Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems

For S being a symplectic orthogonal matrix on R2n, the S-periodic orbits in Hamiltonian systems are a solution which satisfies x(0)=Sx(T) for some period T. This paper is devoted to establishing the theory of conditional Fredholm determinant in studying the S-periodic orbits in Hamiltonian systems. First, we study the property of the conditional Fredholm determinant, such as the Fréchet differentiability, the splittingness for the cyclic type symmetric solutions. Also, we generalize the Hill formula originally gotten by Hill and Poincaré. More precisely, let M be the monodromy matrix of the S-periodic orbits, then we get the formula relating the characteristic polynomial of the matrix SM and the conditional Fredhom determinant. Moreover, we study the relation of the conditional Fredholm determinant and the relative Morse index. Applications to the problem of linear stability for the S-periodic orbits are given.