Frequency Extraction from Invariant Flows

TL;DR

This paper introduces a novel invariant-flow method for extracting fundamental frequencies in integrable Hamiltonian systems, offering a direct, accurate, and scalable alternative to traditional spectral and action-angle techniques.

Contribution

The paper presents an invariant-flow formulation that directly computes frequencies from flow generators, extending applicability to multi-degree systems with high precision.

Findings

Achieves machine-precision accuracy on the McMillan map

Provides a direct numerical procedure for frequency extraction

Extends naturally to systems with multiple degrees of freedom

Abstract

In non-degenerate integrable Hamiltonian systems, invariant tori can be parameterized equivalently by action variables or by their fundamental frequencies. We introduce an invariant-flow formulation for extracting fundamental frequencies of integrable Hamiltonian systems. By treating invariants as generators of commuting Hamiltonian flows, the frequencies are obtained from time-of-flight parameters along these flows, providing a direct alternative to action-angle constructions and spectral methods based on long time series. The approach yields an explicit numerical procedure that extends naturally to systems with multiple degrees of freedom. Its effectiveness is demonstrated using the McMillan map, where machine-precision accuracy is achieved.