Laplace–Runge–Lenz Vector - Context

Context

A single particle moving under any conservative central force has at least four constants of motion, the total energy E and the three Cartesian components of the angular momentum vector L. The particle's orbit is confined to a plane defined by the particle's initial momentum p (or, equivalently, its velocity v) and the vector r between the particle and the center of force (see Figure 1, below).

As defined below (see Mathematical definition), the Laplace–Runge–Lenz vector (LRL vector) A always lies in the plane of motion for any central force. However, A is constant only for an inverse-square central force. For most central forces, however, this vector A is not constant, but changes in both length and direction; if the central force is approximately an inverse-square law, the vector A is approximately constant in length, but slowly rotates its direction. A generalized conserved LRL vector can be defined for all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form.

The plane of motion is perpendicular to the angular momentum vector L, which is constant; this may be expressed mathematically by the vector dot product equation r·L = 0; likewise, since A lies in that plane, A·L = 0.

The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does not exist such a coordinate for the LRL vector. Thus, the conservation of the LRL vector must be derived directly, e.g., by the method of Poisson brackets, as described below. Conserved quantities of this kind are called "dynamic", in contrast to the usual "geometric" conservation laws, e.g., that of the angular momentum.