The Third-Order Eigenvalue Problem With The Speed Energy And The C.Neumann System

In this paper, the C. Neumann system for the third-order eigenvalue problem with the speed energyLφ=(-(?)~3+p(?)+(?)p+r)φ=λφ_xis discussed. Based on the Euler-Lagrange function and Legendre transformations, the reasonable Jacobi-Ostrogradsky coordinate systems are found. Then the original systems can be transformed into a finite-dimensional Hamiltonian canonical system.