Higherderivative harmonic oscillators: stability of classical dynamics and adiabatic invariants
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Abstract
The status of classical stability in higherderivative systems is still subject to discussions. In this note, we argue that, contrary to general belief, many higherderivative systems are classically stable. The main tool to see this property are Nekhoroshev’s estimates relying on the actionangle formulation of classical mechanics. The latter formulation can be reached provided the Hamiltonian is separable, which is the case for higherderivative harmonic oscillators. The Pais–Uhlenbeck oscillators appear to be the only type of higherderivative 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 timedependent perturbation.
1 Introduction
 1.
Explicit construction of classically stable and unstable HD dynamical systems: A class of classical HD harmonic oscillators was proposed in Sect. 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, 11, 12, 13] and references therein for recent contributions to the field;
 2.
Renormalizability of HD field theories: In their pioneering work [1], Pais and Uhlenbeck addressed the issue of renormalizability 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 [14, 15] and recent references in [16, 17]. These HD models bring in the Einstein–Hilbert term upon radiative corrections, see e.g. [18]. They are also interesting in the context of cosmology and supergravity, see [19, 20, 21, 22] 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. [23, 24, 25, 26, 27]. In this case, higher derivative terms may be thought of as a way to account for an intrinsic nonlocality (in time) of planified motion: Human motor control may indeed add memory effects to standard Newtonian dynamics, which may be translated into a HD effective action.
The present paper is organised as follows. The Lagrangian (Sect. 2) and Hamiltonian (Sect. 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 actionangle coordinates in Sect. 4 and adiabatic invariants are computed. Finally, classical stability against timedependent perturbations is discussed by using Nekhoroshev’s estimates [32, 33].
2 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 Ostrogradki [28]. We then review the Pais–Uhlenbeck parametrisation [1] of HD Lagrangians.
2.1 Generalities
2.2 Toy model

unbounded when \(\mathrm{Re}\, \beta _j >0\,\). This occurs for j such that \(0<2j+1<N\) and \(3N<2j+1\leqslant 4N1\,\);

damped when \(\mathrm{Re}\, \beta _j<0\,\). This occurs for j such that \(N<2j+1< 3N\,\);

periodic with period \(2\pi /\beta \) if \(\mathrm{Re}\, \beta _j=0\): This can happen for N odd and the two values \(j=(N1)/2\) and \(j=(3N1)/2\,\).
2.3 General case: Pais–Uhlenbeck oscillator
The solution of the equation of motion related to (10), \(F_N\left( \frac{d}{dt}\right) x=0\,\), reads \(x(t)=\sum ^N_{k=1} A_k\, \sin (\omega _k\, t+\varphi _k)\) with \(A_k, \varphi _k\in {\mathbb {R}}\,\). 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 \(\omega _i\): The formal analogy between the P–U oscillator and the dynamics of an Nbody springmass system has been explored in the \(N=2\) case in Ref. [36].
3 Hamiltonian formalism
3.1 Ostrogradski’s approach
3.2 Link with Pais–Uhlenbeck variables
The equivalence of Ostrogradski and P–U formalisms at Hamiltonian level is not obvious in the sense that the canonical transformation between the Ostrogradski and the P–U variables has not been explicitly given in the general case, as far as we could see. It was shown for \(N=2\) in [31]. Here we show it for the \(N=3\) case and leave the explicit form of the canonical transformation relating the Ostrogradski to the P–U variables for future works. Below we simply give the general expression for the generating function of the canonical transformation for arbitrary \(N\,\), without solving the partial differential equations that explicitly relate the two sets of phase space variables.
3.2.1 The case N = 3
3.2.2 Arbitrary N
4 Adiabatic invariants and Nekhoroshev estimates
4.1 Action variables
4.2 Nekhoroshev estimates
5 Concluding comments
Higherderivative action principles generally lead to unstable classical dynamics. However, all the classical trajectories allowed by the Pais–Uhlenbeck oscillator (10) with distinct and nonresonant frequencies are bounded: it is an explicit realisation of a stable classical theory with higherderivatives. Therefore the problem can be formulated in actionangle 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 higherderivative 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 or higherderivative terms may also lead to stable classical dynamics. For example it is shown in Ref. [31] that a \(N=2\) Pais–Uhlenbeck oscillator with cubic and quartic potential terms is stable too, except for very special values of the parameters. The stable models of [31] should give, after generalisation to field theory Lagrangians, further examples (compared to the one reviewed in [10]) of wellbehaved dynamical systems with infinite number of degrees of freedom.
One can recover a Lagrangian function from a Hamiltonian function provided that the regularity condition det \(\frac{\partial ^2 H}{\partial p_{N1}\partial p_{N1}}\ne 0\,\) is satisfied – which is the case for a free system that is regular when the added interactions in the perturbed Hamiltonian are algebraic functions in the P–U variables Q’s, as we assumed implicitly here. This is clear for N odd since the latter variables do not imply Ostrogradski’s last momentum \(p_{N1}\,\). In the even N cases where the Q variables involve \(p_{N1}\,\), the added perturbations cannot ruin the regularity property of the unperturbed P–U Hamiltonian as long as the perturbation is of polynomiality degree in Q higher than two, as we assume here. However, it is notoriously difficult (see e.g. Section 8 of [10]) to find the analytical expression of the perturbed Lagrangian corresponding to a given Hamiltonian perturbation. It it usually nonlocal, even for simple (e.g. quartic) Hamiltonian perturbation [10], which suggests considering as a starting point the \(N\rightarrow \infty \) limit studied by Pais and Uhlenbeck, as it would encompass all the possible Hamiltonian perturbations.
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 higherderivative theories does not a priori forbids wellbehaved quantum dynamics. In fact, a quantization technique was proposed in [6] that keeps the higherderivative dynamics stable at the quantum level. In fact, we propose that the positivedefinite 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 perturbatively perturbed classical motion, as long as the trajectories are bounded, the action variables are positivedefinite quantities and conserved to the approximation given. More recently, it was conjectured in [10] that indeed, when all the classical trajectories of a given higherderivative model are bounded, its quantum dynamics only contains socalled benign ghosts, i.e. negativeenergy quantum states with a normalisable wave function and preserved unitarity of the evolution. The present work aimed at clarifying the conditions for higherderivative 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 higherderivative Lagrangians are discussed for example in [48, 49, 50].
Finally it has to be noticed that the necessary conditions for adiabatic invariants and Nekhoroshev estimates to be computed are the separability of the higherderivative 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 higherderivative systems may be studied by resorting to the methods we have presented.
Footnotes
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