Lagrangian dynamics by nonlocal constants of motion

TL;DR

This paper introduces a general theorem to generate nonlocal constants of motion for Lagrangian systems, demonstrating its usefulness across various physical models and proving a new explosion criterion for certain mechanical systems.

Contribution

It presents a novel theorem for deriving nonlocal constants of motion and applies it to multiple physical systems, including a new result on explosion in mechanical systems with hydraulic resistance.

Findings

Constants useful in systems with homogeneous potentials of degree -2

Application to Maxwell-Bloch equations in laser dynamics

New explosion criterion for hydraulic fluid resistance systems

Abstract

A simple general theorem is used as a tool that generates nonlocal constants of motion for Lagrangian systems. We review some cases where the constants that we find are useful in the study of the systems: the homogeneous potentials of degree~$-2$, the mechanical systems with viscous fluid resistance and the conservative and dissipative Maxwell-Bloch equations of laser dynamics. We also prove a new result on explosion in the past for mechanical system with hydraulic (quadratic) fluid resistance and bounded potential.