Hill's formula

In his study of periodic orbits of the three-body problem, Hill obtained a formula connecting the characteristic polynomial of the monodromy matrix of a periodic orbit with the infinite determinant of the Hessian of the action functional. A mathematically rigorous definition of the Hill determinant and a proof of Hill's formula were obtained later by Poincaré. Here two multidimensional generalizations of Hill's formula are given: for discrete Lagrangian systems (symplectic twist maps) and for continuous Lagrangian systems. Additional aspects appearing in the presence of symmetries or reversibility are discussed. Also studied is the change of the Morse index of a periodic trajectory upon reduction of order in a system with symmetries. Applications are given to the problem of stability of periodic orbits.
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