  It is widely accepted that conservation laws, especially energy-momentum
conservation, have fundamental importance for both classical and quantum
systems in physics. A widely used method to derive the conservation laws is
based on Noether's theorem. However, for classical relativistic particle-field
systems, this process is still impeded. Different from the quantum situation,
the obstruction emerged when we regard the particle's field as a classical
world line. The difficulties come from two aspects. One is the mass-shell
constraint and the other comes from the heterogeneous-manifolds that particles
and fields reside on. This study develops a general geometric (manifestly
covariant) field theory for classical relativistic particle-field systems. In
considering the mass-shell constraint, the Euler-Lagrange-Barut (ELB) equation
as a geometric version of the Euler-Lagrange (EL) equation is applied to
determine the world lines of the relativistic particles. As a differential
equation in the standard field theory, the infinitesimal criterion of the
symmetry condition is converted into an integro-differential equation. To
overcome the second difficulty, we develop a weak ELB equation on the 4D
space-time. The weak version of the ELB equation will play an essential role in
establishing the connections between symmetries and local conservation laws.
Using field theory together with the weak ELB equation developed here, the
conservation laws can be systematically derived from the symmetries that the
systems admit.
