a cyclic coordinate

Sometimes the Lagrangian simply does not care about one of the coordinates: only its velocity appears in L, never the coordinate itself. Such a coordinate is called cyclic (or ignorable), and it hands you a conservation law for free.

The coordinate q_j is cyclic when partial L / partial q_j = 0, meaning q_j is absent from the Lagrangian even though its velocity q_dot_j may appear. Then the Euler-Lagrange equation collapses to d/dt (partial L / partial q_dot_j) = 0, so the conjugate momentum p_j = partial L / partial q_dot_j is conserved.

Cyclic coordinates are the practical everyday face of Noether's theorem: an ignorable coordinate is precisely a continuous symmetry of the Lagrangian. If L does not depend on x, linear momentum is conserved (translation symmetry); if it does not depend on an azimuthal angle phi, angular momentum p_phi is conserved (rotation symmetry). Each cyclic coordinate also lets you eliminate one degree of freedom from the problem (Routhian reduction).