Hill’s formula for $g$-periodic trajectories of Lagrangian systems

In this paper some results of a work by Bolotin and Treshchëv are generalized to the case of $g$-periodic trajectories of Lagrangian systems. Formulae connecting the characteristic polynomial of the monodromy matrix with the determinant of the Hessian of the action functional are obtained both for the discrete and continuous cases. Applications to the problem of stability of $g$-periodic trajectories are given. Hill’s formula can be used to study $g$-periodic orbits obtained by variational methods.