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
This is a special case of Noether’s theorem. Such coordinates are called “cyclic” or “ignorable”.
— Wikipedia on Lagrangian mechanics
If only the cyclic coordinate varies with time (if it doesn’t, is superfluous), the Lagrangian, or the essential physical situation, doesn’t vary. Hence the initial value of doesn’t determine the path, which is only possible if the path is closed.
— edited Jul 28, 2014 at 15:55
— answered Jul 28, 2014 at 15:50
— Pieter Kockx
— Why are they called “cyclic” coordinates?
— Physics Stack Exchange
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