This paper studies integration theory for Vacco dynamical equation of nonlinear nonholonomic system. First, the first integrals of Vacco equation are given. Secondly, using cyclic integral and energy integral respectively, the order of Vacco equation is reduced, generalized Routh equation and generalized whittaker equation of Vacco form are obtained.
By using higher order Lagrange equations of holonomic mechanical system,higher order Hamilton's canonical equations of holonomic mechanical system are obtained,by which the higher order cyclic integral and the higher order generalized energy integral of holonomic potential mechanical system are obtained,and the physical meaning of higher order Hamilton's function is explained.
Motivated by concepts of Jacobi integral and cyclic integral in analytical me-chanics and energy-momentum tensor in electro-magntic field,the conservation laws forpiezoelectric materials were derived, from which the path independent integrals can be ob-tained.
On this basis, the higher order Lagrange function is introduced,the higher order Lagrange equations of holonomic potential mechanical system are derived, and the higher order cyclic integral and the integral of higher order generalized energy of the system are obtained.
For nonconservative systems, the cycle integrals of Lagranges equations which represent the theorems of generalized impulse are deduced, and the calculation methods of generalized impulse are also given, so that a new way of application is provided for Lagranges equations.
In this paper, the sum of equal powers is discussed. Bernowlli's numbers B72-B100and the cycle integrating formula are obtained. 107 expressions on the sum of equal powers are given by the formula above, and calculate formula of M-N expression on the sum of equal powers is obtained.
Based on the generalized Routh equation and generalized chaptygin equation of relativistic nonlinear nonholonomic system, the generalized cyclic integrals, generalized energy integrals and local energy integrals and their conditions of existence for the system are gived. The former conclasion for the first integrals of holonomic system and classical mechanics system are special cases in this paper.
Secondly,the order of the equation of motion is reduced by using cyclic integrals and energy integrals,and thus the generalized Routh equation and generalized Whittaker equation are obtained. Thirdly,the canonical equation and variational equation of the system are established,and the integral invariant is constructed by using the first integrals.