Abstract:The Lagrange's method is one of the general methods to derive the dynamic equations for multibody systems, which are in the form of ordinary differential equations or differential-algebraic equations. Numerical analysis is an important way to investigate the behaviors of the dynamics of multibody systems. In this paper, the first kind and the second kind of Lagrange's equations and the modified Lagrange's equations for multibody systems with their canonical forms are introduced, together with the characteristics of their numerical solutions. The advances are reviewed in the following numerical methods, symplectic algorithms and the implicit algorithms for the dynamic equations of multibody systems, as well as other algorithms for dynamic behaviors of multibody systems, such as Poincaré maps and Lyapunov exponents.