On the energy partition in oscillations and waves

Area of Science:

• Physics
• Applied Mathematics
• Nonlinear Dynamics

Background:

• Investigates nonlinear dynamical systems where the Lagrangian is a sum of homogeneous functions.
• Focuses on the energy partition within these systems.

Purpose of the Study:

• To demonstrate that an energy partition relation directly follows from the general Euler-Lagrange equation.
• To establish that this partition is determined by homogeneity orders of energy functions.

Main Methods:

• Utilizes the general form of the Euler-Lagrange equation to derive an energy partition relation.
• Applies the Derrick-Pohozaev identity for cases involving solitary waves with potential energy from functions of different orders.
• Considers various physical systems including discrete and continuous bodies, and waveguides.

Main Results:

• An energy partition relation is derived directly from the Euler-Lagrange equation, uniquely defined by homogeneity orders.
• For systems with single homogeneous kinetic and potential energy functions, the partition is unique.
• The Derrick-Pohozaev identity provides an additional relation for solitary waves with complex potential energy structures.

Conclusions:

• The energy partition in a broad class of nonlinear dynamical systems is governed by the homogeneity orders of the energy functions.
• The derived methods are applicable to a wide range of phenomena, including linear and nonlinear waves and oscillations.
• This work provides a unified approach to understanding energy distribution in diverse physical systems.