In this paper, the polynomial type Lagrange equation of finite dimensional con-
strained dynamics is considered, and four possibilities of Euler-Lagrange equations
possibilities are analysed. Based on Wu elimination method and Wu differential
characteristic set method, two algorithms are presented for determining the four
possibilities. Using the two algorithms, without calculating the rank of Hessian
matrix, which case of the four possibilities of the Euler-Lagrange equations is de-
termined, and the corresponding result is obtained. On the symbolic computation
software platform, the two algorithms can be executed in computers.