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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.

    This page titled 7.5: Cyclic Coordinates is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.