Generating functions of the Cauchy operator of a hamiltonian system

The article is related to the mathematical apparatus for describing the phase trajectories of a hamiltonian system. An approach related to the construction of generating functions for the Cauchy operator is proposed. It is shown that one -parameter families of generating functions satisfy the Hamilton - Jacobi equation or its modifications. Using the example of small oscillations of a mathematical pendulum, it is shown that the description of the Cauchy operator for sufficiently long periods of time requires the use of generating functions of various types. With the help of generating functions, a variational principle similar to the principle of least action is formulated. The efficiency of using generating functions in the development of conservative methods of numerical integration is also noted.