Wheels within wheels: Hamiltonian dynamics as a hierarchy of action variables

TL;DR

This paper reveals that in systems with periodic oscillations, the net displacement of other coordinates can be derived from the action integral, which also functions as a Hamiltonian for slow variables, with applications across physics.

Contribution

It introduces a novel framework linking action integrals to Hamiltonian dynamics for slow variables in oscillatory systems, supported by diverse physical examples.

Findings

Net displacement derived from action integral in oscillatory systems

Action integral acts as a Hamiltonian for slow coordinates

Applications include charged particle drifts and relativistic motion

Abstract

In systems where one coordinate undergoes periodic oscillation, the net displacement in any other coordinate over a single period is shown to be given by differentiation of the action integral associated with the oscillating coordinate. This result is then used to demonstrate that the action integral acts as a Hamiltonian for slow coordinates providing time is scaled to the ``tick-time'' of the oscillating coordinate. Numerous examples, including charged particle drifts and relativistic motion, are supplied to illustrate the varied application of these results.