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Physics LibreTexts

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

ddtL˙qkLqk=0

then a cyclic coordinate qk, is one for which Lqk=0. Thus

ddtL˙qk=˙pk=0

that is,  pk is a constant of motion if the conjugate coordinate qk 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.

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