Periodic Solution And Chord Solution Of Hamiltonian System

In general, we can get a nonlinear ordinary differential equation from laws of motion in celestial mechanics. This is Hamiltonian system. Then, the motion trajectory of celestial body can be described by this system. So, seeking invariants of Hamiltonian system, which can be used to study solutions of this system, especially the existence of periodic solution, is an important problem.In this article, the main results are:1, Let be an arbitrary compact smooth hypersurface and let be a parameterized family of hypersurface modeled on and let be a family of sub manifolds as intersections of a linear Lagrangian subspace with .Then there exists a constant such that for there is an in for which the hypersurface carries a characteristic chord connecting and satisfying the estimate:Here.2, Let be the standard symplectic vector space and be a contact manifold of restricted type in with induced contact form and its Reeb vector field, a closed Legendrian submanifold in which is an intersection of with the standard Lagrangian submanifold in , then there exists an least one characteristic chord for .