Cyclic coordinate

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A generalized coordinate component that does not appear explicitly in the Lagrangian is called a cyclic coordinate. The generalized momentum component conjugate to any cyclic coordinate is a constant of the motion.

— 1.8 Conserved Quantities

— Structure and Interpretation of Classical Mechanics

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This is a special case of Noether’s theorem. Such coordinates are called “cyclic” or “ignorable”.

— Wikipedia on Lagrangian mechanics

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If only the cyclic coordinate \displaystyle{q(t)} varies with time (if it doesn’t, \displaystyle{q} is superfluous), the Lagrangian, or the essential physical situation, doesn’t vary. Hence the initial value of \displaystyle{q} doesn’t determine the path, which is only possible if the path is closed.

— edited Jul 28, 2014 at 15:55

— ACuriousMind

— answered Jul 28, 2014 at 15:50

— Pieter Kockx

— Why are they called “cyclic” coordinates?

— Physics Stack Exchange

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2022.07.22 Friday ACHK