A cyclic coordinate \(q_{k}\) is a coordinate absent from the Lagrangian (even though \(\dot{q}_{k}\) is present in L.)

(a) Show that a cyclic coordinate is likewise absent from the Hamiltonian.

(b) Show from the Hamiltonian formalism that the momentum \(p_{k}\) canonical to a cyclic coordinate \(q_{k}\) is conserved, so \(p_{k}=\alpha=\) constant. Therefore one can ignore both \(q_{k}\) and \(p_{k}\) in the Hamiltonian. This led E. J. Routh to suggest a procedure for dealing with problems having cyclic coordinates. He carries out a transformation from the \(q, \dot{q}\) basis to the \(q, p\) basis only for the cyclic coordinates, finding their equations of motion in the Hamiltonian form, and then uses Lagrange's equations for the noncyclic coordinates. Denote the cyclic coordinates by \(q_{\mathrm{s}+1} \ldots q_{n}\); then define the Routhian as

\[R\left(q_{1}, \ldots q_{n} ; \dot{q}_{1} \ldots . \dot{q}_{s} ; p_{\mathrm{s}+1} \ldots p_{n} ; t\right)=\sum_{\mathrm{i}=\mathrm{s}+1}^{n} p_{i} \dot{q}_{1}-L\]

Show then (using \(R\) rather than \(H\) ) that one obtains Hamilton-type equations for the \(n-s\) cyclic coordinates, while (using \(R\) rather than \(L\) ) one obtains Lagrange-type equations for the non-cyclic coordinates. The Hamilton-type equations are trivial, showing that the momenta canonical to the cyclic coordinates are constants of the motion. In this procedure one can in effect "ignore" the cyclic coordinates, so "cyclic" coordinates are also "ignorable" coordinates.