Frequency Extraction from Invariant Flows

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.

Figures (1)

• Figure 1: Comparison between time-series-based frequency extraction and the invariant-flow method for the McMillan map, top: $|\Delta\nu_1|$, bottom: $|\Delta\nu_2|$. As the tracking length increases, both numerical estimates converge to the invariant-flow results. Reducing the tracking length by one order of magnitude leads to a loss of three to four significant digits in $\nu_1$, while the estimate of $\nu_2$ converges more slowly and exhibits residual oscillatory deviations. The invariant-flow method achieves machine-level precision without relying on long time series.