Discussion Of Some Problems In Linear And Nonlinear Hamiltonian Systems

In this paper, we study some problems of linear and nonlinear Hamilton system.The full text is divided into two parts.The first part discusses the moving rules about the eigenvalues of the fundamen-tal solution of convex linear Hamiltonian on the unit circle. Suppose A(t)(t ? 0) is a continuous symmetric positive definite 2n-order matrix, J is called the standard symplectic matrix,then (?)(t) = JA(t)R(t), R(0) = I2n.Suppose ???(R(t0))?U, where t0 > 0,U is unit circle, mt is the number of eigenvalues of R(t) near ? and on U. We will check the conjecture that when t ?t0±,mt is a constant by numerical calculation methods.The second part discusses the existence of solutions for second order Hamilton system (?)+V'(t,x) = 0,x(1)-x(0) = 0,(?)(1) - (?)(0) = M1,where V ? C1([0,1]× Rn,R). We transform the problem into the homotopy problem:(?)+(1-?)B(t)x(t)+?V'(t,x)=0,t?[0,1],x(1)-x(0)=0,(?)(1)-(?)(0)=?M1.where M1 is short for M1(x(0),(?)(0),x(1),(?)(1))(??(0,1)). Thus, a proof of the existence of nonzero solutions is presented.