Introduction to higher-order classical dynamics: Pais–Uhlenbeck model and coupled oscillators

Most of the laws of nature involve derivatives up to the second order. Ostrogradsky 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–Ostrogradsky formulation served as the basis for later studies with higher-order derivatives. However, the Hamilton–Ostrogradsky formalism is rarely discussed in textbooks or the pedagogical literature. This motivated us to show how the Hamilton–Ostrogradsky formalism can be applied 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.