Trace formula in general Hamiltonian mechanics

The variational equation corresponding to a fixed interval of the trajectory of a Bamiltonian system of classical dynamics generates a linear canonical differential operator. If a connection consistent with the sympleetic structure is defined on the tangent bundle of the phase space, it is possible to introduce a regularized determinant of such an operator. The trace formula expresses this determinant in terms of the Jacobian of a transformation that is determined by the motion of the classical system and acts on a space with dimension equal to the number of degrees of freedom. A connection between the relations that are obtained and the semielassical asymptotic behavior for the functional integral that describes the dynamics of the corresponding quantum system is noted.