L along the -axis, and the major semiaxis as the -axis, yields the locus equation for p, In other words, the momentum vector p is confined to a circle of radius centered on . The eccentricity corresponds to the cosine of the angle shown in Figure 3. In the degenerate limit of circular orbits, and thus vanishing A, the circle centers at the origin (0,0). For brevity, it is also useful to introduce the variable . This circular hodograph is useful in illustrating the symmetry of the Kepler problem. Constants of motion and superintegrability The seven scalar quantities E, A and L (being vectors, the latter two contribute three conserved quantities each) are related by two equations, and , giving five independent constants of motion. (Since the magnitude of A, hence the eccentricity e of the orbit, can be determined from the total angular momentum L and the energy E, only the direction of A is conserved independently; moreover, since A must be perpendicular to L, it contributes only one additional conserved quantity.) This is consistent with the six initial conditions (the particle's initial position and velocity vectors, each with three components) that specify the orbit of the particle, since the initial time is not determined by a constant of motion. The resulting 1-dimensional orbit in 6-dimensional phase space is thus completely specified. A mechanical system with d degrees of freedom can have at most constants of motion, since there are 2d initial conditions and the initial time cannot be determined by a constant of motion. A system with more than d constants of motion is called superintegrable and a system with constants is called maximally superintegrable. Since the solution of the Hamilton–Jacobi equation in one coordinate system can yield only d constants of motion, superintegrable systems must be separable in more than one coordinate system. The Kepler problem is maximally superintegrable, since it has three degrees of freedom () and five independent constant of motion; its Hamilton–Jacobi equation is separable in both spherical coordinates and parabolic coordinates, as described below. Maximally superintegrable systems follow closed, one-dimensional orbits in phase space, since the orbit is the intersection of the phase-space isosurfaces of their constants of motion. Consequently, the orbits are perpendicular to all gradients of all these independent isosurfaces, five in this specific problem, and hence are determined by the generalized cross products of all of these gradients. As a