In the Lagrangian formalism, a coordinate q, is ignorable

QUESTION:

In the Lagrangian formalism, a coordinate \(q_{i}\), is ignorable if \(\partial \mathcal{L} / \partial q_{i}=0\); that is, if \(\mathcal{L}\) is independent of \(q_{i}\). This guarantees that the momentum \(p_{i}\), is constant. In the Hamiltonian approach, we say that \((q_{i}\), is ignorable if \(\mathcal{H}\) is independent of \((q_{i}\), and this too guarantees \(p_{i}\) is constant. These two conditions must be the same, since the result \(" p_{i}=\text { const" }\) is the same either way. Prove directly that this is so, as follows:

(a) For a system with one degree of freedom, prove that \(\partial \mathcal{H} / \partial q=-\partial \mathcal{L} / \partial q\) starting from the expression (13.14) for the Hamiltonian. This establishes that \(\partial \mathcal{H} / \partial q=0\) if and only if \(\partial \mathcal{L} / \partial q=0\).

(b) For a system with n degrees of freedom, prove that \(\partial \mathcal{H} / \partial q_{i}=-\partial \mathcal{L} / \partial q_{i}\) starting from the expression (13.24).