The. Groupoid Discrete Field Of Application

On the basis of the theory of groupoids and Lie algebroids, we mainly study two different difference discrete Lagrangian formulas which are defined on the Lie groupoid Q×Q and corresponding discrete variations. Firstly, we introduce a discrete field which was called an analogue of continuous function in the difference discrete mechanics by a groupoid morphism defined on the regular grid and taking value on the Lie groupoid Q×Q. Associated with the groupoid morphism, we define groupoid versions of the difference discrete Lagrangian functionals S =(?)L(u_i,j,u_i+1,j) + L(u_i,j,u_i,j+1) and S =(?)L(u_i,j,△u_i+1,j+L(u_i,j,△u_i,j+1.We also get the discrete Euler-Lagrange's equations respectively by finite variations of groupoids. Finally, in terms of groupoids, we give the geometric meaning of DVPⅡ.