Two different methods of deriving Arnowitt-Deser-Misner constraint equations for Einstein's gravitational fields

In this paper,two different methods are given,by which the famous Arnowitt- Deser-Misner (ADM) constraint equations for Einstein's gravitational fields can be derived. One method is to construct an action of gravitational fiel ds with a parameter in the specetimes of Lorentzian sgnature. Thus,ADM cons traints with a parameter can be obtained and the familiar ADM constraint eq uations can be naturally derived by adjusting the value of this parameter. The other is to apply the double complex function theory to gravitational f ields in Hamiltonian formulation. Hence,the double constraints can be obtai ned,in which the well-known ADM constraint equations are included as a spec ial case. In addition,the Lorentzian and Euclidean gravitational theory can be expressed in a unified way by use of these two different methods.