Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems
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A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is developed. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space-time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles’ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler–Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.
Keywordsrelativistic particle-field system different manifolds mass-shell constraint geometric weak Euler–Lagrange equation symmetry conservation laws
This research was supported by the National Magnetic Confinement Fusion Energy Research Project (Grant Nos. 2015GB111003 and 2014GB124005), the National Natural Science Foundation of China (Grant Nos. NSFC- 11575185, 11575186, and 11305171), JSPS-NRF-NSFC A3 Foresight Program (Grant No. 11261140328), the Key Research Program of Frontier Sciences CAS (QYZDB-SSW-SYS004), Geo- Algorithmic Plasma Simulator (GAPS) Project, and the National Magnetic Confinement Fusion Energy Research Project (Grant No. 2013GB111002B).
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