TY - JOUR

T1 - On the energy partition in oscillations and waves

AU - Slepyan, Leonid I.

N1 - Publisher Copyright:
© 2015 The Author(s) Published by the Royal Society. All rights reserved.

PY - 2015/3/8

Y1 - 2015/3/8

N2 - 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.

AB - 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.

KW - Euler-Lagrange equations

KW - Homogeneous functions

KW - Nonlinear dynamics

KW - Solitary waves

KW - Time-dependent parameters

UR - http://www.scopus.com/inward/record.url?scp=84982126219&partnerID=8YFLogxK

U2 - 10.1098/rspa.2014.0838

DO - 10.1098/rspa.2014.0838

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AN - SCOPUS:84982126219

SN - 0080-4630

VL - 471

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2175

M1 - 20140838

ER -