Abstract:Starting from the analysis of the underdetermined problem of constrained mechanical systems, the fundamental variational principles of analytical mechanics and three kinds of differential equations of motion are introduced and the universality of analytical mechanics is analyzed in this paper. For nonholonomic constrained mechanical systems, the dynamic modeling, geometric structure and key development directions are emphatically analyzed. At the same time, the general symplectic structure and research significance of Birkhoffian systems are briefly introduced, as well as the key problems to be solved. The Noether symmetry of mechanical systems and the symmetry of differential equations of motion are discussed in detail, and corresponding examples are given to illustrate the relationship between the two symmetries and conserved quantities. In the part of geometric mechanics, the symplectic geometric structure and symmetry reduction theory of analytical mechanics are mainly described, including local canonical structure of symplectic manifolds by DarbouxMoserWeinstein theorem, global topological structure and its influence on quantum mechanics, adjoint and coadjoint representation of Lie group and Lie algebra, momentum mapping, Cartan symplectic reduction, MarsdenWeinstein reduction and so on. At the end of the paper, the research methods and results of the integrability of holonomic and nonholonomic mechanical systems are discussed, and the limitations of the existing integrability methods of nonholonomic mechanical systems are pointed out.