Hamiltonian Mechanics of Fields

In relativistic field mechanics one ordinarily introduces the time derivative of a field component as its velocity and the partial derivative of the Lagrangian density with respect to the velocity as its canonically conjugate momentum. In order to treat the time and space equivalently, Born and Weyl once treated the four space-time derivatives of a field component as four velocities and introduced the four partial derivatives of the Lagrangian density with respect to the velocities as four momenta. In the present paper this idea is carried further in order to introduce the generalizations of the point-mechanics ideas of Hamiltonian equations, Lagrange brackets, Poisson brackets, and integrals of motion.