# 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.

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