Field theory and weak Euler-Lagrange equation for classical particle-field systems.
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.
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China and Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543, USA.
Hong
H
Qin
Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543, USA.
Joshua W
JW
Burby
Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543, USA.
Ronald C
RC
Davidson