The variational equation (Jacobi equation) on a fixed trajectory of a natural Lagrangian system leads to a certain linear differential operator. The trace formula expresses a suitably regularized determinant of this operator in terms of the determinant of a finite-dimensional operator generated by the classical motion in the neighborhood of the trajectory. The aim of the paper is to discuss such a formula in a fairly free geometrical framework and establish its connection with the trace formula in general Hamiltonian mechanics, which was the subject of a preceding publication of the authors.