A Summary on Symmetries and Conserved Quantities of Autonomous Hamiltonian Systems,

TL;DR

This paper provides a comprehensive geometric classification of symmetries in autonomous Hamiltonian systems, detailing how to derive their conserved quantities, including new types of non-symplectic symmetries.

Contribution

It introduces new classes of non-symplectic symmetries and explains their associated conserved quantities within a complete geometric framework.

Findings

Classification of symmetries in Hamiltonian systems

Introduction of non-symplectic symmetries

Method to obtain conserved quantities

Abstract

A complete geometric classification of symmetries of autonomous Hamiltonian mechanical systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied.