On potentiality, discretization, and integral invariants of the infinite-dimensional Birkhoff systems

In the study of the equations of motion of systems of various physical nature, there are problems in determining the qualitative indicators and properties of motion according to the known structure and properties of the equations under consideration. Such qualitative indicators for finite-dimensional systems are, in particular, integral invariants  — integrals of some functions that retain their value during the system movement. They were introduced into analytical mechanics by A. Poincaré. In the future, the connection of integral invariants with a number of fundamental concepts of classical dynamics was established. The main purpose of this work is to extend some notions of the theory of integral invariants to broad classes of equations of motion of infinite-dimensional systems. Using a given Hamilton’s action, the equations of motion of potential systems with an infinite number of degrees of freedom are obtained, generalizing the well-known Birkhoff equations. A difference analog with discrete time is constructed for them. Based on it, a difference approximation of the corresponding integral invariant of the first order is found.