Chapter 13: Problem 22
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 " \(p_{i}=\) 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).
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