Cyclic coordinate

Remarks

The generalized momentum conjugate to $q_k$ is $p_k = \partial L / \partial \dot q_k$. If $q_k$ is cyclic, the $k$-th Euler–Lagrange equation reads $\frac{d}{dt} p_k = \partial L/\partial q_k = 0$, so $p_k$ is constant along every motion. Whether a coordinate is cyclic depends on the choice of coordinates: the same symmetry may be hidden in other coordinates.

Example: a particle of mass $m$ in the plane with a central potential, in polar coordinates, has $L = \tfrac12 m(\dot r^2 + r^2\dot\theta^2) - V(r)$. The angle $\theta$ does not occur in $L$, so it is cyclic, and $p_\theta = m r^2 \dot\theta$, the angular momentum, is conserved.

Non-example: in the same Lagrangian $r$ is not cyclic, since $\partial L/\partial r = m r\dot\theta^2 - V'(r)$ is not identically $0$. In Cartesian coordinates the same Lagrangian is $\tfrac12 m(\dot x^2 + \dot y^2) - V(\sqrt{x^2 + y^2})$, and for non-constant $V$ neither $x$ nor $y$ is cyclic.