7.5: Cyclic Coordinates
( \newcommand{\kernel}{\mathrm{null}\,}\)
Translational and rotational invariance occurs when a system has a cyclic coordinate qk. 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
ddt∂L∂˙qk−∂L∂qk=0
then a cyclic coordinate qk, is one for which ∂L∂qk=0. Thus
ddt∂L∂˙qk=˙pk=0
that is, pk is a constant of motion if the conjugate coordinate qk is cyclic. This is just Noether’s Theorem.