In this article, generalized Birkhoff equations are put forward by adding supplementary terms to the Birkhoff equations. A conformal invariance of the Birkhoff equations can be used to study the generalized Birkhoff Equations, and two examples are presented to illustrate the application of the results.

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Operational systems of spacecraft are general variable mass mechanics systems, and their symmetries and conserved quantities imply profound physical rules of the space system. We study the Mei symmetry of Tzénoff equations for a variable mass nonholonomic system and the new conserved quantities derived. The function expression of the new conserved quantities and the criterion equation which deduces these conserved quantities are presented. This result has some theoretical values in further research of conservation laws obeyed by the variable mass system.

For a Boltzmann-Hamel equation of nonholonomic mechanical system, when it meets certain conditions, the Boltzmann-Hamel equation can be transformed into a Birkhoffian system. By constructing the generating function, the system is investigated numerically using the generalized symplectic geometric algorithm of the nonautonomous Birkhoffian system. Compared with the above-mentioned algorithm with the classical Runge-Kutta method, Birkhoffian symplectic scheme is very accurate in a long-term tracing.