On the energy partition in oscillations and waves

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.