Noether's first theorem in Hamiltonian mechanics

TL;DR

This paper explores how Noether's first theorem applies to Hamiltonian mechanics formulated on fibre bundles, establishing a correspondence between symmetries and integrals of motion, with applications to the Kepler problem.

Contribution

It demonstrates that in Hamiltonian mechanics, every integral of motion corresponds to a symmetry current, extending Noether's theorem beyond Lagrangian mechanics.

Findings

In Hamiltonian mechanics, all integrals of motion are symmetry currents.

Energy functions relative to a reference frame are symmetry currents.

The approach is illustrated with a detailed analysis of the Kepler problem.

Abstract

Non-autonomous non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles over the time axis R. Hamiltonian mechanics herewith can be reformulated as particular Lagrangian theory on a momentum phase space. This facts enable one to apply Noether's first theorem both to Lagrangian and Hamiltonian mechanics. By virtue of Noether's first theorem, any symmetry defines a symmetry current which is an integral of motion in Lagrangian and Hamiltonian mechanics. The converse is not true in Lagrangian mechanics where integrals of motion need not come from symmetries. We show that, in Hamiltonian mechanics, any integral of motion is a symmetry current. In particular, an energy function relative to a reference frame is a symmetry current along a connection on a configuration bundle which is this reference frame. An example of the global Kepler problem is analyzed in detail.