A Hamiltonian structure of the interacting gravitational and matter fields

Abstract

We present a Hamiltonian formulation for classical field theories. In a general case we write the Hamilton equation by means of the energy–momentum function E and the symplectic 2-form Ω. We investigate thoroughly an important example, the gravitational field coupled to a matter tensor field. It will be shown that the energy-momentum differential 3-form yields a generalization of the Komar energy formula. We prove that the energy–momentum function E, the symplectic 2-form Ω, the Hamilton equation, and four constraint equations for initial values of canonical variables give rise to the system which is equivalent to the Euler–Lagrange variational equations. We also discuss relations between the Hamilton equation of evolution, the degeneracy of the symplectic 2-form Ω, and the action of the diffeomorphism group of spacetime in the set of solutions.