  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.
