The Third-order Eigenvalue Problem’s Integrable System In Which The Energy Is Dependent On Speed

The 3th-order complex eigenvalue problem whose energy is dependent on speed Lφ-((?)~3+q(?)~2+(?)p+r)φ=λφ and the integrability of Hamilton system which is constrained by the Bargmann constraint condition are discussed in this paper.First,the theoretical background and the research achievements are introduced.Second, some usable concepts are introduced in short.By the eigenvalue problem’s compatible condition,the Hamilton operator K,J are established.Based on Lenard sequence,the constraint relation between the potentials(7) q, p, r(8) and the eigenvector ?,the assocoated Lax pairs are nonlinearized,then we can find the Bargmann system.At the last,using the Lagrange density function,the Euler-Lagarange function and Legendre transforms,a reasonable Jacobi-Ostrogradsky coordinate system could be received. Then the infinite-dimensions Dynamical system can be translated into the finite-dimendions Hamilton canonical system in the symplectic space.Meanwhile,the representations of the solutions for the evolution equations are obtained.