7.5: Cyclic Coordinates
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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.