With the vanishing of the variation in time, as it is required for the Lagrange function in the principle of least action, for the variation of the Hamiltonian we have:
By the principle of least action, the variation of the action must be equal to zero: [delta]S = [integral]L dt = [delta][S.
From the stated above it follows that the action is not only a function by which from the principle of least action the equations of motion are obtained, through the Legendre transformation the Hamiltonian, or the Hamilton-Jacobi equations are defined.
Based on the principle of least action and Euler-Lagrange equations, we presented in (17) the relativistic equation of motion of a substance unit in fundamental fields (for motion along the axis OX of the Cartesian reference frame).
After applying the principle of least action to this Lagrangian we obtain equations for the gravitational and electromagnetic fields (76), the equation of substance motion (80), and the equation for the metric (81) and the relation for the cosmological constant (82).