The differential equations of motion of high order nonholonomic systems constructed by utilizing the general methods not only lack the symplectic structure and Lie algebra structure of holonomic systems, but also lack the self adjoint nature of holonomic systems. In this paper, the high order nonholonomic system was transformed into first order kinetic system by using the reduced order method, which was self adjointized by making use of the Cauchy Kowalevski theorem, then the generalized Birkhoffian equation of high order nonholonomic system was obtained. This will provide a new method for researching a number of high order nonholonomic system's problems, such as the geometry and algebraic structure, geometric numerical integrator and engineering applications, etc.