Field theory and weak Euler-Lagrange equation for classical particle-field systems

TL;DR

This paper develops a generalized weak Euler-Lagrange equation for classical particle-field systems, establishing a formal link between space-time symmetry and energy-momentum conservation laws.

Contribution

It introduces a weak form of the Euler-Lagrange equation to connect particle and field dynamics, enabling systematic derivation of conservation laws.

Findings

Established the weak Euler-Lagrange equation for particle-field systems

Derived energy-momentum conservation laws from space-time symmetry

Introduced the weak Euler-Lagrange current for flux analysis

Abstract

It is commonly believed as a fundamental principle that energy-momentum conservation of a physical system is the result of space-time symmetry. However, for classical particle-field systems, e.g., charged particles interacting through self-consistent electromagnetic or electrostatic fields, such a connection has only been cautiously suggested. It has not been formally established. The difficulty is due to the fact that the dynamics of particles and the electromagnetic fields reside on different manifolds. We show how to overcome this difficulty and establish the connection by generalizing the Euler-Lagrange equation, the central component of a field theory, to a so-called weak form. The weak Euler-Lagrange equation induces a new type of flux, called the weak Euler-Lagrange current, which enters conservation laws. Using field theory together with the weak Euler-Lagrange equation developed here, energy-momentum conservation laws that are difficult to find otherwise can be systematically derived from the underlying space-time symmetry.