  In his study of periodic orbits of the 3 body problem, Hill obtained a
formula relating the characteristic polynomial of the monodromy matrix of a
periodic orbit and an infinite determinant of the Hessian of the action
functional. A mathematically correct definition of the Hill determinant and a
proof of Hill's formula were obtained later by Poincar\'e. We give two
multidimensional generalizations of Hill's formula: to discrete Lagrangian
systems (symplectic twist maps) and continuous Lagrangian systems. We discuss
additional aspects which appear in the presence of symmetries or reversibility.
We also study the change of the Morse index of a periodic trajectory after the
reduction of order in a system with symmetries. Applications are given to the
problem of stability of periodic orbits.
