will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian (the Staeckel conditions). For illustration, several examples in orthogonal coordinates are worked in the next sections. Examples in various coordinate systems Spherical coordinates In spherical coordinates the Hamiltonian of a free particle moving in a conservative potential U can be written The Hamilton–Jacobi equation is completely separable in these coordinates provided that there exist functions: such that can be written in the analogous form Substitution of the completely separated solution into the HJE yields This equation may be solved by successive integrations of ordinary differential equations, beginning with the equation for where is a constant of the motion that eliminates the dependence from the Hamilton–Jacobi equation The next ordinary differential equation involves the generalized coordinate where is again a constant of the motion that eliminates the dependence and reduces the HJE to the final ordinary differential equation whose integration completes the solution for . Elliptic cylindrical coordinates The Hamiltonian in elliptic cylindrical coordinates can be written where the foci of the ellipses are located at on the -axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that has an analogous form where : , and are arbitrary functions. Substitution of the completely separated solution into the HJE yields Separating the first ordinary differential equation yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator) which itself may be separated into two independent ordinary differential equations that, when solved, provide a complete solution for . Parabolic cylindrical coordinates The Hamiltonian in parabolic cylindrical coordinates can be written The Hamilton–Jacobi equation is completely separable in these coordinates provided that has an analogous form where , , and are arbitrary functions. Substitution of the completely separated solution into the HJE yields Separating the first ordinary differential equation yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator) which itself may be separated into two independent ordinary differential equations that, when solved, provide a complete solution for . Waves and particles Optical wave fronts and trajectories The HJE establishes a duality between trajectories and wave fronts. For example, in geometrical optics, light can be considered either as “rays” or waves. The wave front can be defined