Trace formula in Hamiltonian mechanics

The equation in variations, corresponding to a fixed interval of the trajectory of the Hamiltonian system of classical mechanics, generates a linear canonical differential operator. It is shown that for the ratio of such operators there exists a regularized determinant. The trace formula expresses this determinant in terms of the Jacobian of a certain transformation, given by the motion of the classical system and acting in a space having dimension equal to the number of degrees of freedom of the system. One notes the connection between the obtained relations and the quasiclassical asymptotics for the continual integral, describing the dynamics of the corresponding quantum system.