Lambda-symmetries of Dynamical Systems, Hamiltonian and Lagrangian equations

TL;DR

This paper explores Lambda-symmetries in dynamical systems, introducing Lambda-constants of motion for Hamiltonian equations, and compares Lagrangian and Hamiltonian reduction methods, highlighting how Lambda-invariance transfers between frameworks.

Contribution

It introduces the concept of Lambda-constants of motion and demonstrates how Lambda-invariance in Lagrangian systems translates into Hamiltonian systems, providing a unified symmetry framework.

Findings

Lambda-symmetries define new constants of motion.

Lambda-invariance in Lagrangian systems induces Lambda-symmetry in Hamiltonian equations.

Comparison of Lagrangian and Hamiltonian reduction methods for Euler-Lagrange equations.

Abstract

After a brief survey of the definition and the properties of Lambda-symmetries in the general context of dynamical systems, the notion of "Lambda-constant of motion'' for Hamiltonian equations is introduced. If the Hamiltonian problem is derived from a Lambda-invariant Lagrangian, it is shown how the Lagrangian Lambda-invariance can be transferred into the Hamiltonian context and shown that the Hamiltonian equations turn out to be Lambda-symmetric. Finally, the "partial'' (Lagrangian) reduction of the Euler-Lagrange equations is compared with the reduction obtained for the corresponding Hamiltonian equations.