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A generalized coordinate,\(q_k\), is called cyclic if the Lagrangian is independent of the coordinate \(q_k\). It is shown that the corresponding canonical momentum is a constant of motion. A simple example of cyclic coordinate is given.
A generalized coordinate \(q\) is called cyclic if the Lagrangian \(L\) is independent of \(q\). Therefore, if \(q\) is cyclic, we have \[\frac{\partial L}{\partial q}=0.\] In such a case, the equation of motion for \(q\) \begin{equation} \frac{d}{dt}\Big(\frac{\partial L}{\partial \dot {q}} \Big) - \frac{\partial L}{\partial q} = 0 \end{equation}implies a conservation law.\begin{equation} \frac{d}{dt}\Big(\frac{\partial L}{\partial \dot q} \Big)=0. \end{equation} In other words the momentum canonically conjugate to \(q\) is a constant of motion.
An Example
The Lagrangian for a free particle in plane polar coordinates \((r, \theta)\) is given by \begin{equation} L = \frac{1}{2} m \Big(\dot{r}^2 + r^2 \dot{\theta}^2.\Big) \end{equation} It is seen that \(\theta \) does not appear in the Lagrangian and is a cyclic coordinate. The corresponding canonical momentum \[\dfrac{\partial L}{\partial \dot\theta}= mr^2 \dot \theta\] is a constant of motion.
