Homoclinic Orbits Of Non-periodic High-dimensional Second-order Singular Hamiltonian System

Hamiltonian system theory is both classical and modern research area, which is studied by different methods. In this paper, we use variational method since Hamiltonian system has a variational structure .The solutions of Hamiltonian system will be obtained as critical point of the corresponding functional. During the past twenty years there has been a great deal of progress in the use of methods from the calculus of variations to find periodic,homoclinic,heteroclinic,and other kinds of orbits for Hamiltonian systems.In this paper, the existence of nontrivial homoclinic orbits of Hamiltonian systems without periodicity:is proved, where is a potential with a singularity, i.e..Our main assumptions are Gordon-Strong Force condion and the uniqueness of a global maximum of V(t,q) and assume that V(t,q) satisfies:uniformly for all t∈ R . uniformly for all t∈R.(V4) There is a neighbourhood W Rn of e and a function such that (V5) There is function and constants m > 0,M > 0 such thatis negative definition,(V7) There is a function and constant T > 0, such thatThen (HS2) possesses at least one nontrivial homoclinic orbit, namely At first we consider the approximate problem:Solutions of this approximate problem will be obtained as critical points of the functional fN(q).we also get some uniform estimates with respect to N≥1,which permit us to choose a subsequence of qN (t) converging to a nontrivialhomoclinic orbit of (HS2) as N â†'∞.