On the energy partition in oscillations and waves

TL;DR

This paper derives a general energy partition relation for nonlinear dynamical systems based on the homogeneity of their Lagrangian, applicable to various oscillations and waves, including linear, nonlinear, steady-state, and transient cases.

Contribution

It establishes a universal energy partition relation directly from the Euler-Lagrange equation, linking energy distribution to the homogeneity orders of the system's Lagrangian.

Findings

Energy partition relation depends solely on homogeneity orders.

Partition is uniquely defined when potential and kinetic energies are homogeneous functions.

Applicable to diverse systems including finite, continuous, and waveguide structures.

Abstract

A class of generally nonlinear dynamical systems is considered, for which the Lagrangian is represented as a sum of homogeneous functions of the displacements and their derivatives. It is shown that an energy partition as a single relation follows directly from the Euler-Lagrange equation in its general form. It is defined solely by the homogeneity orders. If the potential energy is represented by a single homogeneous function, as well as the kinetic energy, the partition between these energies is defined uniquely. Finite discrete systems, finite continual bodies, homogeneous and periodic-structure waveguides are considered. The general results are illustrated by examples of various types of oscillations and waves, linear and nonlinear, homogeneous and forced, steady-state and transient, periodic, non-periodic and solitary, regular, parametric and resonant. The reduced energy partition relation for statics is also presented.