Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants

The status of classical stability in higher-derivative systems is still subject to discussions. In this note, we argue that, contrary to general belief, many higher-derivative systems are classically stable. The main tool to see this property are Nekhoroshev's estimates relying on the action-angle formulation of classical mechanics. The latter formulation can be reached provided the Hamiltonian is separable, which is the case for higher-derivative harmonic oscillators. The Pais-Uhlenbeck oscillators appear to be the only type of higher-derivative harmonic oscillator with stable classical dynamics. A wide class of interaction potentials can even be added that preserve classical stability. Adiabatic invariants are built in the case of a Pais-Uhlenbeck oscillator slowly changing in time; it is shown indeed that the dynamical stability is not jeopardised by the time-dependent perturbation.


Introduction
Variational principles based on action functionals of the form S[x] = L(x,ẋ) dt have a special status in the sense that Newtonian mechanics can be recovered from Lagrangians L(x,ẋ) depending only on position and velocity. Still, Lagrangians functions depending on higher time derivatives of the position, i.e, higher-derivative (HD) Lagrangians, are also worth of interest. Let us mention three areas in which HD models are encountered: 1. Explicit construction of classically stable and unstable HD dynamical systems: A class of classical HD harmonic oscillators was proposed in Sec. II of [1] and is still actively studied nowadays under the name of Pais-Uhlenbeck (P-U) oscillator, see e.g. Refs [2][3][4][5][6][7][8][9][10] and references therein for recent contributions to the field; 2. Renormalisability of HD field theories: In their pioneering work [1], Pais and Uhlenbeck addressed the issue of renormalisability in field theory through the inclusion of HD terms. HD gravities, like Weyl gravity, are promising renormalisable models of quantum gravity, see the seminal paper [11] and recent references in [12,13]. These HD models bring in the Einstein-Hilbert term upon radiative corrections, see e.g. [14]. They are also interesting in the context of cosmology and supergravity, see [15][16][17][18] and Refs. therein; 3. HD effective dynamics of voluntary human motions: The underlying dynamics of such motions is expected to involve HD variational principles such as minimal jerk, see e.g. Refs [19]. In this case, higher derivative terms may be thought of as a way to account for an intrinsic nonlocality (in time) of planified motion: Motor control may indeed add memory effects to standard Newtonian dynamics, which may be translated into a HD effective action.
A key feature of classical HD dynamics is that the energy has no definite sign, as it is readily observed from the general structure of HD Hamiltonians [20]. The presence of HD terms may lead to unbounded trajectories at the classical level -an explicit case is built in [2] -and to loss of unitarity at the quantum level [10]. However several cases are known for which classical trajectories are bounded and unitarity is preserved at the quantum level [10]. Having these recent results in mind, we think that providing a general method in order to assess classical stability of HD models is worth of interest. This is the main goal of the present paper, in which we focus on the case of HD harmonic oscillators. In the case of perturbed harmonic oscillators, an important body of works concerning their stability has been produced [21,22] that seems to have gone unnoticed by the HD community, so far. In a very specific case of a P-U oscillator with at most two time derivatives in the Lagrangian, Pagani et al. proved stability under a general class of cubic and quartic interactions [23].
The present paper is organised as follows. The Lagrangian (Sec. 2) and Hamiltonian (Sec. 3) formulations of HD harmonic oscillators are reviewed and the necessary conditions for their classical trajectories to be bounded are established. The dynamics is then formulated in terms of the action-angle coordinates in Sec. 4 and adiabatic invariants are computed. Finally, classical stability against time-dependent perturbations is discussed by using Nekhoroshev's estimates [24].

Lagrangian formulation
In this section we review the Lagrangian formulation of HD classical systems with finitely many degrees of freedom, in essentially the way that was presented long ago by Ostrogradsky [20]. We then review the Pais-Uhlenbeck parametrisation [1] of HD Lagrangians. 1/ 12

Generalities
Let L(x (0) , x (1) , . . . , x (N ) ) be a Lagrangian depending on the N first derivatives of the dynamical variables x(t) := x (0) (t), x (1) (t) :=ẋ(t) etc. possibly upon adding total derivatives in order to lower as much as possible the order of derivatives of x(t) . The action, evaluated between time t 1 and time t 2 , is S[x] = t2 t1 L dt . Hamilton's variational principle δS = 0 implies the equations of motion together with the vanishing of the boundary terms One chooses to cancel the above boundary terms by imposing the following conditions at the boundaries of the integration domain: It amounts to declaring that the initial data needed for solving the equation (1) is given by the values of x (0) , x (1) , . . . , x (N −2) and x (2N −1) at initial time t 1 . Indeed, provided one assumes the regularity condition the above initial data give a well-posed Cauchy problem for the ordinary differential equation (1).

Toy model
The simple HD harmonic oscillator with N ∈ N 0 and λ, β ∈ R + can be used to illustrate some features of HD dynamics. It reduces to the standard harmonic oscillator for N = 1, in which case λ is a mass-parameter. The case N = 2 has already been used as a toy model in [25].
The equation of motion (1) reads and its classical solution is given by A j e βj t with β j = β e i θj and θ j = π 2N + jπ N .
It can be observed from x(t) that the allowed trajectories go beyond a standard periodic motion since, up to an appropriate choice of A j , there may appear: • unbounded when Re β j > 0 . This occurs for j such that 0 < 2j + 1 < N and 3N < 2j + 1 4N − 1 ; • damped when Re β j < 0 . This occurs for j such that N < 2j + 1 < 3N ;
The standard case N = 1 is the only value for which all the possible trajectories are periodic. When N > 1, damping or "blowing-up" phenomena occur at time scales of order β cos(π/2N ) .

General case: Pais-Uhlenbeck oscillator
Lagrangian (3) and the corresponding equation of motion reads The characteristic polynomial of the above differential equation is with ω j ∈ R 0 and all the trajectories x(t) will be bounded. The further choice a 0 < 0 is such that a standard potential energy is recovered for N = 1 .
Therefore, replacing (x (j) ) 2 by (−) j x x (2j) up to total derivatives, Lagrangian (6) can be rewritten as Lagrangian (8) is nothing but the P-U oscillator [1], originally written under the equivalent form where the frequencies ω i are assumed to be real and distinct. It has been shown in [1] that equal or imaginary frequencies lead to unbounded trajectories so these cases will not be considered in the following.
The solution of the equation of motion related to (9) All classical trajectories are therefore bounded. Note that x(t) may describe the motion of a given mass in an N −body coupled harmonic oscillator whose normal modes have frequencies ω i : The formal analogy between the P-U oscillator and the dynamics of a N −body spring-mass system has been explored in the N = 2 case in Ref. [27].
An equivalent writing of (9) makes use of the oscillator variables and shows that (9) can formally be written as a Lagrangian describing N decoupled harmonic oscillators: 1 Adding degrees of freedom coupled to x is another way of addressing the problem, see. e.g. [26] 3/ 12 where it can be deduced from [1] that The signs of η i are alternating, which is a typical signature of HD dynamics, eventually leading to a total energy whose sign is undefined. The equation of motion for the Q i 's reads Generalisation of the P-U oscillator in more than one spatial dimension is straightforward and will be left for future works. New integrals of motion such as HD angular momenta are naturally expected: we refer the interested reader to [28] for explicit definitions.

Hamiltonian formalism
The Ostrogradsky construction that we have reviewed in Sec. 2.1 canonically leads to Hamiltonians that are not separable in the variables p i (2) and In this section we show in some particular cases that a suitable change of canonical variables allows to recast the P-U Hamiltonian under a separable form.

Ostrogradsky's approach
Details about singular HD Lagrangians can be found in [29]. The Hamiltonian function H is defined as The symplectic structure, as already apparent from the structure of the boundary terms in the variation δS in the equation above (2), is given by the two-form In particular, the Poisson-Ostrogradsky bracket between any two functions in the phase space T * Q locally coordinatised by the 2N variables (q i , p i ) i=0,...,N −1 is given by

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In particular, the symplectic structure (17) amounts to writing and the field equations are given bẏ The original equation of motion (1) is reproduced fromṗ 0 = {p 0 , H} , while all the other equations in (20) reproduce the relations (2) between the momenta-and the position-like variables.
Starting from Lagrangian (6) and applying Ostrogradsky's procedure one gets and the Hamiltonian reads The equations of motion (20) are given bẏ

Link with Pais-Uhlenbeck variables
The equivalence of Ostrogradsky and P-U formalisms at Hamiltonian level is not obvious. It was shown for N = 2 in [23]. Here we show it for the N = 3 case and then generalise our conclusions to arbitrary N .

The case N=3
A convenient parameterisation in this case is with Note that Lagrangian (24) can be recast under the form provided that

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The above constraints are fulfilled with The momenta associated with (24) read ..
x Ω 6 , and the Hamiltonian is given by The constants η i satisfy the relations If one denotes by X the 6-vector with components (q 0 , q 1 , q 2 , p 0 , p 1 , p 2 ) and introduces the 6 × 6 matrix the Hamiltonian can then be written as the quadratic form Let Z be the 6-vector with components (Q 0 , Q 1 , Q 2 , P 0 , P 1 , P 2 ) linked to X by the transformation Z = AX.
Then it can be checked that the symplectic matrix diagonalizes Λ and that Hamiltonian (33) eventually reads -recalling that the η i are of alternating sign: Let us observe that, from the expressions (29) for the Ostrogradsky momenta and the definition (26) of the P-U oscillator variables Q i , i = 0, 1, 2 , one has the relation

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Finally, a last canonical transformation given by the mere rescaling gives the desired form of the Hamiltonian One can easily derive a generating function for the canonical transformation from (q 0 , q 1 , q 2 , p 0 , p 1 , p 2 ) to (Q 0 , Q 1 , Q 2 , P 0 , P 1 , P 2 ) between the Ostrogradsky and P-U variables. It is given by It leads to the following system of partial differential equations: The first three equations are trivial, whereas the last three can be obtained from the relations (31) as well as In particular, it yields that are easily seen to be true from (34).

Arbitrary N
The N = 2 and N = 3 cases show that Ostrogradsky's procedure leads to P-U Hamiltonian and variables by successive canonical transformations. It can therefore be safely stated that Hamiltonian (22) can always be recast in a separable equivalent form. By introducing the positive quantities E i , i = 0, 1, 2 , the Hamiltonian reads once expressed in terms of the canonical coordinates It is clear that the frequencies ω i can be expressed as functions of the a i as in the N = 2 and N = 3 cases, but finding the explicit link at arbitrary N would lead to unnecessary long calculations. Indeed the form (44) can also be obtained from Lagrangian (11) in which only the ω i appear without resorting to Ostrogradsky's approach.

Action variables
The P-U Hamiltonian (44) is separable and admits elliptic trajectories in the planes (Q i , P i ) , i ∈ {0, 1, . . . , N − 1} , these fixed-E i cycles being denoted as Γ i . Hence a set of N action variables can be defined: The (−) j factor is such introduced in such a way that the action variables {I i } i=0,...,N −1 are all positive. It can be checked that the Hamiltonian (44) reads and that the relations holds as well. The action variable I 0 reduces to the average kinetic energy for N = 1 , as expected.
The action variables can be expressed in terms of the classical trajectory x(t) through the definition (45).
x cannot be expressed in terms ofẍ 2 with the use of a total derivative since x(t) is not a priori periodic with frequency ω 0 or ω 1 unless ω 1 /ω 0 = n/m ∈ Q . In the latter case, after a time T = m2π/ω 0 = n2π/ω 1 , the action variables can be recast under the form Γ0 ẋ 2 + cẍ 2 + d ... x 2 dt with c, d real coefficients. At the same time, the commensurability condition on ω 1 /ω 0 implies instability of the N = 2 dynamics under small perturbations so it is not relevant for the present study [23].
The I j are constant of motion provided that H does not explicitly depend on time. It is nevertheless possible that some external parameter is time-dependent: we set ω i = ω i (t) in (44). Under the assumption that the ω i (t) vary slowly enough with respect to the typical duration of a cycle -and despite the fact that no rigorous definition of "enough" can be given [30] -, the quantities I j given by (47) are adiabatic invariants. It can be deduced from [31] that their small rate of change is given by: where φ k are the angle variables conjugated to I k : {φ j , I k } = δ jk . The interested reader may find general computations related to time-varying harmonic oscillators in [32].
As an illustration of the above relations, suppose that the time-dependent parameter is a small perturbation of the frequencies: the only dependence on time being contained in the real functions g i . The smallness ofġ k has to be assumed for adiabatic invariants; therefore, at the lowest order in , Eqs. (51) becomė The solution of (53) at order is given by

Nekhoroshev estimates
The P-U Hamiltonian (48) with the time-dependent perturbation (52) can be formally written under the form where and where the vectors ( I) k = I k , ( φ) k = φ k , ( g) k = g k and ( ) k = (−) k k have been introduced. Nekhoroshev's theorem [24] states that if the nearly integrable Hamiltonian (55) is analytic and the unperturbed part h( I) is steep (or convex, or quasiconvex) on some domain, then there is a threshold 0 > 0 and positive constants R, T , a and b such that whenever | | < 0 , for all initial actions variables I(0) in the domain (and far enough from the boundary), one has This result has been particularized to several explicit examples, among which the harmonic oscillator in Refs. [21,22]. The fact that some components of are negative is allowed by the formalism of the latter references.
From the study [21] in particular, it can be deduced that unstable behaviours in the P-U oscillator appears at exponentially large time in , i.e. the dynamics is classically stable. Since 0 ∼ N −2N [21], the more the dynamics contains HD, the more the perturbation must be small to preserve stability. Moreover, there must exist two real positive constants σ, τ such that | · n| ≥ σ N −1 j=0 |n j | −τ for all n ∈ Z N 0 otherwise 0 becomes arbitrarily large and the system is unstable. In other words the frequencies must define a non resonant harmonic oscillator. Such an instability can only occur for N > 1 one-dimensional dynamics; it is trivially avoided when N = 1 because no energy transfer between the different components E i are de facto absent in this case.

Concluding comments
Higher-derivative action principles generally lead to unstable classical dynamics. However, all the classical trajectories allowed by the Pais-Uhlenbeck oscillator (9) with distinct and nonresonant frequencies are bounded: It is an explicit realisation of a stable classical theory with higher-derivatives. Therefore the problem can be formulated in action-angle variables formalism, allowing a computation of adiabatic invariants and a proof of the classical stability based on Nekhoroshev estimates. Although the Pais-Uhlenbeck oscillator has been widely studied as a prototypal higher-derivative physical theory, it is the first time, to our knowledge, that such results are obtained. Emphasis has been put on harmonic potentials in the present study. Other types of potentials 9/ 12 or higher-derivative terms may also lead to stable classical dynamics. For example it is shown in Ref. [23] that a N = 2 Pais-Uhlenbeck oscillator with cubic and quartic potential terms is stable too, except for very particular values of the parameters. The stable models of [23] should give, after generalization to field theory Lagrangians, further examples (compared to the one reviewed in [10]) of well-behaved dynamical systems with infinite number of degrees of freedom.
It is worth making comments about quantization. In a first approximation a Bohr-Sommerfeld quantization rule can be applied since action variables exist. The fact that the energy spectrum is unbounded both from below and above in higher-derivative theories does not a priori forbids well-behaved quantum dynamics. A quantization technique was proposed in [6] that aims at keeping the higher-derivative dynamics stable at the quantum level. In fact, we propose that the positive-definite quantities suggested in [6] and that are responsible for a stable dynamics at the quantum level are nothing but the action variables. Indeed, even for a perturbed classical motion, as long as the trajectories are bounded, the action variables are positive-definite quantities and conserved to the approximation given. More recently, it was conjectured in [10] that indeed, when all the classical trajectories of a given higher-derivative model are bounded, its quantum dynamics only contain so-called benign ghosts, i.e. negative-energy quantum states with a normalisable wave function and preserved unitarity of the evolution. The present work aimed at clarifying the conditions for higher-derivative models to exhibit bounded classical dynamics; hence it is a step toward the identification of quantum models with unitary quantum dynamics, to which the Pais-Uhlenbeck oscillator belongs. Further issues about quantization of higher-derivative Lagrangians are discussed for example in [33].
Finally it has to be noticed that the necessary conditions for adiabatic invariants and Nekhoroshev estimates to be computed are the separability of the higher-derivative Hamiltonian and the existence of bounded classical trajectories. Both conditions are met in the Pais-Uhlenbeck oscillator case after appropriate choice of canonical variables, but we believe that other classes of higher-derivative systems may be studied by resorting to the methods we have presented.