On the energy partition in oscillations and waves

Leonid I. Slepyan*

*Corresponding author for this work

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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. The partition 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. For a steady-state solitary wave, where the potential energy consists of two functions of different orders, the Derrick-Pohozaev identity is involved as an additional relation to obtain the partition. Finite discrete systems, finite continuous 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 and non-periodic, parametric and resonant, regular and solitary.

Original languageEnglish
Article number20140838
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume471
Issue number2175
DOIs
StatePublished - 8 Mar 2015

Funding

FundersFunder number
Seventh Framework Programme284544

    Keywords

    • Euler-Lagrange equations
    • Homogeneous functions
    • Nonlinear dynamics
    • Solitary waves
    • Time-dependent parameters

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