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

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.

A gauge-symmetrization method for energy-momentum tensors in high-order electromagnetic field theories

“…We emphasize that, different from the situation in standard field theories, this symmetry group simultaneously translates both the spatial coordinate x for the field and particle's position X a [25][26][27]. The infinitesimal criterion of this symmetry is…”

“…The equation of motion for particles is also derived from the variational principle. However, because particles and field reside on different manifolds, the equation of motion for particles will be the weak EL equation [20,[25][26][27]]…”

Discovering exact, gauge-invariant, local energy-momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

“…We start from the action of particle-field systems and revisit the field theory on heterogeneous manifolds developed in Refs. [70][71][72]. We extend the theory to include high order field derivatives and use noncanonical phase space coordinates (X a , U a ) for particles.…”

“…Recently, this difficulty is overcome by the development of an alternative field theory for the particle-field system [70][71][72]. This new field theory embraces the fact that different components, i.e., particles and electromagnetic field, reside on heterogeneous manifolds, and a weak Euler-Lagrange equation was derived to replace the standard Euler-Lagrange equation for particles.…”

“…The field theory on heterogeneous manifolds has been successfully applied to find local conservation laws in the Vlasov-Poisson system and the Vlasov-Darwin system that were previously unknown [70,72]. In particular, the previous well-known momentum conservation law for the Vlasov-Darwin system written down by Kaufman and Rostler [73] in 1971 without derivation was found to be erroneous, and a correct momentum conservation was systematically derived using the the field theory for particle-field system on heterogeneous manifolds .…”