[NOTES/CM-05001] Cyclic coordinates and constants of motion

Finding conserved quantities
Two general methods are available to find conserved quantities.
Firstly, if there are continuous symmetries under which the Lagrangian is invariant which can be can be arrived at by inspection. The existence of conservation laws follows by appealing to Noether's Theorem. Also the expressions of constants of motion can be found. However, this requires some amount of practice in most cases. As an example, for a spherically symmetric potential, the angular momentum is a constant of motion in addition to the total energy.

Second method works by inspection. If the Lagrangian is independent of a particular coordinate $q_{r}$, {\it i.e.} it depends only on generalized velocity $\dot{q}_{r}$, we have a conservation law. Such a coordinate is called {\bf cyclic coordinate or ignorable coordinate.} It follows from the Euler Lagrange equations of motion that the canonical momentum \(p_r\), conjugate to $q_{r}$, is a constant of motion. However, this method works by inspection only in specific particular choice of coordinates. For a spherically symmetric potential, the Lagrangian in spherical polar coordinates \(r,\theta,\phi\) the \(\phi\) turns pout to be cyclic coordinate.

How does a conservation law help?
Newton's EOM, and also the Euler Lagrange equations, are second order differential equations. The knowledge of conservation laws and of constants of motion greatly simplifies the task of obtaining solutions to the equations of motion. This is most clearly seen in one dimension, where use of conservation law for energy reduces the problem to quadrature.

In case of spherically symmetric potential problems in three dimensions, conservation of angular momentum implies that the orbits lie in a plane. It is, therefore, sufficient to work in two dimensions using plane polar coordinates. The canonical momentum conjugate to \(\phi\) turns out to be the magnitude of angular momentum. In this case, the existence of conservation laws again means that only one first order differential equation needs to be solved to find the orbit.