7.5: Cyclic Coordinates

Translational and rotational invariance occurs when a system has a cyclic coordinate $q_{k}.$ A cyclic coordinate is one that does not explicitly appear in the Lagrangian. The term cyclic is a natural name when one has cylindrical or spherical symmetry. In Hamiltonian mechanics a cyclic coordinate often is called an ignorable coordinate . By virtue of Lagrange’s equations

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_{k}}-\frac{\partial L}{ \partial q_{k}}=0$$

then a cyclic coordinate $q_{k},$ is one for which $\frac{\partial L}{ \partial q_{k}}=0$. Thus

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_{k}}=\dot{p}_{k}=0$$

that is, $\ p_{k}$ is a constant of motion if the conjugate coordinate $q_{k}$ is cyclic. This is just Noether’s Theorem.