Jacobi multipliers and Hojman symmetry

TL;DR

This paper develops a geometric approach to Hojman symmetry using Jacobi last multipliers, enabling the identification of conserved quantities in various dynamical systems, including nonautonomous cases.

Contribution

It introduces a novel geometric framework combining Jacobi multipliers with Hojman symmetry, extending analysis to non divergence-free and nonautonomous systems.

Findings

Derived conserved quantities for non divergence-free vector fields.

Extended Hojman symmetry to nonautonomous systems.

Applied the framework to autonomous Lagrangian and Hamiltonian systems.

Abstract

The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed.