The idea of the gradient method for integrating the dynamical equations of a nonconservative system presented by Vujanovic is transplanted to a Birkhoffian system, which is a new method for the integration of Birkhoff＇s equations. First, the differential equations of motion of the Birkhoffian system are written out. Secondly, 2n Birkhoff＇s variables are divided into two parts, and assume that a part of the variables is the functions of the remaining part of the variables and time. Thereby, the basic quasi-linear partial differential equations are established. If a complete solution of the basic partial differential equations is sought out, the solution of the problem is given by a set of algebraic equations. Since one can choose n arbitrary Birkhoff＇s variables as the functions of n remains of variables and time in a specific problem, the method has flexibility. The major difficulty of this method lies in finding a complete solution of the basic partial differential equation. Once one finds the complete solution, the motion of the systems can be obtained without doing further integration. Finally, two examples are given to illustrate the application of the results.
