Ultimate generalization of Noether's theorem in the realm of Hamiltonian point dynamics

TL;DR

This paper extends Noether's theorem within Hamiltonian point dynamics to include transformations involving time, providing a comprehensive framework that relates invariants like the Runge-Lenz vector to symmetry transformations.

Contribution

It presents the most general form of Noether's theorem in Hamiltonian formalism, incorporating time transformations via extended Hamiltonian formalism.

Findings

Derived a generalized Noether theorem including time transformations.

Provided a symmetry representation for the Runge-Lenz invariant.

Established a unified framework for invariants and symmetries in Hamiltonian systems.

Abstract

Noether's theorem in the realm of point dynamics establishes the correlation of a constant of motion of a Hamilton-Lagrange system with a particular symmetry transformation that preserves the form of the action functional. Although usually derived in the Lagrangian formalism, the natural context for deriving Noether's theorem for first-order Lagrangian systems is the Hamiltonian formalism. The reason is that the class of transformations that leave the action functional invariant coincides with the class of canonical transformations. As a result, any invariant of a Hamiltonian system can be correlated with a symmetry transformation simply by means of the canonical transformation rules. As this holds for any invariant, we thereby obtain the most general representation of Noether's theorem. In order to allow for symmetry mappings that include a transformation of time, we must refer to the extended Hamiltonian formalism. This formalism enables us to define generating functions of canonical transformations that also map time and energy in addition to the conventional mappings of canonical space and momentum variables. As an example for the generalized Noether theorem, a manifest representation of the symmetry transformation is derived that corresponds to the Runge-Lenz invariant of the Kepler system.