Extracting Dynamical Frequencies from Invariants of Motion in Finite-Dimensional Nonlinear Integrable Systems

TL;DR

This paper introduces a method to determine the frequencies of motion in integrable systems directly from known integrals of motion without requiring explicit action-angle variables, aiding analysis in physics applications.

Contribution

It presents a novel approach to extract dynamical frequencies from invariants, bypassing the need for explicit transformation to action-angle coordinates.

Findings

Method successfully applied to multiple examples

Allows frequency determination without explicit action-angle variables

Enhances analysis of integrable systems in physics

Abstract

Integrable dynamical systems play an important role in many areas of science, including accelerator and plasma physics. An integrable dynamical system with $n$ degrees of freedom (DOF) possesses $n$ nontrivial integrals of motion, and can be solved, in principle, by covering the phase space with one or more charts in which the dynamics can be described using action-angle coordinates. To obtain the frequencies of motion, both the transformation to action-angle coordinates and its inverse must be known in explicit form. However, no general algorithm exists for constructing this transformation explicitly from a set of $n$ known (and generally coupled) integrals of motion. In this paper we describe how one can determine the dynamical frequencies of the motion as functions of these $n$ integrals in the absence of explicitly-known action-angle variables, and we provide several examples.