more complicated, where is the natural oscillation frequency, and Proofs that the Laplace–Runge–Lenz vector is conserved in Kepler problems The following are arguments showing that the LRL vector is conserved under central forces that obey an inverse-square law. Direct proof of conservation A central force acting on the particle is for some function of the radius . Since the angular momentum is conserved under central forces, and where the momentum and where the triple cross product has been simplified using Lagrange's formula The identity yields the equation For the special case of an inverse-square central force , this equals Therefore, A is conserved for inverse-square central forces A shorter proof is obtained by using the relation of angular momentum to angular velocity, , which holds for a particle traveling in a plane perpendicular to . Specifying to inverse-square central forces, the time derivative of is where the last equality holds because a unit vector can only change by rotation, and is the orbital velocity of the rotating vector. Thus, A is seen to be a difference of two vectors with equal time derivatives. As described elsewhere in this article, this LRL vector A is a special case of a general conserved vector that can be defined for all central forces. However, since most central forces do not produce closed orbits (see Bertrand's theorem), the analogous vector rarely has a simple definition and is generally a multivalued function of the angle θ between r and . Hamilton–Jacobi equation in parabolic coordinates The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates , which are defined by the equations where r represents the radius in the plane of the orbit The inversion of these coordinates is Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations where Γ is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta px and py shows that Γ is equivalent to the LRL vector Noether's theorem The connection between the rotational symmetry described above and the conservation of the LRL vector can be made quantitative by way of Noether's theorem. This theorem, which is used for finding constants of motion, states that any infinitesimal variation of the generalized coordinates of a physical system that causes the Lagrangian to vary to first order by a total time derivative corresponds to a conserved quantity