Various disguises of the Pais-Uhlenbeck oscillator

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.


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 One of the crucial observations [1] is that the Hamiltonian of the 2nth order PU oscillator can be written as a combination of n decoupled harmonic oscillators with an alternating sign 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 momenta 1 and the associated Lagrangian begins with a square of nth 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 4th 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 4th order PU oscillator span 2 copies of the Heisenberg algebra for generic λ i s [14]. In the case of odd frequencies the algebra becomes richer: It is the ℓ = 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 Here, for the first time, we will relate those transverse motions in the Lukash and the complete plane waves to a 1 + 1 dimensional 4th 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 6th order PU oscillator. The physical parameters of the Penning trap, namely modified cyclotron, magnetron and axial frequencies turn out be frequencies λ 1 , λ 2 , λ 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 θ, 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 4th order 1 + 1 dimensional PU oscillator with θ-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.
where M is a non-zero parameter. The corresponding Hamiltonian is of the form Note that by the replacement H → −H, ω → −ω we can restrict to M > 0; moreover using the coordinate change x ↔ y we may choose ω > 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 { ′ } is w.r.t. to NR time. We assume no prior relations between ω and Ω ± so that our findings can be applied to various cases. Note that without the Coriolis force i.e., ω = 0, the motions will be uncoupled and they will be described by hyperbolic functions 2 . It is this Coriolis force or magnetic field that makes trapping possible (within a suitable parameter range for ω, Ω ± ). 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 Comparing eq. (2.3) with the equation of motion following from (1.1) (for n = 2) we conclude that to obtain the PU model λ 1,2 should satisfy the following conditions In the particular case of ω 2 = 2(Ω 2 + + Ω 2 − ) the equation (2.3) automatically takes the PU form [19]. These equations are symmetric under λ 1 ↔ λ 2 ; for λ 2 2 we found the algebraic relation that yields two roots We make the choice However, we should consider only real and positive λ 2 's, thus some additional relations on Ω 2 ± and ω are needed. First, In view of this for ω > Ω + + Ω − we obtain real and positive λ 2 ′ 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 4th 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 Ω − , is vanishing, the solution (2.8) again works with λ 2 = 0 and λ 2 1 = ω 2 − Ω 2 + . This case will be separately handled at section 9.2 when we discuss Hill's equations [34]. Let us also mention that when ∆ < 0, λ 2 1,2 are complex conjugate to each other. 2 It is not excluded that those motions can be described by trigonometric functions when Ω 2 ± < 0. In that case one should be careful about reality/hermiticity of the related Hamiltonian. Here we consider the case Ω± > 0.

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 Π i = (Π x , Π y ) as and write the associated symplectic structure in terms of that non-canonical coordinates (x, y, Π x , Π y ) 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, Π x , Π y ) with the new ones (X 1,2 and solve for the following equations Omitting details, below we present our solutions for the constants β ± , α ± in terms of diagonalization frequencies (the PU frequencies) λ 1,2 (2.8) as Substituting them into eqs. (2.10), we obtain a symplectic structure which is decomposed into + and − sectors Let us remind that in the above formulae, we assume that λ 2 i 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 λ 1,2 (2.8) 3 When we pass to Darboux coordinates by means of the following simple redefinition we see that the Hamiltonian consists of two harmonic oscillators, surprisingly with a relative minus sign between them: , (2.18a) To sum up, the Hamiltonian (2.2), for the case of minus sign, can lead to the PU oscillator with λ 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.

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.

ED lift and symmetries
When the NR system (2.10) is ED lifted, we get the following plane-wave metric in rotating Brinkmann coordinates x µ = (x, y, u, v). Conversely, we can use the null geodesic condition 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 λ 1,2 as 4 has an extra isometry which is a u-translation ∂ 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 symmetry 5 . 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 6th order PU oscillator.
5 Extra isometry ∂u corresponds to the conserved Hamiltonian of the PU oscillator (2.19) after null projection [19].
The trace of the 2 × 2 K matrix is Tr(K) = Ω 2 + + Ω 2 − − ω 2 2 . When 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 ω, Ω ± 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 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.

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 metric 6 in Brinkmann coordinates (X i , U, V ) which includes both (3.1) and (3.6) as special cases (ǫ is the antisymmetric matrix with off-diagonal entries ±1). The Lagrangian of a relativistic particle in an electromagnetic potential A µ is of the form where dot refers to the derivative w.r.t. an arbitrary parameter τ and we adopt the conventions (−, +, +, +).
In our further considerations we will restrict to the form 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 τ = τ (X µ ). Let us fix the gauge demanding τ = τ (U ) = U . At the Lagrangian level this leads to where prime stands for the derivative w.r.t. U . Then one of e.q.m.
where P V is the canonical momentum conjugated to V coordinate; moreover, we see that τ = U becomes the affine parametrization. Now, we can form the Hamiltonian formalism. The phase space obtained is six-dimensional, (V, P V ) and (X, P) with the canonical Poisson brackets, and the Hamiltonian, for positive P V , is of the following form 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 (X, 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) satisfy the condition 2(−P U + eA + K 2 P V )P V − (P + P V ωǫX/2) 2 = m 2 ; moreover, sign(P V ) = sign(U ) . In consequence, we arrive at the constraint (3.14) Due to the above constraint we can only express the velocities V,Ẋ in terms of momenta P V , P (there remains not specified functionU ). Namely, we obtaiṅ and U is an arbitrary function of τ . In view of the above the Hamiltonian thus φ is a first class constraint. We break the reparametrisation symmetry by adding a new constraint, Then {ψ, φ} = 1 and thus ψ and φ belong to the second class constraints. Moreover, demanding Unfortunately the constraint ψ depends on τ 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 variableŨ = U − τ (P µ = P µ ,Ṽ = V ,X = X). The corresponding generating function W is of the form Then ψ = 0 is equivalent toψ =Ũ = 0 (τ independent) and the Hamiltonian takes the form (with U replaced byŨ + τ ). Now, we are in the position to perform the standard Dirac procedure i.e. restrict dynamics to the constraint surface (eliminatingŨ andPŨ = P U variables). Then the Hamiltonian (3.20) coincides with (3.12); moreover, the Dirac bracket between X, V and the corresponding canonical momenta take the standard form. In the case of negative P V , after redefinition of τ → −τ , the HamiltonianH 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 = 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,H = −P U (not P U , despite the fact that U takes the role of time). Finally, we can skip the constant term m 2 2P V (using the canonical transformation generated by W = − 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 ).

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, X) = K ij (U )X i X j (ω = 0 and A µ = 0), there is an alternative coordinate system called the BJR coordinates in which the metric takes the form ds 2 = a ij (u)dα i dα 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 ω = 0). This will allow us to describe the PU model in the BJR framework. At the level of metrics, the following coordinate transformations The BJR profile a ij is a 2 × 2 symmetric matrix depending only on the coordinate u. Then, Killing vectors of (3.21) can be found easily as cf. (3.3). In these coordinates, s and α 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) = u a ij dũ is the Souriau's matrix with a ij a jk = δ i k . The BJR metric (3.21) is also a Bargmann manifold i.e., it is endowed with a covariantly constant null Killing vector ∂ v . Therefore, by a null reduction the Hamiltonian in BJR coordinates will be of the form moreover, we have p s = P V . First, let us note that we should have where the generating function F is F = ps 4 α i α j a ′ ij ; this can be confirmed using (3.22). Rewriting (3.26) as where k i = p s a ij (α j ) ′ are the conserved canonical momenta. Next choosing W as F = W (X i , k j , u) − k i α i , we obtain the canonical transformation of the associated Hamiltonians augmented with the definitions 29) and the generating function

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 ℓ = 3 2 conformal NH group.

Relation to PU oscillator
The Brinkmann profile (3.6) of CPP GW is with amplitude b > 0 and frequency ω. It is a vacuum solution of the Einstein equations. Let us consider the U -dependent canonical transformation to the new coordinates x, p generated by the function: where R(U ) is given by (3.5). Then the Hamiltonian (3.20) for the profile K = K (1) transforms into the following one so it takes the form of (2.2) with minus sign after identification M = P V and Thus for parameters ω 2 > 4b the frequencies λ 2 1,2 (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, ω 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 5) and the Souriau matrix is .
(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

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 Ω 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 where Y c i and Y b i denote the Carroll translations and Carroll boosts, respectively. Their commutators with ∂ u are found as According to Noether's theorem, the substitution of basis vectors ∂ µ in (4.8) with the associated canonical momenta p µ 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 c 1 yields a conserved charge with explicit time-dependence 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]. ℓ = 3 2 NH symmetry: On the other hand, it is known that the PU oscillator enjoys ℓ = 3 2 conformal NH symmetry provided that Ω + = 3Ω − = 3Ω [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 with ω 2 = 20Ω 2 . A straightforward calculation leads extra charges We complete it with the explicit expression of H geo (3.2) and canonical momenta p x =ẋ − √ 5Ωyu, p y =ẏ + √ 5Ωxu. 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.

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 The coordinates are well defined either for U > or U < 0 but singular at U = 0. One can consider a nonnegative wave amplitude, i.e., C ≥ 0. Parameter κ 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 In addition to 5-parameter Carroll isometry and the homothety Thus, Lukash plane-wave is maximally symmetric. In order to bring the Lukash metric (5.2) to the form (3.1), Y κ leads us to do the following transformations [27] U = e T , X i = e T /2 ξ i , V = V, ξ i = (ξ, η).

(5.4)
In the new coordinates (ξ, η, T, V ) Lukash metric g L (5.2) is conformally related to another metric g as g L = e T g (5.5) and the extra isometry Y κ (5.3) turns to be a combination of a T translation, a rotation and homothety: 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 κ (5.7) of g L leads a screw isometry for the conformally related metric g . We can even redefine 7 V → V − 1 4 ξ i ξ i which puts g (5.6) into standard Brinkmann coordinates (3.6) as 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 Then, renaming T = u, V = v we end up with rotating Brinkmann coordinates as where 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 where { ′ } denotes derivation w.r.t. u. These equations are exactly in the form of (1.3). Next, we compute the diagonalization frequencies (2.8) For κ > 1 2 and κ < C < κ 2 + 1/4 the quantities Ω 2 ± and λ 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 where the right hand side is expressed in terms of the tilde coordinates. In our case the Hamiltonian (3.20) transforms into the following oneH (5.17) where K (1) given by (4.1) with ω = −2κ. Next, performing the canonical transformation generated by (4.2) we arrive at the Hamiltonian H Now, the above Hamiltonian takes the form of (2.10) ((2.2) with minus sign) with Ω ± given by (5.12). The related NR symplectic structure (2.10) can be written as (5.20) with Π x = p x − κy, Π y = p y + κx 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 and finally arrive at the PU Hamiltonian with frequencies (5.14).

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 ω = 0 and the profile [28] K (2) (U, X) = a (U 2 + ε 2 ) 2 X T cos(φ(U )) sin(φ(U )) sin(φ(U )) − cos(φ(U )) X, and we assume a, ε, γ > 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 (X,p,ũ) = 1 εp T X cos(ũ) + P V 2ε X 2 sin(ũ) cos(ũ), (6.3) together with the time change U = ε tan(ũ). Then the new Hamiltonian readsH(x,p,ũ) = H dU dũ + ∂F ∂ũ . After direct calculations we find that the Hamiltonian (3.20) (with K = K (2) ) transforms intõ withM = εP V and K (1) is defined by the parameters b = a/ε 2 , ω = 2ω 0 = 2γ/ε. Thus, using the Rtransformation we can perform the canonical transformation generated by (4.2) to the new coordinates x, p (the termx Tx takes the same form). At the end we arrive at the Hamiltonian The above Hamiltonian is of the form (2.10) and thus is separable. Namely, for P V > 0 and 1 − ω 2 0 < −a/ε 2 we obtain, after direct computations, the PU Hamiltonian with the frequencies where ∆ = 4ω 2 0 + a 2 ε 4 . Of course, we may ask what happens for other values of parameters. One can check that for |1−ω 2 0 | < a/ε 2 we obtain the sum of the inverted oscillator and ordinary one, while for 1 − ω 2 0 > a/ε 2 we obtain the sum of two oscillators. The case P V < 0 can be analysed by the change H → −H and ω 0 → −ω 0 . Finally, one can easily find suitable values of parameters for which the odd frequencies and additional integrals of motion appear.

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 B = Bẑ and an anisotropic but time independent quadrupole potential V (x, y, z) = V 0 thus they satisfy ω + + ω − = ω c and ω + − ω − = ω 2 c − 2ω 2 z . We note here an interesting fact that the condition for the system to be a PU oscillator (7.6), i.e., ω 2 c = ω 2 + + ω 2 − + ω 2 z , 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 In those coordinates, both the Hamiltonian and symplectic form for the Penning trap decompose A final transformation to canonical coordinates provides us with the following Hamiltonian of 3 oscillators with alternating sign 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 ω ± , ω 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 ω = ω c and Ω 2 + = Ω 2 − = ω 2 z 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 ∂ v ) via modifying (3.3) as (ω + − ω − )(x cos ω + u − y sin ω + u) + p x sin ω + u + p y cos ω + u, (7.11b) They satisfy dQ du = {Q, H P } + ∂Q ∂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 ℓ = 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 ℓ = 5 2 conformal NH algebra or not. To this end, we need to impose the odd frequency condition [15] on the Penning trap frequencies ω ± , ω z . As it is mentioned, in a typical Penning trap those frequencies follow the order ω c > ω + ≫ ω z ≫ ω − . This physical condition forces us to set However, the above choice contradicts (7.6). In a Penning trap, it is also possible that the magnetron frequency ω − can be larger than ω 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 ω − and ω 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 ℓ = 5 2 conformal NH algebra.

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 while keeping Hamiltonian the same. θ is a constant non-commutativity parameter. Poisson brackets can be easily obtained from the inverse of the modified symplectic matrix σ⋆ (8.1a) as where x and y are no more commuting. θ also modifies Hamilton's equations as Poisson brackets (8.2) are identical to the ones in the non-commutative Landau problem [53]. When θ 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 cf. (2.12). Assuming no relation between parameters ω, Ω ± as before , we may solve for θ-dependent coefficients α ± , β ± 8 . However, the expressions become rather long in that case. Thus we continue with symbols α ± , β ± without substituting their actual values in terms of ω, Ω ± .
Solutions of (8.4) satisfy the following identities For instance, we find cf. (2.13). In order to have simpler expressions, one may choose a specific parametrization such as ω 2 = 2(Ω 2 + + Ω 2 − ) [19]. Then the remaining coefficients can be easily found. and they decompose the symplectic structure (8.1) as Both the symplectic structure (8.6) and diagonalization frequencies (8.7) go back to the previous expressions (2.14) when θ = 0. Solving Hamilton's equations, we obtain Associated equations of motion are Thus, we may pass to the Darboux coordinates via   as in (4.11). Their Poisson brackets read (8.14) and span the exotic NH algebra with two central extensions [50]. Obviously, there is a solution for the system (8.4) with α + = α − that necessarily yields β − = β + . This case is excluded above as it leads to a vanishing symplectic structure (8.6). We emphasize that θ is a small parameter at hand and it is possible to find realizations such that β − > β + and α − > α + 9 . When θ = 0, the usual NH symmetry with one central extension M is recovered. Above results are in line with [19] where θ = 0 and where parameters obey λ 1 = Ω + , λ 2 = Ω − , ω 2 = 2(Ω 2 + + Ω 2 − ), see section 4.

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

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 One can easily see that so the generated electromagnetic fields are singular 10 . Maxwell's equations are where ∇ 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 = η (the Minkowski metric) and ω = 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 Comparing it with (3.12) we see that the above Hamiltonian is equal to the one in the gravitational case with the replacement K → 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 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 → 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 ω. 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 ω 2 → 2eω 2 P V and Ω 2 ± → 2eΩ 2 ± 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.

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 and they correspond to "zero frequency condition": one of the frequencies, say Ω − = 0. Diagonalization frequencies (2.8) can be found easily Solving for (2.12), we get 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 + 2ω(xdy − ydx) + 3ω 2 x 2 du 2 , (9.9) which is a Bargmann manifold 11 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 cos ωu∂ x + 4 ω sin ωu∂ y + 2(x sin ωu − y cos ωu)∂ 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. 11 Comparing with Lukash example (5.11), we see that this is a particular case of it with x ↔ β, y ↔ α, κ = ω, and a simple calculation yields

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 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] 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.

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), where γ is a constant. Depending on the sign of it, one can assign Ω ± and calculate λ 1,2 from (2.8) with λ 2 1 > λ 2 2 . Then, it is easy to observe that the diagonalization frequencies happen to be the same as the original frequencies λ 2 1,2 = Ω 2 ± = (1/4 ∓ γ)ω 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.

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 Ω ± and λ 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., λ 1 = λ 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 derivative 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 ω + , the magnetron frequency ω − and the axial frequency ω 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 θ 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.