Jacobi multipliers, non-local symmetries and nonlinear oscillators

TL;DR

This paper explores the use of geometric formalism and jet bundle techniques to identify constants of motion, Lagrangians, and non-local symmetries in nonlinear oscillators, extending existing theories and proving new symmetry results.

Contribution

It introduces a novel application of the Jacobi last multiplier and jet bundle formalism to find non-local symmetries and constants of motion in nonlinear oscillators.

Findings

Constants of motion and Lagrangians derived for nonlinear oscillators

Non-local symmetries identified using extended formalism

Existence of non-local symmetries proved for specific oscillators

Abstract

Constants of motion, Lagrangians and Hamiltonians admitted by a family of relevant nonlinear oscillators are derived using a geometric formalism. The theory of the Jacobi last multiplier allows us to find Lagrangian descriptions and constants of the motion. An application of the jet bundle formulation of symmetries of differential equations is presented in the second part of the paper. After a short review of the general formalism, the particular case of non-local symmetries is studied in detail by making use of an extended formalism. The theory is related to some results previously obtained by Krasil'shchi, Vinogradov and coworkers. Finally the existence of non-local symmetries for such two nonlinear oscillators is proved.