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Cyclic coordinate, a useful concept in Lagrangian dynamics, is defined and is shown to give rise to a conservation law. $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
A generalized coordinate \(q\) is called cyclic if the Lagrangian \(L\) is independent of \(q\).
Therefore, if \(q\) is cyclic, we have \[\pp[L]{q}=0.\] In such a case, the equation of motion for \(q\) \begin{equation} \dd{t}\Big(\pp[L]{\dot {q}} \Big) - \pp[L]{q} = 0 \end{equation} implies a conservation law.\begin{equation} \dd{t}\Big(\pp[L]{\dot {q}} \Big)=0. \end{equation} In other words the momentum canonically conjugate to \(q\) is a constant of motion.
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