# General field theory and weak Euler-Lagrange equation for classical particle-field systems in plasma physics

## TL;DR

This paper develops a general field theory for classical particle-field systems in plasma physics, introducing a weak Euler-Lagrange equation to connect symmetries with conservation laws despite the particles and fields residing on different manifolds.

## Contribution

It introduces a novel weak Euler-Lagrange equation for particles, enabling the derivation of conservation laws in systems where particles and fields are on different manifolds.

## Key findings

- Established relations between rotational symmetry and angular momentum conservation.
- Applied the theory to Klimontovich-Poisson and Klimontovich-Darwin systems.
- Extended Noether's procedure to particle-field systems with different manifolds.

## Abstract

A general field theory for classical particle-field systems is developed. Compared with the standard classical field theory, the distinguish feature of a classical particle-field system is that the particles and fields reside on different manifolds. The fields are defined on the 4D space-time, whereas each particle's trajectory is defined on the 1D time-axis. As a consequence, the standard Noether's procedure for deriving local conservation laws in space-time from symmetries is not applicable without modification. To overcome this difficulty, a weak Euler-Lagrange equation for particles is developed on the 4D space-time, which plays a pivotal role in establishing the connections between symmetries and local conservation laws in space-time. Especially, the non-vanishing Euler derivative in the weak Euler-Lagrangian equation generates a new current in the conservation laws. Several examples from plasma physics are studied as special cases of the general field theory. In particular, the relations between the rotational symmetry and angular momentum conservation for the Klimontovich-Poisson system and the Klimontovich-Darwin system are established.