Nonlocal constants of motion in Lagrangian Dynamics of any order

TL;DR

This paper presents a method to generate nonlocal constants of motion for Lagrangian systems of any order, enabling analysis of dissipative and higher-order systems with potential extensions to PDEs.

Contribution

It introduces a general recipe for nonlocal constants of motion applicable to higher-order Euler-Lagrange systems, including dissipative and complex oscillators.

Findings

Derived nonlocal constants for dissipative systems.

Established first integrals for higher-order Euler-Lagrange equations.

Provided estimates and global existence results for solutions.

Abstract

We describe a recipe to generate "nonlocal" constants of motion for ODE Lagrangian systems. As a sample application, we recall a nonlocal constant of motion for dissipative mechanical systems, from which we can deduce global existence and estimates of solutions under fairly general assumptions. Then we review a generalization to Euler-Lagrange ODEs of order higher than two, leading to first integrals for the Pais-Uhlenbeck oscillator and other systems. Future developments may include adaptations of the theory to Euler-Lagrange PDEs.