Advances in research on Birkhoffian mechanics

The Birkhoffian mechanics is a natural development of Hamiltonian mechanics, and is a new development stage of analytical mechanics. This mechanics can be widely applied to mechanics, physics and engineering. In this paper, the formation and development of Birkhoffian mechanics, especially the achievements in the last twenty years are summarized. First, to start with the relevant paragraphs of Birkhoff's "Dynamical Systems", the origin of the Birkhoffian mechanics is narrated. Next, the formations and developments of the basic principle, the Pfaff-Birkhoff principle, and the basic equations, Birkhoff equations for the mechanics, are described. After that, a brief introduction to some special problems of the mechanics is given, including constrained Birkhoffian systems, integration methods for Birkhoff equations, inverse problems of Birkhoffian dynamics, stability of motion of Birkhoff equations, geometric methods for Birkhoffian systems, global analysis of Birkhoffian systems and so on. Finally, some suggestions are put forward for the future research of Birkhoffian mechanics.