A constructive method for computing generalized Manley--Rowe constants of motion

TL;DR

This paper introduces a constructive method to compute generalized Manley--Rowe constants for resonance clusters, extending conservation laws beyond simple triads, with a practical Mathematica implementation for analyzing complex dynamical systems.

Contribution

It presents a novel constructive approach to derive generalized Manley--Rowe constants for arbitrary resonance clusters, expanding the analytical tools for dynamical systems.

Findings

Provides a Mathematica implementation of the method

Enables qualitative and numerical analysis of resonance clusters

Extends conservation laws to complex resonance structures

Abstract

The Manley--Rowe constants of motion (MRC) are conservation laws written out for a dynamical system describing the time evolution of the amplitudes in resonant triad. In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley--Rowe constants (gMRC) and give a constructive method how to compute them. We also give details of a \emph{Mathematica} implementation of this method. While MRC provide integrability of the underlying dynamical system, gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.