This paper studies a type of integral and reduction of the generalized Birkhoffian system. An existent condition and the form of the integral are obtained. By using the integral, the dimension of the system can be reduced two degrees. An example is given to illustrate the application of the results.
The Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhofflan systems. Finally, simulation results of the given example indicate that structure-preserving algorithms have great advantage in stability and energy conserving.
A new kind of weak Noether symmetry is obtained that arises in the study of Birkhoffian systems. The conserved quantities, which includes Noether conserved quantity, Hojman type conserved quantity and new type conserved quantity, are also calculated from this symmetry. Finally, two examples are presented to illustrated the application of these new results.
