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

TL;DR

This paper introduces the Hamilton-Ostrogradski formalism and applies it to the Pais-Uhlenbeck oscillator to enhance understanding of higher-order classical dynamics.

Contribution

It demonstrates how to apply the Hamilton-Ostrogradski formalism to higher-order derivatives, specifically the Pais-Uhlenbeck model, for educational purposes.

Findings

Hamilton-Ostrogradski formalism can be effectively applied to higher-order systems.

The approach aids in teaching advanced classical mechanics.

Provides a pedagogical framework for higher-order derivative systems.

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.