{\pm}^{2}$ given by (4.4). We will
analyse symmetries and integrals of motion for this case and next compare them
with the ones for the PU model.

The Carroll symmetry of CPP GW is a particular case of (3.3) as

$\displaystyle Y^{c}_{1}$
$\displaystyle=\cos\Omega_{-}u\frac{\partial}{\partial
x}-\frac{\omega}{2\Omega_{-}}\sin\Omega_{-}u\frac{\partial}{\partial
y}-\frac{(\Omega_{+}^{2}-\Omega_{-}^{2})}{2\Omega_{-}}x\sin\Omega_{-}u\frac{\partial}{\partial
v},$ (4.8a) $\displaystyle Y^{c}_{2}$
$\displaystyle=\frac{\omega}{2\Omega_{+}}\sin\Omega_{+}u\frac{\partial}{\partial
x}+\cos\Omega_{+}u\frac{\partial}{\partial
y}+\frac{(\Omega_{+}^{2}-\Omega_{-}^{2})}{2\Omega_{+}}y\sin\Omega_{+}u\frac{\partial}{\partial
v},$ (4.8b) $\displaystyle Y^{b}_{1}$
$\displaystyle=-\frac{2}{(\Omega_{+}^{2}-\Omega_{-}^{2})}\left(\Omega_{-}\sin\Omega_{-}u\frac{\partial}{\partial
x}+\frac{\omega}{2}\cos\Omega_{-}u\frac{\partial}{\partial
y}\right)-x\cos\Omega_{-}u\frac{\partial}{\partial v},$ (4.8c) $\displaystyle
Y^{b}_{2}$
$\displaystyle=-\frac{2}{(\Omega_{+}^{2}-\Omega_{-}^{2})}\left(\frac{\omega}{2}\cos\Omega_{+}u\frac{\partial}{\partial
x}-\Omega_{+}\sin\Omega_{+}u\frac{\partial}{\partial
y}\right)-y\cos\Omega_{+}u\frac{\partial}{\partial v},$ (4.8d)

where $Y^{c}_{i}$ and $Y^{b}_{i}$ denote the Carroll translations and Carroll
boosts, respectively. Their commutators with $\partial_{u}$ are found as

$[\partial_{u},Y_{i}^{c}]=\pm\frac{(\Omega_{+}^{2}-\Omega_{-}^{2})}{2}Y_{i}^{b},\quad[\partial_{u},Y^{i}_{b}]=\mp\frac{2\Omega_{\mp}^{2}}{(\Omega_{+}^{2}-\Omega_{-}^{2})}Y_{i}^{c}.$
(4.9)

According to Noether’s theorem, the substitution of basis vectors
$\partial_{\mu}$ in (4.8) with the associated canonical momenta $p_{\mu}$
yields charges $Q(x,p)$ which are linear in momenta. Those charges are
conserved along the geodesic motion

$\\{Q,H_{geo}\\}=0,$ (4.10)

where geodesic Hamiltonian is (3.2). When projected, they become explicitly
time dependent as $u$ plays the role of time for NR physics. For instance,
Carroll translation $Y_{1}^{c}$ yields a conserved charge with explicit time-
dependence

$Q_{1}^{c}=\cos\Omega_{-}up_{x}-\frac{\omega}{2\Omega_{-}}\sin\Omega_{-}up_{y}-\frac{(\Omega_{+}^{2}-\Omega_{-}^{2})}{2\Omega_{-}}Mx\sin\Omega_{-}u,\qquad\frac{dQ_{1}^{c}}{du}=\\{Q_{1}^{c},H\\}+\frac{\partial
Q_{1}^{c}}{\partial u}=0,$ (4.11)

where $H$ is the NR Hamiltonian in (2.10). After canonical transformations
given at Section 2, the projected Carroll vector fields (4.8) generate NH
symmetry of the PU oscillator with $p_{v}=M$ being the central extension, see
eqn. $\\#$ (4.27) in [19].

$\ell=\frac{3}{2}$ NH symmetry: On the other hand, it is known that the PU
oscillator enjoys $\ell=\frac{3}{2}$ conformal NH symmetry provided that
$\Omega_{+}=3\Omega_{-}=3\Omega$ [15]. In this particular case, the PU
oscillator has two more symmetries, namely the dilation $D$ and special
conformal transformation $K$. Now, we can go in the opposite direction and
reversing the transformations at Section 2, we carry those charges to the
related CPP GW metric

$ds^{2}=\delta_{ij}dx^{i}dx^{j}+2dudv+2\sqrt{5}\Omega\epsilon_{ij}x^{i}dx^{j}du+\Omega^{2}\Big{(}9x^{2}+y^{2}\Big{)}du^{2},$
(4.12)

with $\omega^{2}=20\Omega^{2}$. A straightforward calculation leads extra
charges

$\displaystyle D$ $\displaystyle=-\frac{1}{2\Omega}\left(A\cos{2\Omega
u}-B\sin{2\Omega u}\right),$ (4.13a) $\displaystyle K$
$\displaystyle=-\frac{1}{2\Omega^{2}}\left(A\sin{2\Omega u}+B\cos{2\Omega
u}+p_{u}\right).$ (4.13b)

with

$\displaystyle A$
$\displaystyle=4\Omega^{2}\left(xy+\frac{(\sqrt{5}+1)}{8\Omega^{2}p_{v}^{2}}p_{x}p_{y}+\frac{(2+\sqrt{5})}{4\Omega
p_{v}}xp_{x}+\frac{\sqrt{5}}{4\Omega p_{v}}yp_{y}\right)p_{v},$ (4.14a)
$\displaystyle B$
$\displaystyle=4\Omega^{2}\left(x^{2}+\frac{(3+2\sqrt{5})}{4\Omega
p_{v}}xp_{y}+\frac{(3\sqrt{5}+5)}{16\Omega^{2}p_{v}^{2}}p_{y}^{2}+\frac{1}{4\Omega
p_{v}}yp_{x}+\frac{(\sqrt{5}-1)}{16\Omega^{2}p_{v}^{2}}p_{x}^{2}\right)p_{v}.$
(4.14b)

We complete it with the explicit expression of $H_{geo}$ (3.2)

$H_{geo}=\frac{\Big{(}p_{i}+\sqrt{5}\Omega\epsilon_{ij}x^{j}p_{v}\Big{)}^{2}}{2}+p_{u}p_{v}-\frac{\Omega^{2}}{2}\Big{(}9x^{2}+y^{2}\Big{)}p_{v}^{2},$
(4.15)

and canonical momenta

$p_{x}=\dot{x}-\sqrt{5}\Omega y\dot{u},\quad p_{y}=\dot{y}+\sqrt{5}\Omega
x\dot{u}.$ (4.16)

The conservation of (4.13) can be verified via

$\\{D,H_{geo}\\}=0=\\{K,H_{geo}\\}.$ (4.17)

Their Poisson brackets with the conserved momentum $p_{u}$ yield

$\\{p_{u},D\\}=2\Omega^{2}K+p_{u},\quad\\{p_{u},K\\}=-2D.$ (4.18)

It is straightforward to calculate the algebra of the charges $D$ and $K$ with
the Carroll generators (4.8). Finally, we note that the form of (4.13) is
similar to the distorted symmetries given in [45, 46] and deserves a further
analysis.

##  5 Lukash plane wave and PU oscillator

As a new example, here we study the underlying NR system of the Lukash plane
wave [26] and link it to the PU oscillator. Like CPP GW, Lukash plane wave is
also a vacuum solution of Einstein’s equations and it enjoys a $7$-dimensional
symmetry algebra. In complex coordinates, $1+3$ dimensional Lukash metric is
given by

$g_{L}=2dUdV+2d\bar{Z}dZ-2C\ \text{Re}[U^{2(i\kappa-1)}Z^{2}]dU^{2},\quad
Z=\frac{X^{1}+iX^{2}}{\sqrt{2}}.$ (5.1)

The coordinates are well defined either for $U>$ or $U<0$ but singular at
$U=0$. One can consider a non-negative wave amplitude, i.e., $C\geq 0$.
Parameter $\kappa$ is the frequency of the wave. We will work in $U>0$ region
where the Lukash metric (5.1) can be written in terms of the real Brinkmann
coordinates

$g_{L}=dX^{i}dX^{i}+2dUdV-\frac{C}{U^{2}}\Big{(}\big{(}(X^{1})^{2}-(X^{2})^{2}\big{)}\cos(2\kappa\ln
U)-2X^{1}X^{2}\sin(2\kappa\ln U)\Big{)}dU^{2}.$ (5.2)

In addition to 5-parameter Carroll isometry and the homothety
$Y_{h}=2V\partial_{V}+X^{i}\partial_{i}$, Lukash metric (5.2) is endowed with
an extra isometry $Y_{\kappa}$

$Y_{\kappa}=U\partial_{U}-V\partial_{V}-\kappa\epsilon_{ij}X^{i}\partial_{j}=U\partial_{U}+\frac{1}{2}(X^{i}+2\kappa\epsilon_{ij}X^{j})\partial_{i}-\frac{1}{2}Y_{h}.$
(5.3)

Thus, Lukash plane-wave is maximally symmetric. In order to bring the Lukash
metric (5.2) to the form (3.1), $Y_{\kappa}$ leads us to do the following
transformations [27]

$U=e^{T},\quad X^{i}=e^{T/2}\xi^{i},\quad V=V,\qquad\xi^{i}=(\xi,\eta).$ (5.4)

In the new coordinates $(\xi,\eta,T,V)$ Lukash metric $g_{L}$ (5.2) is
conformally related to another metric $g$ as

$\displaystyle g_{L}$ $\displaystyle=$ $\displaystyle e^{T}g$ (5.5)
$\displaystyle g$ $\displaystyle=$ $\displaystyle
d\xi^{i}d\xi^{i}+\xi^{i}d\xi^{i}dT+2dTdV+\left(-C\Big{(}(\xi^{2}-\eta^{2})\cos(2\kappa
T)-2\xi\eta\sin(2\kappa T)\Big{)}+\frac{1}{4}\xi^{i}\xi^{i}\right)dT^{2},$
(5.6)

and the extra isometry $Y_{\kappa}$ (5.3) turns to be a combination of a $T$
translation, a rotation and homothety:

$Y_{\kappa}=\partial_{T}-V\partial_{V}+(\kappa\epsilon_{ij}\xi^{j}-\frac{1}{2}\xi^{i})\frac{\partial}{\partial\xi^{i}}=\partial_{T}-\kappa\epsilon_{ij}\xi^{i}\partial_{j}-\frac{1}{2}Y_{h}.$
(5.7)

Now, let us make some observations: Firstly, in its new form (5.5), Lukash
metric is no more a Bargmann manifold because of the conformal factor $e^{T}$.
However, conformally related metrics share identical null geodesics. In order
to investigate the NR dynamics accompanying the Lukash metric $g_{L}$, we can
simply consider $g$ (5.6) which is a Bargmann manifold, and perform a null
reduction there. A simple calculation shows that $Y_{\kappa}$ (5.7) of $g_{L}$
leads a screw isometry for the conformally related metric $g$

$Y_{g}=\partial_{T}-\kappa\epsilon_{ij}\xi^{i}\partial_{j},$ (5.8)

cf. (3.7). We can even redefine999$\xi^{i}d\xi^{i}dT$ part of the metric has
no effect on geodesic equations and on NR dynamics as it is a total
derivative. $V\to V-\frac{1}{4}\xi^{i}\xi^{i}$ which puts $g$ (5.6) into
standard Brinkmann coordinates (3.6) as

$g=d\xi^{i}d\xi^{i}+2dTdV+\left(-C\Big{(}(\xi^{2}-\eta^{2})\cos(2\kappa
T)-2\xi\eta\sin(2\kappa T)\Big{)}+\frac{1}{4}\xi^{i}\xi^{i}\right)dT^{2}.$
(5.9)

While the Lukash metric $g_{L}$ is a vacuum solution, its conformal
counterpart $g$ (5.9) is not. In that, it also differs from the CPP GW. As in
the case of CPP GW, extra isometry (5.8) suggests us to make a rotation

$\displaystyle x$ $\displaystyle=\cos(\kappa T)\xi-\sin(\kappa T)\eta,$
(5.10a) $\displaystyle y$ $\displaystyle=\cos(\kappa T)\eta+\sin(\kappa
T)\xi.$ (5.10b)

Then, renaming $T=u$, $V=v$ we end up with rotating Brinkmann coordinates as

$g=dx^{2}+dy^{2}+2\kappa(ydx-
xdy)du+2dudv+(\Omega_{+}^{2}x^{2}+\Omega_{-}^{2}y^{2})du^{2},$ (5.11)

where

$\Omega_{+}^{2}=\kappa^{2}+\frac{1}{4}-C,\quad\Omega_{-}^{2}=\kappa^{2}+\frac{1}{4}+C,$
(5.12)

cf. (3.1).

In order to analyse the dynamics, first, we follow the null geodesic argument.
Namely, the geodesic equations for $x$ and $y$ with $u$ being the affine
parameter become

$\displaystyle x^{\prime\prime}+2\kappa y^{\prime}-\Omega_{+}^{2}x$
$\displaystyle=0,$ (5.13a) $\displaystyle y^{\prime\prime}-2\kappa
x^{\prime}-\Omega_{-}^{2}y$ $\displaystyle=0,$ (5.13b)

where $\\{^{\prime}\\}$ denotes derivation w.r.t. $u$. These equations are
exactly in the form of (1.3). Next, we compute the diagonalization frequencies
(2.8)

$\lambda_{1}^{2}=\kappa^{2}-\frac{1}{4}+\sqrt{C^{2}-\kappa^{2}},\qquad\lambda_{2}^{2}=\kappa^{2}-\frac{1}{4}-\sqrt{C^{2}-\kappa^{2}}.$
(5.14)

For $\kappa>\frac{1}{2}$ and $\kappa<C<\kappa^{2}+1/4$ the quantities
$\Omega_{\pm}^{2}$ and $\lambda^{2}$’s (5.14) are real and positive and thus
the transverse dynamics can be described by the PU oscillator. However, due to
the bi-Hamiltonian aspects of the PU model we confirm it by the Hamiltonian
approach.

First, we compare (5.2) with (3.8) and identify the profile $K$. Then, by
virtue of (3.20), we obtain the corresponding Hamiltonian $H$. Next, the
following generating function

$W({\bm{X}},\tilde{\bf p},\tilde{u})=\tilde{\bf
p}^{T}{\bm{X}}e^{-\tilde{u}/2}+\frac{P_{V}}{4}e^{-\tilde{u}}{\bm{X}}^{2},$
(5.15)

with new time $\tilde{u}=\ln(U)$ leads to the new coordinates
$\tilde{\bm{x}},\tilde{\bf p}$ and Hamiltonian $\tilde{H}$

$\tilde{H}=H\frac{dU}{d\tilde{u}}+\frac{\partial F}{\partial\tilde{u}},$
(5.16)

where the right hand side is expressed in terms of the tilde coordinates. In
our case the Hamiltonian (3.20) transforms into the following one

$\tilde{H}=\frac{\tilde{\bf
p}^{2}}{2P_{V}}-\frac{P_{V}}{8}\tilde{\bm{x}}^{2}+\frac{P_{V}C}{2}\tilde{\bm{x}}^{T}K^{(1)}(\tilde{u}){\bm{x}},$
(5.17)

where $K^{(1)}$ given by (4.1) with $\omega=-2\kappa$. Next, performing the
canonical transformation generated by (4.2) we arrive at the Hamiltonian $H$

$H=\frac{{\bf
p}^{2}}{2P_{V}}-\frac{P_{V}}{2}\left(x^{2}(1/4-C)+y^{2}(1/4+C)\right)-\kappa(yp_{x}-xp_{y}).$
(5.18)

Now, the above Hamiltonian takes the form of (2.10) ((2.2) with minus sign)
with $\Omega_{\pm}$ given by (5.12). The related NR symplectic structure
(2.10) can be written as

$\displaystyle H_{NR}$ $\displaystyle=$
$\displaystyle\frac{\Pi_{x}^{2}}{2}+\frac{\Pi_{y}^{2}}{2}-\frac{1}{2}(\Omega_{+}^{2}x^{2}+\Omega_{-}^{2}y^{2}),$
(5.19) $\displaystyle\sigma$ $\displaystyle=$ $\displaystyle d\Pi_{x}\wedge
dx+d\Pi_{y}\wedge dy-2\kappa dx\wedge dy,$ (5.20)

with $\Pi_{x}=p_{x}-\kappa y,\quad\Pi_{y}=p_{y}+\kappa x$ where NR mass $M$
which is reminiscent of conserved momentum $P_{V}$ can be restored trivially.
We find the chiral decomposition (2.11) and the coefficients (2.13) as

$\displaystyle\beta_{-}$
$\displaystyle=\frac{C-2\kappa^{2}-\sqrt{C^{2}-\kappa^{2}}}{2\kappa},\quad\quad\beta_{+}=\frac{2\kappa(\kappa^{2}+\frac{1}{4}-C)}{C-2\kappa^{2}-\sqrt{C^{2}-\kappa^{2}}}$
(5.21a) $\displaystyle\alpha_{+}$
$\displaystyle=\frac{-C-2\kappa^{2}+\sqrt{C^{2}-\kappa^{2}}}{2\kappa},\quad\quad\alpha_{-}=\frac{2\kappa(\kappa^{2}+\frac{1}{4}+C)}{-C-2\kappa^{2}+\sqrt{C^{2}-\kappa^{2}}};$
(5.21b)

and finally arrive at the PU Hamiltonian with frequencies (5.14).

##  6 Complete gravitational wave pulse and PU model

In this section we consider the third example of plane GWs whose transversal
dynamics can be related to the PU model. Namely, the metric is defined by
(3.8) with $\omega=0$ and the profile [28]

$K^{(2)}(U,{\bm{X}})=\frac{a}{(U^{2}+\varepsilon^{2})^{2}}{\bm{X}}^{T}\left(\begin{array}[]{cc}\cos(\phi(U))&\sin(\phi(U))\\\
\sin(\phi(U))&-\cos(\phi(U))\end{array}\right){\bm{X}},$ (6.1)

where

$\phi(U)=\frac{2\gamma}{\varepsilon}\tan^{-1}(U/\varepsilon),$ (6.2)

and we assume $a,\varepsilon,\gamma>0$. In contrast to the Lukash case, such a
plane GW is a complete pulse and exhibits proper conformal symmetry (besides
five Killing vectors and homothety). As above we consider the Hamiltonian
approach. To this end let us take the generating function

$W({\bm{X}},\tilde{\bf p},\tilde{u})=\frac{1}{\varepsilon}\tilde{\bf
p}^{T}{\bm{X}}\cos(\tilde{u})+\frac{P_{V}}{2\varepsilon}{\bm{X}}^{2}\sin(\tilde{u})\cos(\tilde{u}),$
(6.3)

together with the time change $U=\varepsilon\tan(\tilde{u})$. Then the new
Hamiltonian reads $\tilde{H}(\tilde{\bm{x}},\tilde{\bf
p},\tilde{u})=H\frac{dU}{d\tilde{u}}+\frac{\partial F}{\partial\tilde{u}}$.
After direct calculations we find that the Hamiltonian (3.20) (with
$K=K^{(2)}$) transforms into

$\tilde{H}=\frac{1}{2\tilde{M}}\tilde{\bf
p}^{2}+\frac{\tilde{M}}{2}\tilde{\bm{x}}^{T}\tilde{\bm{x}}-\frac{\tilde{M}}{2}{\tilde{\bm{x}}^{T}}K^{(1)}(\tilde{u})\tilde{\bm{x}},$
(6.4)

with $\tilde{M}=\varepsilon P_{V}$ and $K^{(1)}$ is defined by the parameters
$b=a/\varepsilon^{2}$, $\omega=2\omega_{0}=2\gamma/\varepsilon$. Thus, using
the $R$-transformation we can perform the canonical transformation generated
by (4.2) to the new coordinates ${\bm{x}},{\bf p}$ (the term
$\tilde{\bm{x}}^{T}\tilde{\bm{x}}$ takes the same form). At the end we arrive
at the Hamiltonian

$H=\frac{{\bf
p}^{2}}{2\tilde{M}}+\frac{\tilde{M}}{2}\left(x^{2}(1-\frac{a}{\varepsilon^{2}})+y^{2}(1+\frac{a}{\varepsilon^{2}})\right)+\omega_{0}(yp_{x}-xp_{y}).$
(6.5)

The above Hamiltonian is of the form (2.10) and thus is separable. Namely, for
$P_{V}>0$ and $1-\omega_{0}^{2}<-a/\varepsilon^{2}$ we obtain, after direct
computations, the PU Hamiltonian with the frequencies

$\quad\lambda_{1}^{2}=1+\omega_{0}^{2}+\sqrt{\Delta},\quad\lambda_{2}^{2}=1+\omega_{0}^{2}-\sqrt{\Delta};$
(6.6)

where $\Delta=4\omega_{0}^{2}+\frac{a^{2}}{\varepsilon^{4}}$.

Of course, we may ask what happens for other values of parameters. One can
check that for $|1-\omega^{2}_{0}|<a/\varepsilon^{2}$ we obtain the sum of the
inverted oscillator and ordinary one, while for
$1-\omega^{2}_{0}>a/\varepsilon^{2}$ we obtain the sum of two oscillators. The
case $P_{V}<0$ can be analysed by the change $H\rightarrow-H$ and
$\omega_{0}\rightarrow-\omega_{0}$. Finally, one can easily find suitable
values of parameters for which the odd frequencies and additional integrals of
motion appear.

##  7 Ideal Penning trap and $n=3$ PU oscillator

The Penning traps are used for mass spectroscopy and for measuring properties
of charged particles (see the Nobel Lecture by Dehmelt [29] and the reviews
[47, 48]). The ideal Penning trap is a $3+1$ dimensional NR ion trap where
charged ions are subject to a constant magnetic field in the 3rd direction
$\bm{B}=B\hat{z}$ and an anisotropic but time independent quadrupole potential
$V(x,y,z)=\frac{V_{0}}{4}(2z^{2}-x^{2}-y^{2})$ satisfying
${\bm{\nabla}}^{2}V(x,y,z)=0$ with constant $V_{0}$. It is this quadrupole
potential that leads to the inverted oscillators. The Lagrangian of a single
charged particle with mass M in an ideal Penning trap is

$L_{P}=\frac{1}{2}M(x^{\prime 2}+y^{\prime 2}+z^{\prime
2})+\frac{M\omega_{c}}{2}(xy^{\prime}-yx^{\prime})+\frac{1}{4}M\omega_{z}^{2}(x^{2}+y^{2}-2z^{2}),$
(7.1)

where we set $\omega_{c}=\frac{qB}{M}$ and $\omega_{z}=\frac{qV_{0}}{M}$ and
$q$ is the charge of the particle. The equations of motion are

$\displaystyle
x^{\prime\prime}-\omega_{c}y^{\prime}-\frac{1}{2}\omega_{z}^{2}x$
$\displaystyle=0,$ (7.2a) $\displaystyle
y^{\prime\prime}+\omega_{c}x^{\prime}-\frac{1}{2}\omega_{z}^{2}y$
$\displaystyle=0,$ (7.2b) $\displaystyle z^{\prime\prime}+\omega_{z}^{2}z$
$\displaystyle=0.$ (7.2c)

$x$-$y$ part of the dynamics is a special case of (1.3) with
$\Omega_{+}^{2}=\Omega_{-}^{2}=\frac{1}{2}\omega_{z}^{2}$ and
$\omega=\omega_{c}$. The motion in the $z-$direction is a decoupled harmonic
oscillator with frequency $\omega_{z}$. The symplectic structure regarding
(7.2) differs from (2.10) only with a trivial addition of that harmonic
motion.

It is known that the motions in the Penning trap can be separated into three
independent parts [47, 48]. The first motion is the harmonic trapping one
along the $z$-direction with axial frequency $\omega_{z}$. The second motion
is called the circular cyclotron motion with the modified cyclotron frequency
$\omega_{+}$ and the last one is the circular magnetron motion at magnetron
frequency $\omega_{-}$ where

$\omega_{\pm}=\frac{\omega_{c}\pm\sqrt{\omega_{c}^{2}-2\omega_{z}^{2}}}{2}.$
(7.3)

Typically in Penning traps $\omega_{c}>\omega_{+}\gg\omega_{z}\gg\omega_{-}$.
Interestingly, the energy of the magnetron motion is negative which is
relevant for the buffer gas cooling technique [48]. And, this motivates us to
study Penning trap in our PU setting outlined at Section 2.

Diagonalization frequencies (2.8) are found to be

$\lambda_{1,2}^{2}=\frac{(\omega_{c}^{2}-\omega_{z}^{2})\pm\omega_{c}\sqrt{\omega_{c}^{2}-2\omega_{z}^{2}}}{2},$
(7.4)

and we immediately realize that they are equal to the modified cyclotron and
magnetron frequencies as

$\lambda_{1}=\omega_{+},\qquad\lambda_{2}=\omega_{-}.$ (7.5)

This attributes direct physical meaning to the PU oscillator frequencies
$\lambda_{1}$ and $\lambda_{2}$. We may rewrite (2.5) in terms of
$\omega_{\pm},\omega_{z}$ as

$\omega_{+}\omega_{-}=\frac{\omega_{z}^{2}}{2},\quad\omega_{c}^{2}=\omega_{+}^{2}+\omega_{-}^{2}+\omega_{z}^{2},$
(7.6)

thus they satisfy $\omega_{+}+\omega_{-}=\omega_{c}$ and
$\omega_{+}-\omega_{-}=\sqrt{\omega_{c}^{2}-2\omega_{z}^{2}}$. We note here an
interesting fact that the condition for the system to be a PU oscillator
(7.6), i.e., $\omega_{c}^{2}=\omega_{+}^{2}+\omega_{-}^{2}+\omega_{z}^{2}$,
holds not only for the ideal trap but also for the real Penning traps
suffering from imperfections [49].

Coefficients for chiral decomposition (2.11) are computed as

$\beta_{-}=M\omega_{+}=\alpha_{-},\quad\alpha_{+}=M\omega_{-}=\beta_{+}.$
(7.7)

In those coordinates, both the Hamiltonian and symplectic form for the Penning
trap decompose

$\displaystyle H_{P}$
$\displaystyle=\frac{M(\omega_{+}-\omega_{-})}{2}\left(\omega_{+}(X_{-}^{1}X_{-}^{1}+X_{-}^{2}X_{-}^{2})-\omega_{-}(X_{+}^{2}X_{+}^{2}+X_{+}^{1}X_{+}^{1})\right)+\frac{1}{2}(p_{z}^{2}+\omega_{z}^{2}z^{2}),$
(7.8a) $\displaystyle\sigma_{P}$
$\displaystyle=M(\omega_{+}-\omega_{-})(dX_{+}^{1}\wedge
dX_{+}^{2}-dX_{-}^{1}\wedge dX_{-}^{2})+dp_{z}\wedge dz.$ (7.8b)

A final transformation to canonical coordinates

$\displaystyle\big{(}M(\omega_{+}-\omega_{-})\omega_{+}\big{)}^{1/2}X_{-}^{1}$
$\displaystyle=-p_{1},\quad\left(\frac{M(\omega_{+}-\omega_{-})}{\omega_{+}}\right)^{1/2}X_{-}^{2}=x^{1},$
(7.9a)
$\displaystyle\big{(}M(\omega_{+}-\omega_{-})\omega_{-}\big{)}^{1/2}X_{+}^{2}$
$\displaystyle=-p_{2},\quad\left(\frac{M(\omega_{+}-\omega_{-})}{\omega_{-}}\right)^{1/2}X_{+}^{1}=x^{2}.$
(7.9b)

provides us with the following Hamiltonian of 3 oscillators with alternating
sign

$H_{P}=\frac{1}{2}\left[\big{(}p_{1}^{2}+\omega_{+}^{2}(x^{1})^{2}\big{)}-\big{(}p_{2}^{2}+\omega_{-}^{2}(x^{2})^{2}\big{)}+\big{(}p_{z}^{2}+\omega_{z}^{2}z^{2}\big{)}\right].$
(7.10)

Comparison with (1.2) shows that Penning trap becomes the $1+1$-dimensional
$n=3$ (the 6th order) PU oscillator composed of separate physical motions of
Penning trap with $\omega_{\pm},\omega_{z}$.

We can lift the Penning trap (7.1) and the associated plane wave is a
straightforward extension of (3.1) with the addition of $z$-motion (7.2c) and
with $\omega=\omega_{c}$ and
$\Omega_{+}^{2}=\Omega_{-}^{2}=\frac{\omega_{z}^{2}}{2}$. This is a $1+4$
dimensional non-vacuum solution of Einstein’s equations as noted in [30].
Therefore, we can derive conserved quantities for the Penning trap using the
Carroll symmetries (3 translations, 3 boosts and $\partial_{v}$) via modifying
(3.3) as

$\displaystyle Q_{1}^{c}$
$\displaystyle=-\frac{M}{2}(\omega_{+}-\omega_{-})(x\sin\omega_{-}u+y\cos\omega_{-}u)+p_{x}\cos\omega_{-}u-p_{y}\sin\omega_{-}u,$
(7.11a) $\displaystyle Q_{2}^{c}$
$\displaystyle=-\frac{M}{2}(\omega_{+}-\omega_{-})(x\cos\omega_{+}u-y\sin\omega_{+}u)+p_{x}\sin\omega_{+}u+p_{y}\cos\omega_{+}u,$
(7.11b) $\displaystyle Q_{1}^{b}$
$\displaystyle=-\frac{M}{2}(x\cos\omega_{-}u-y\sin\omega_{u})-\frac{1}{(\omega_{+}-\omega_{-})}(p_{x}\sin\omega_{-}u+p_{y}\cos\omega_{-}u),$
(7.11c) $\displaystyle Q_{2}^{b}$
$\displaystyle=-\frac{M}{2}(x\sin\omega_{+}u+y\cos\omega_{+}u)-\frac{1}{(\omega_{+}-\omega_{-})}(p_{x}\cos\omega_{+}u-p_{y}\sin\omega_{+}u)$
(7.11d) $\displaystyle Q_{3}^{c}$
$\displaystyle=p_{z}\cos\omega_{z}u+M\omega_{z}z\sin\omega_{z}u,$ (7.11e)
$\displaystyle Q_{3}^{b}$
$\displaystyle=\frac{p_{z}}{\omega_{z}}\sin\omega_{z}u-Mz\cos\omega_{z}u.$
(7.11f)

They satisfy

$\frac{dQ}{du}=\\{Q,H_{P}\\}+\frac{\partial Q}{\partial u}=0,$ (7.12)

where the conserved Hamiltonian $H_{P}$ can be obtained from the screw
symmetry. Charges (7.11) satisfy 3 independent Heisenberg algebras with the
same central extension mass $p_{v}=M$ and they can be derived from the NH
symmetry of the PU oscillator [15] as well.

Comments on $\ell=\frac{5}{2}$ conformal NH algebra: As in Section 4, we may
proceed the other way around. By using the correspondence with the PU
oscillator, we are immediately tempted to investigate whether a Penning trap
can be a realization of the $\ell=\frac{5}{2}$ conformal NH algebra or not. To
this end, we need to impose the odd frequency condition [15] on the Penning
trap frequencies $\omega_{\pm},\omega_{z}$. As it is mentioned, in a typical
Penning trap those frequencies follow the order
$\omega_{c}>\omega_{+}\gg\omega_{z}\gg\omega_{-}$. This physical condition
forces us to set

$\omega_{+}=5\Omega,\quad\omega_{z}=3\Omega,\quad\omega_{-}=\Omega.$ (7.13)

However, the above choice contradicts (7.6). In a Penning trap, it is also
possible that the magnetron frequency $\omega_{-}$ can be larger than
$\omega_{z}$. But this comes at a price because the trap is now operated close
to stability point. In that case, one interchanges the values of $\omega_{-}$
and $\omega_{z}$ in (7.13) but again fails to satisfy (7.6). Thus, we come to
the conclusion that Penning trap is not a realization of the
$\ell=\frac{5}{2}$ conformal NH algebra.

##  8 Non-commuting coordinates and the PU oscillator

From a theoretical point of view, our $2+1$ dimensional NR equations (1.3)
provide a proper arena for exotic dynamics (see e.g., [50, 51, 31] and
references therein). In solid state physics, non-commuting coordinates and
their exotic algebra arise in anomalous Hall problem as a semiclassical effect
of Berry’s phase [32]. A particular case of (1.3), namely the planar Hill’s
equations were already carried to the non-commutative plane where symmetries
span two independent Heisenberg algebras with different central extensions
[11, 12]. Just recently, Horvathy and his collaborators investigated $2+1$
dimensional Carrolian dynamics which admits two central charges [52]. Here, we
will be dealing with its Galilean analogue and relate it to the PU oscillator
explicitly.

To postulate non-commutativity of the coordinates $(x,y)$, we modify the
symplectic 2-form in (2.10) as

$\displaystyle\sigma\star$ $\displaystyle=\sigma+\theta d\Pi_{x}\wedge
d\Pi_{y},$ (8.1a) $\displaystyle H$
$\displaystyle=\frac{\Pi_{x}^{2}}{2M}+\frac{\Pi_{y}^{2}}{2M}-\frac{M}{2}(\Omega_{+}^{2}x^{2}+\Omega_{-}^{2}y^{2}),$
(8.1b)

while keeping Hamiltonian the same. $\theta$ is a constant non-commutativity
parameter. Poisson brackets can be easily obtained from the inverse of the
modified symplectic matrix $\sigma\star$ (8.1a) as

$\\{\Pi_{i},\Pi_{j}\\}=\frac{\epsilon_{ij}M\omega}{1-M\omega\theta},\qquad\\{x^{i},\Pi_{j}\\}=\frac{\delta_{ij}}{1-M\omega\theta},\qquad\\{x^{i},x^{j}\\}=\frac{\epsilon_{ij}\theta}{1-M\omega\theta},$
(8.2)

where $x$ and $y$ are no more commuting. $\theta$ also modifies Hamilton’s
equations as

$\displaystyle(x^{i})^{\prime}$
$\displaystyle=\frac{1}{1-M\omega\theta}\left(\frac{\Pi_{i}}{M}-\theta
M\epsilon_{ij}(\Omega_{+}^{2}x\delta_{j1}+\Omega_{-}^{2}y\delta_{j2})\right),$
(8.3a) $\displaystyle(\Pi_{i})^{\prime}$
$\displaystyle=\frac{1}{1-M\omega\theta}\Big{(}\omega\epsilon_{ij}\Pi_{j}+M(\Omega_{+}^{2}x\delta_{i1}+\Omega_{-}^{2}y\delta_{i2})\Big{)}.$
(8.3b)

Poisson brackets (8.2) are identical to the ones in the non-commutative Landau
problem [53]. When $\theta$ is switched off, we recover (1.3), as expected.

Decomposing our exotic system, we see that only the equations related to the
symplectic form alter

$\displaystyle\beta_{+}\beta_{-}$
$\displaystyle=M^{2}\Omega_{+}^{2},\quad\quad\alpha_{+}\alpha_{-}=M^{2}\Omega_{-}^{2},$
(8.4a) $\displaystyle\alpha_{+}+\beta_{-}-\theta\alpha_{+}\beta_{-}$
$\displaystyle=M\omega,\quad\quad\alpha_{-}+\beta_{+}-\theta\alpha_{-}\beta_{+}=M\omega,$
(8.4b)

cf. (2.12). Assuming no relation between parameters $\omega,\Omega_{\pm}$ as
before , we may solve for $\theta$-dependent coefficients
$\alpha_{\pm},\beta_{\pm}$101010For instance, we find
$\displaystyle\beta_{-}$ $\displaystyle=$
$\displaystyle\frac{M(\omega^{2}+\Omega_{+}^{2}-\Omega_{-}^{2})+\omega\sqrt{\Delta}-\theta
M\Omega_{-}^{2}(\theta
M^{2}\Omega_{+}^{2}+\sqrt{\Delta})}{2(\omega-M\theta\Omega_{-}^{2})},$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{M^{2}}{(\omega-M\theta\Omega_{-}^{2})^{2}}(\theta^{2}M^{2}\Omega_{+}^{2}\Omega_{-}^{2}+\Omega_{-}^{2}-\Omega_{+}^{2}-\omega^{2})^{2}-4M^{2}\Omega_{+}^{2},$
cf. (2.13). In order to have simpler expressions, one may choose a specific
parametrization such as $\omega^{2}=2(\Omega_{+}^{2}+\Omega_{-}^{2})$ [19].
Then the remaining coefficients can be easily found.. However, the expressions
become rather long in that case. Thus we continue with symbols
$\alpha_{\pm},\beta_{\pm}$ without substituting their actual values in terms
of $\omega,\Omega_{\pm}$.

Solutions of (8.4) satisfy the following identities

$\displaystyle(\beta_{-}-\beta_{+})(1-\theta\alpha_{+})$
$\displaystyle=(\alpha_{-}-\alpha_{+})(1-\theta\beta_{+}),$ (8.5a)
$\displaystyle(\beta_{-}-\beta_{+})(1-\theta\alpha_{-})$
$\displaystyle=(\alpha_{-}-\alpha_{+})(1-\theta\beta_{-}),$ (8.5b)

and they decompose the symplectic structure (8.1) as

$\displaystyle H$
$\displaystyle=\scriptstyle{\frac{M}{2}\left[\frac{\beta_{-}}{M^{2}}(\beta_{-}-\beta_{+})(X_{-}^{1})^{2}+\frac{(\alpha_{-}-\alpha_{+})}{\beta_{-}}(1-\theta\alpha_{-})(1-\theta\beta_{-})\lambda_{1}^{2}(X_{-}^{2})^{2}-\frac{\alpha_{+}}{M^{2}}(\alpha_{-}-\alpha_{+})(X_{+}^{2})^{2}-\frac{(\beta_{-}-\beta_{+})}{\alpha_{+}}(1-\theta\beta_{+})(1-\theta\alpha_{+})\lambda_{2}^{2}(X_{+}^{1})^{2}\right]},$
(8.6a) $\displaystyle\sigma$
$\displaystyle=(\beta_{-}-\beta_{+})\left[(1-\theta\alpha_{+})dX_{+}^{1}\wedge
dX_{+}^{2}-(1-\alpha_{-})dX_{-}^{1}\wedge dX_{-}^{2}\right],$ (8.6b)

where

$\lambda_{1}^{2}=\frac{\alpha_{-}\beta_{-}}{M^{2}(1-\theta\alpha_{-})(1-\theta\beta_{-})},\quad\lambda_{2}^{2}=\frac{\alpha_{+}\beta_{+}}{M^{2}(1-\theta\beta_{+})(1-\theta\alpha_{+})}$
(8.7)

Both the symplectic structure (8.6) and diagonalization frequencies (8.7) go
back to the previous expressions (2.14) when $\theta=0$.

Solving Hamilton’s equations, we obtain

$\displaystyle X_{+}^{1}$
$\displaystyle=A\cos\lambda_{2}t+B\sin\lambda_{2}t,\qquad\qquad\qquad\qquad
X_{-}^{1}=C\cos\lambda_{1}t+D\sin\lambda_{1}t,$ (8.8a) $\displaystyle
X_{+}^{2}$
$\displaystyle=-\frac{\lambda_{2}M(1-\theta\beta_{+})}{\alpha_{+}}(A\sin\lambda_{2}t-B\cos\lambda_{2}t),\quad
X_{-}^{2}=-\frac{\lambda_{1}M(1-\theta\beta_{-})}{\alpha_{-}}(C\sin\lambda_{1}t-D\cos\lambda_{1}t).$
(8.8b)

Associated equations of motion are

$(X_{-}^{1,2})^{\prime\prime}+\lambda_{1}^{2}X_{-}^{1,2}=0,\quad(X_{+}^{1,2})^{\prime\prime}+\lambda_{2}^{2}X_{+}^{1,2}=0.$
(8.9)

Thus, we may pass to the Darboux coordinates via

$\displaystyle p_{1}$
$\displaystyle=-\sqrt{\frac{\beta_{-}(\beta_{-}-\beta_{+})}{M}}X_{-}^{1},\qquad
x^{1}=\sqrt{\frac{M(\alpha_{-}-\alpha_{+})(1-\theta\alpha_{-})(1-\theta\beta_{-})}{\beta_{-}}}X_{-}^{2},$
(8.10a) $\displaystyle p_{2}$
$\displaystyle=-\sqrt{\frac{\alpha_{+}(\alpha_{-}-\alpha_{+})}{M}}X_{+}^{2},\qquad
x^{2}=\sqrt{\frac{M(\beta_{-}-\beta_{+})(1-\theta\beta_{+})(1-\theta\alpha_{+})}{\alpha_{+}}},$
(8.10b)

so that our Hamiltonian becomes a combination of two harmonic oscillators with
a minus sign between them (2.17) where frequencies $\lambda_{1,2}$ are
$\theta$ dependent (8.7). With the help of a final canonical transformation
(2.18) we obtain the Ostrogradski Hamiltonian for the PU oscillator

$H_{PU}^{\theta}=p_{x}v+\frac{p_{v}^{2}}{2}+\frac{\lambda_{1}(\theta)^{2}+\lambda_{2}(\theta)^{2}}{2}v^{2}-\frac{\lambda_{1}(\theta)^{2}\lambda_{2}(\theta)^{2}}{2}x^{2},$
(8.11)

$\lambda_{1,2}$ are positive for sufficiently small values of $\theta$. The
conserved charges take the form

$\displaystyle Q_{1}^{c}$
$\displaystyle=\sqrt{\frac{M(\beta_{-}-\beta_{+})(1-\theta\beta_{+})}{\alpha_{+}(1-\theta\alpha_{+})(\lambda_{1}^{2}-\lambda_{2}^{2})}}\left((p_{x}+\lambda_{2}^{2}v)\cos\lambda_{2}u-(p_{v}+\lambda_{1}^{2}x)\lambda_{2}\sin\lambda_{2}u\right),$
(8.12a) $\displaystyle Q_{2}^{c}$
$\displaystyle=\sqrt{\frac{(\beta_{-}-\beta_{+})}{(\lambda_{1}^{2}-\lambda_{2}^{2})}}\left(\sqrt{\frac{M}{\beta_{-}}}\lambda_{1}(p_{v}+\lambda_{2}^{2}x)\cos\lambda_{1}u+\sqrt{\frac{\beta_{-}}{M}}\frac{\alpha_{-}}{M\lambda_{1}^{2}(1-\theta\beta_{-})(1-\theta\alpha_{-})}(p_{x}+\lambda_{1}^{2}v)\sin\lambda_{1}u\right),$
(8.12b) $\displaystyle Q_{1}^{b}$
$\displaystyle=-\sqrt{\frac{M\alpha_{+}}{(\beta_{-}-\beta_{+})(1-\theta\beta_{+})(1-\theta\alpha_{+})(\lambda_{1}^{2}-\lambda_{2}^{2})}}\left((p_{v}+\lambda_{1}^{2}x)\cos\lambda_{2}u+\frac{1}{\lambda_{2}}(p_{x}+\lambda_{2}^{2}v)\sin\lambda_{2}u\right),$
(8.12c) $\displaystyle Q_{2}^{b}$
$\displaystyle=-\sqrt{\frac{M}{(\beta_{-}-\beta_{+})(\lambda_{1}^{2}-\lambda_{2}^{2})}}\left(\frac{\sqrt{\beta_{-}}}{\lambda_{1}(1-\theta\alpha_{-})}(p_{x}+\lambda_{1}^{2}v)\cos\lambda_{1}u-\sqrt{\frac{1}{\beta_{-}}}\frac{M^{2}\lambda_{1}^{2}(1-\theta\beta_{-})}{\alpha_{-}}(p_{v}+\lambda_{2}^{2}x)\sin\lambda_{1}u\right),$
(8.12d)

and when $\theta=0$, they are projected versions of (3.3) subject to
coordinate changes outlined above. Indeed, we have

$\frac{dQ_{1,2}^{c,b}}{du}=\\{Q_{1,2}^{c,b},H_{PU}^{\theta}\\}+\frac{\partial
Q_{1,2}^{c,b}}{\partial u}=0,$ (8.13)

as in (4.11). Their Poisson brackets read

$\\{Q_{1}^{c},Q_{1}^{b}\\}=\frac{M}{1-\theta\alpha_{+}},\qquad\\{Q_{2}^{c},Q_{2}^{b}\\}=\frac{M}{1-\theta\alpha_{-}},$
(8.14)

and span the exotic NH algebra with two central extensions [50]. Obviously,
there is a solution for the system (8.4) with $\alpha_{+}=\alpha_{-}$ that
necessarily yields $\beta_{-}=\beta_{+}$. This case is excluded above as it
leads to a vanishing symplectic structure (8.6). We emphasize that $\theta$ is
a small parameter at hand and it is possible to find realizations such that
$\beta_{-}>\beta_{+}$ and $\alpha_{-}>\alpha_{+}$111111For instance suitable
modifications of the Penning trap with (7.7).. When $\theta=0$, the usual NH
symmetry with one central extension $M$ is recovered. Above results are in
line with [19] where $\theta=0$ and where parameters obey
$\lambda_{1}=\Omega_{+},\quad\lambda_{2}=\Omega_{-},\quad\omega^{2}=2(\Omega_{+}^{2}+\Omega_{-}^{2})$,
see section 4.

##  9 Other examples

Below, we present other examples which fit in our framework.

###  9.1 Double copy and electromagnetic configurations for the PU oscillator

In this section, based on the classical double copy conjecture [54, 55], we
will show that the PU dynamics appears also in the motion of the charged
particle in some electromagnetic fields. Namely, in the Minkowski spacetime
and the light-cone coordinates we take the potential $A=K(U,X^{1},X^{2})dU$
where $K$ is the profile of the pp-wave metric (3.8). Then the related field
strength is

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\partial_{\mu}K\delta^{U}_{\nu}-\partial_{\nu}K\delta^{U}_{\mu}\Big{)}dX^{\mu}\wedge
dX^{\nu}\equiv\frac{1}{2}F_{\mu\nu}dX^{\mu}dX^{\nu}.$ (9.1)

One can easily see that

$F\wedge F=0=\star F\wedge F,$ (9.2)

so the generated electromagnetic fields are singular121212Such configurations
sometimes called as null.. Maxwell’s equations are

$dF=0,\quad d\star F={\bm{\nabla}}^{2}K\ dX^{1}\wedge dX^{2}\wedge dU,$ (9.3)

where ${\bm{\nabla}}^{2}$ is $2$-dimensional Laplacian. While the first set of
equations are identity $d^{2}=0$, the second set (here written in a compact
manner) relates the original plane-wave (3.6) to the Maxwell fields. Next, we
put $g=\eta$ (the Minkowski metric) and $\omega=0$ in the Lagrangian (3.9).
Then by virtue of (3.12) we obtain that the transversal dynamics, i.e.
$X^{1},X^{2}$ direction, is governed by the Hamiltonian

$H=\frac{{\bf P}^{2}+m^{2}}{2P_{V}}-eA_{U}.$ (9.4)

Comparing it with (3.12) we see that the above Hamiltonian is equal to the one
in the gravitational case with the replacement
$K\rightarrow\frac{2eK}{P_{V}}$. In view of this the transversal dynamics in
such electromagnetic fields reduce, after above identification, to the one
presented for the gravitational backgrounds. In particular, we can use the
examples discussed in the above sections. We start with the CPP GW and its
double copy, namely BB’s vortex [56] as pointed out by Ilderton [33].
Specifically, when it comes to the charged particle motion with charge $e$ via
double copy, diagonalization frequencies become

$\lambda_{1,2}^{2}=\frac{\omega^{2}}{4}\pm\frac{2eb}{P_{V}},$ (9.5)

and for suitable values of parameters (cf. Section 4) they are real and
positive and thus they become the frequencies of the PU oscillator. The same
concerns the electromagnetic counterparts of the Lukash waves after
replacement $C\rightarrow 2eC/P_{V}$ in the frequencies (5.14) as well as the
complete pulse discussed in Section 6. In summary, for all the electromagnetic
fields constructed (via double copy) from our gravitational examples the
transversal dynamics, for suitable parameters, are described by the PU
oscillator.

At the end let us consider more general case, i.e. the metric given by (3.1)
with non-vanishing $\omega$. To make a double copy, first, we transform it
into the form (3.6) and next take the corresponding electromagnetic potential
(i.e. given by the electromagnetic field defined by $A_{U}=K$ with (3.6)).
Then, the transverse dynamics of such a system is equivalent to the
gravitational one with the replacement
$\omega^{2}\rightarrow\frac{2e\omega^{2}}{P_{V}}$ and
$\Omega^{2}_{\pm}\rightarrow\frac{2e\Omega^{2}_{\pm}}{P_{V}}$ (for
$e/P_{V}>0$). In view of this the same replacement should be made in
frequencies (2.8), this yields a suitable scaling factor. In consequence, we
have a (in general non-vacuum) solution to the Maxwell equations whose
transverse dynamics can be modeled by the PU oscillator.

###  9.2 Hill’s equations

In [34], the symmetries of Hill’s equations and their relation to the Landau
problem were discussed. They also studied the ED lift of the planar Hill
problem. Here, we review this old problem in our framework presented at
Sections 2 and 3.

In the center of mass frame, Hill’s equations are given as

$\displaystyle x^{\prime\prime}-2\omega y^{\prime}-3\omega^{2}x$
$\displaystyle=0,$ (9.6a) $\displaystyle y^{\prime\prime}+2\omega x^{\prime}$
$\displaystyle=0,$ (9.6b)

and they correspond to “zero frequency condition”: one of the frequencies, say
$\Omega_{-}=0$. Diagonalization frequencies (2.8) can be found easily

$\lambda_{1}=\omega,\lambda_{2}=0.$ (9.7)

Solving for (2.12), we get

$\beta_{-}=2M\omega,\quad\beta_{+}=\frac{3}{2}M\omega,\quad\alpha_{-}=\frac{1}{2}M\omega,\quad\alpha_{+}=0.$
(9.8)

It is now straightforward decompose this system as in [34, 11] with suitable
modifications of Section 2. Therefore, the Hill dynamics can be considered as
very a special case of the PU oscillator, though strictly speaking positive
frequencies are usually assumed in the latter.

When we lift Hill’s equations (9.6), we obtain

$ds^{2}=dx^{2}+dy^{2}+2dudv+2\omega(xdy-ydx)+3\omega^{2}x^{2}du^{2},$ (9.9)

which is a Bargmann manifold131313Comparing with Lukash example (5.11), we see
that this is a particular case of it with $x\leftrightarrow\beta,\quad
y\leftrightarrow\alpha,\quad\kappa=\omega,$ and a simple calculation yields
$\omega^{2}=\frac{1}{2},\quad C=\frac{3}{4}.$ as stated in [34] and it is a
non-vacuum solution. Having solved the SL problem for the relativistic plane
wave (9.9) or finding constants of motion for the NR motion (9.6), one can
readily obtain the Carroll vector fields

$\displaystyle Y_{c}^{1}$ $\displaystyle=\partial_{x}-\frac{3}{2}\omega
u\partial_{y}+\frac{\omega}{2}(y-3\omega ux)\partial_{v},$ (9.10a)
$\displaystyle Y_{c}^{2}$ $\displaystyle=\frac{\sin\omega
u}{2}\partial_{x}+\cos\omega u\partial_{y}+\frac{\omega}{2}(x\cos\omega
u+y\sin\omega u)\partial_{v},$ (9.10b) $\displaystyle Y_{1}^{b}$
$\displaystyle=-\frac{2}{\omega}(\partial_{y}+\omega x\partial_{v}),$ (9.10c)
$\displaystyle Y_{2}^{b}$ $\displaystyle=-\frac{2}{\omega}\cos\omega
u\partial_{x}+\frac{4}{\omega}\sin\omega u\partial_{y}+2(x\sin\omega
u-y\cos\omega u)\partial_{v},$ (9.10d)

which satisfy (3.4). It is an easy exercise to find the related BJR metric and
the canonical transformation that leads to a new time-dependent NR
Hamiltonian.

###  9.3 Lagrange points and Rydberg atoms

Equations (1.3) also define orbits of trojan asteroids in the Sun-Jupiter
system near Lagrange points [35]. Specifically, equations in the $x-y$ plane
are

$\displaystyle x^{\prime\prime}-2\omega y^{\prime}-\omega^{2}(1-a)x$
$\displaystyle=0,$ (9.11a) $\displaystyle y^{\prime\prime}+2\omega
x^{\prime}-\omega^{2}(1-b)y$ $\displaystyle=0.$ (9.11b)

It is straightforward to derive diagonalization frequencies (2.8). The motion
in the third direction is a harmonic one and it decouples. In [35], it was
also argued that the motion of electrons in diatomic molecules interacting
with circularly polarized electromagnetic wave is defined with similar
equations. Likewise, three dimensional motion of electrons under the effect of
fields of rotating molecules are governed by the Hamiltonian in a canonical
form, namely equations $\\#$ (31,32) in [57]

$\displaystyle H_{Q}$ $\displaystyle=$
$\displaystyle\omega_{+}a_{+}^{\dagger}a_{+}-\omega_{-}a_{-}^{\dagger}a_{-}+\omega_{z}a_{z}^{\dagger}a_{z}+E$
$\displaystyle\omega_{\pm}$ $\displaystyle=$
$\displaystyle\Omega\sqrt{\frac{2-q\pm\sqrt{9q^{2}-8q}}{2}},\quad\omega_{z}=\Omega\sqrt{q},$
(9.12)

where $E$ is a constant. For stability, one demands $8/9<q<1$. In that manner,
both the systems are related to the Penning trap which can be mapped to a 6th
order PU oscillator, see section 7.

###  9.4 Gravitational trapping

Lastly, we revisit trapping of particles via gravitational waves with angular
momentum [36] where the motion of a test particle in the Bessel gravitational
wave background was found from the geodesic deviation equations. Skipping the
details, we present their result consisting of coupled planar equations,
namely their eqn. $\\#$ (14),

$\displaystyle x^{\prime\prime}-\omega y^{\prime}-(1/4-\gamma)\omega^{2}x$
$\displaystyle=0$ (9.13a) $\displaystyle y^{\prime\prime}+\omega
x^{\prime}-(1/4+\gamma)\omega^{2}y$ $\displaystyle=0,$ (9.13b)

where $\gamma$ is a constant. Depending on the sign of it, one can assign
$\Omega_{\pm}$ and calculate $\lambda_{1,2}$ from (2.8) with
$\lambda_{1}^{2}>\lambda_{2}^{2}$. Then, it is easy to observe that the
diagonalization frequencies happen to be the same as the original frequencies

$\lambda_{1,2}^{2}=\Omega_{\pm}^{2}=(1/4\mp\gamma)\omega^{2},$ (9.14)

as in Section 4 where circularly polarized periodic gravitational wave was
worked out. Following the same steps, one can map this system to a PU
oscillator in a straightforward manner.

##  10 Discussion

Beginning with the NR planar Lagrangian (2.1), we have shown that the dynamics
of several physical systems in different contexts like the geodesic motion in
certain $1+3$ dimensional plane-waves, NR motion of a charged ion in a Penning
trap or the motion of an exotic particle in $2+1$ dimensions etc. all boil
down to the $1+1$ dimensional PU oscillator. We explicitly relate the
(conformal) NH symmetries of the PU oscillator to the Carroll symmetry of
plane waves and to the conserved charges of the Penning trap and the exotic
particle. For our physical examples, it was enough to consider real
frequencies $\Omega_{\pm}$ and $\lambda_{1,2}$. However, our computations can
be extended to a wider parameter range. One challenging question for the
future is how to deal with the degenerate case of the PU oscillator
frequencies i.e., $\lambda_{1}=\lambda_{2}$. See [58] for a realization of
this case.

Next, at Section 3, we discuss the ED lift of our NR planar dynamics and write
the generic $1+3$ dimensional plane wave (3.1) whose underlying NR motion can
be described by the $n=2$ PU oscillator. The integrability of the NR dynamics
(1.3) allows us to solve the SL equation of the geodesic motion and find out
the Carroll symmetry of that plane wave (3.3). As an alternative to null-
geodesics argument, we also provide a Hamiltonian approach because the PU
oscillator is, in fact, a bi-Hamiltonian system.

On the other hand, the ED lift for higher derivatives theories was worked out
by Galajinsky and Masterov in [59]. Having related its underlying NR mechanics
to the PU oscillator, as a follow up problem we would like to find out the
relation between our plane wave metric (3.1) and the one presented in [59]. As
a warm-up exercise, at Section 3.3 we study the coordinate transformations
between Brinkmann and BJR metrics and their underlying NR dynamics.

Our findings lead us to the plane waves with extra symmetries. We revisit the
previously studied CPP GW [19] with our Hamiltonian approach. We provide the
Killing vectors of this plane wave and derived 2 additional charges originated
from the dilation and special conformal transformation of the PU oscillator. A
deeper understanding of this new charges e.g., their derivation from a Killing
tensor is reserved for a future work where we aim to understand the ED lift of
the conformal NH symmetry.

At Section 5, quite non-trivially we show that the underlying/transverse NR
dynamics of the Lukash plane wave [26] can be mapped to a PU oscillator when
it is of Bianchi type VI. One can explicitly write down the Carroll Killing
vectors of this Lukash plane wave. It would be interesting to extend our
parameter range so that we can include other Bianchi type Lukash solutions in
our framework. Section 6 is devoted to another plane wave example exhibiting
an extra conformal symmetry. Using our Hamiltonian approach, we find that the
transverse dynamics, with a suitable parameter choice, is related to the PU
oscillator.

Having completed our discussion of plane waves, we focus on NR dynamics. Our
first example is the motion of a charged ion in a $3+1$ dimensional Penning
trap. We find that this motion can be mapped to $1+1$ dimensional but the 6th
order PU oscillator. Quite nicely, the physical parameters defining the motion
inside the trap, namely the modified cyclotron frequency $\omega_{+}$, the
magnetron frequency $\omega_{-}$ and the axial frequency $\omega_{z}$ turn out
to be actual frequencies of the PU oscillator. The conserved charges of the
motion in a Penning trap are found via Carroll symmetry of the related $1+4$
dimensional plane wave.

It is known that the PU oscillator has different Hamiltonian descriptions. Our
framework has the advantage that its symplectic form can be easily modified to
obtain $2+1$ dimensional exotic dynamics (8.1). The non-commutative dynamics
emerge in condensed matter systems like Hall effect because of momentum
dependent semiclassical the Berry phases [32]. We show that our system can
still be mapped to a $1+1$ dimensional PU oscillator but with $\theta$
dependent frequencies. As a result, the NH symmetry of the PU oscillator
admits a second central extension. As a future project, we may search for
possible condensed matter realizations of the PU oscillator based on this new
approach. It would be also interesting to apply the conformal bridge
transformation [60, 61] to our system and derive the corresponding theory.
Concerning symmetries of the PU model and integrals of motion, another
perspective for further investigations has been opened by the recent studies
in optics and dark energy models [9, 10].

Lastly, we point out other examples like the motion under electromagnetic
fields created via double copy of the plane waves, Hill’s problem etc.. The
role of the Carroll symmetries for the motion under such electromagnetic
configurations has been pointed out in [33, 13]. On the other hand, Maxwell
equations in vacuum has a duality symmetry and the related conserved charge is
called the optical helicity. It would be interesting to understand the meaning
(if any) of the duality symmetry and its charge in the context of
gravitational waves. We also would like to find the relation between the gauge
fields derived from conformally related metrics like in the Lukash plane wave
case.

## Acknowledgements

We thank J. Balog, N. Dimakis, Jarah Evslin, Ilmar Gahramanov and his group,
P. Horvathy, P. Kosiński, M. Plyushchay, Dieter Van den Bleeken and Utku Zorba
for discussions and valuable comments. ME is currently supported by TÜBİTAK
under 2236-Co-Funded Brain Circulation Scheme2 (CoCirculation2) with project
number 121C356.

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