A unified theory of generalized classical mechanics and nonholonomic mechanics

A unified theory of generalized classical mechanics and nonholonomic mechanics,or the theory of generalized nonholonomic mechanics, and its basic theoretical frame are constructed. Firstly, the generalized graphics ( Chetaev) condition, Routh equations and canonical equations of the generalized nonholonomic mechanical system are given. Secondly, the equations of nonsimultaneous variation of the generalized nonholonomic mechanical system are established by using the relation between the simultaneous variation and the nonsimultaneous variation. And it is proved that using a first integral we can costruct an integral invariant of the system. Thirdly, the nonsimultaneous variation of Hamilton action of the generalized nonholonomic mechanical system is studied, and the integral variants and integral invariants with Poincare-Cartan type and Poincare type are obtained. Finally, some deductions are given. The results show that the relevant conslusions of the generalized classical mechanics and the first-order or higher-order nonholonomic mechanics are the deductions of the theory of generalized nonholonomic mechanics.