Generalized Newtonian Equations of Motion

We generalize Newtonian equations of motion to equations in three (and higher) dimensional phase space, which we denominate “Elliptic Equations of Motion.” The new system of equations contains the scale parameter of energy. We find that the total energy of the conservative system described by the elliptic equations is the product of the constants of motion related to the family of Hamilton–Nambu functions. For the case of a stationary potential we establish the equivalence between elliptic equations in n-dimensional phase space with the potential V(x) and Newtonian equations with the potential given by the polynomial of V(x). We show that the equations of motion for a relativistic particle in a stationary potential field can be reformulated via elliptic equations of motion in three-dimensional phase space.