Slepyan Leonid I
School of Mechanical Engineering , Tel Aviv University, PO Box 39040 , Ramat Aviv, 69978 Tel Aviv, Israel.
Proc Math Phys Eng Sci. 2015 Mar 8;471(2175):20140838. doi: 10.1098/rspa.2014.0838.
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.
考虑一类一般的非线性动力系统，其拉格朗日量表示为位移及其导数的齐次函数之和。结果表明，能量分配作为一个单一关系直接由其一般形式的欧拉 - 拉格朗日方程得出。该分配仅由齐次阶数定义。如果势能由单个齐次函数表示，动能也是如此，则这些能量之间的分配是唯一确定的。对于稳态孤立波，其中势能由两个不同阶数的函数组成，德里克 - 波霍扎耶夫恒等式作为一个额外的关系用于获得分配。考虑了有限离散系统、有限连续体、均匀和周期结构波导。通过各种类型的振荡和波的例子来说明一般结果：线性和非线性、均匀和受迫、稳态和瞬态、周期和非周期、参数和共振、规则和孤立。
