The Eigenvalue Problem That A Type Of The Energy Which Dependented On Speed And Integerable System

This paper first describes the development origin of soliton,the research significance of the soliton theory,and the development of the integrable system.Secondly,a second order eigenvalue problem:(?)The problem is that the complete integrability of Hamilton canonical system under its corresponding Bargmann constraint is investigated.The second order eigenvalue problem is composed of the whole spectrum,and calculate the double Hamiltonian operator K and J by using the method of Lax nonlinear method,and the Lenard recursion sequence is used to obtain multiple development equation groups.Then we calculate the functional gradient of the eigenvalue problem and get the Bargmann system corresponding to the second order eigenvalue problem.The suitable Jacobi-Ostrogradsky coordinates are introduced in the symplectic manifold,and the equivalent finite dimensional Hamiltonian regular system of Bargmann system is obtained by the generalized momentum obtained by the Lagrange density function the Euler-Lagrange equation.Then it is proved that Hamilton’s regular system is integrable by confocal involution and Liouville theorem.