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.

18 pages

References in corpus (5)

• On Landau damping
• Energetic Consistency and Momentum Conservation in the Gyrokinetic Description of Tokamak Plasmas
• The Einstein-Vlasov system/Kinetic theory
• Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations
• The Hamiltonian structure and Euler-Poincaré formulation of the Vlasov-Maxwell and gyrokinetic systems

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• Slow manifold reduction for plasma science
• Local Energy Conservation Law for Spatially-Discretized Hamiltonian Vlasov-Maxwell System
• Discovering exact, gauge-invariant, local energy-momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds
• Eulerian variational formulations and momentum conservation laws for kinetic plasma systems
• A gauge-symmetrization method for energy-momentum tensors in high-order electromagnetic field theories
• General field theory and weak Euler-Lagrange equation for classical particle-field systems in plasma physics
• The Eulerian variational formulation of the gyrokinetic system in general spatial coordinates
• Polarization and magnetization in collisional and turbulent transport processes