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arXiv:2605.20094

Introduction to Higher Order Classical Dynamics: Pais-Uhlenbeck Model and Coupled Oscillators

Cássius Anderson Miquele de Melo, Ivan Francisco de Souza

physics.ed-phhep-thmath-phphysics.class-phquant-phSubmitted May 19, 2026• Announced May 20, 2026

Abstract

Most of the laws of Nature involve derivatives up to the second order. Ostrogradski was the first to seek a formulation of the equations of higher-order derivatives. He extended Hamilton's equations by considering Lagrangians that depend on higher-order derivatives of generalized coordinates. The Hamilton-Ostrogradski formulation served as the basis for later studies with higher-order derivatives. However, the Hamilton-Ostrogradski formalism is rarely discussed in textbooks or the pedagogical literature. This motivated us to show how the Hamilton-Ostrogradski formalism can be applied it to the Pais-Uhlenbeck oscillator. We hope that the approach presented in this work can serve as a basis for discussion in advanced classical mechanics courses.

DOI: 10.1119/5.0284311
Comments: Version with expanded references

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Most of the laws of Nature involve derivatives up to the second order. Ostrogradski was the first to seek a formulation of the equations of higher-order derivatives. He extended Hamilton's equations by considering Lagrangians that depend on higher-order derivatives of generalized coordinates. The Hamilton-Ostrogradski formulation served as the basis for later studies with higher-order derivatives. However, the Hamilton-Ostrogradski formalism is rarely discussed in textbooks or the pedagogical literature. This motivated us to show how the Hamilton-Ostrogradski formalism can be applied it to the Pais-Uhlenbeck oscillator. We hope that the approach presented in this work can serve as a basis for discussion in advanced classical mechanics courses.

This preprint presents a stability-based approach to higher-derivative dynamics in non-inertial reference frames. By imposing the non-negativity of the second variation of the action, a dynamically admissible subset of trajectories is selected within the Ostrogradsky formalism. The resulting extended Hamilton–Jacobi structure naturally contains an additional term that coincides with the Bohm quantum potential in the Madelung representation of the Schrödinger equation. The work is related to the author’s contribution to the BSM 2025 conference and is shared here to ensure early dissemination of the results.

We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of uncoupled oscillators. This motivates us to propose a generalized Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation and annihilation operators by including terms quadratic and linear in them. Such a Hamiltonian turns out to be essentially non-Hermitian but has an equivalent Hermitian representation which is reducible to a typically position-dependent reduced mass form.

In this paper, we apply the covariant phase space formalism to the perturbative canonical quantization of the Pais-Uhlenbeck oscillator, with the acceleration-squared term treated as a perturbation. We quantize this model by constructing the symplectic form on the low-energy solution space. We then compute the energy spectrum and the unequal-time commutator in a perturbative way, and obtain the results that agree with the expansion of the exact low-energy theory. The perturbation method bypasses the standard Ostrogradsky construction and naturally decouples the Ostrogradsky ghost. This work extends our previous perturbative quantization scheme to genuine higher-derivative theories.

A generalized canonical form of action of dynamic theories with higher derivatives is proposed, which does not require the introduction of additional dynamic variables. This form is the initial point for the construction of quantum theory, in which the state of motion of the system is described by the wave functional on its trajectories in the configuration space, and the functional itself is an eigenvector of the action operator. The Pais-Uhlenbeck oscillator is considered as the simplest model. For this case, the correspondence between the new form of quantum theory and "ordinary" quantum mechanics has been established in the local limit, when the Pais-Uhlenbeck oscillator is reduced to the "ordinary" harmonic oscillator.