The Hamiltonian is defined as ##\mathcal{H} \equiv \sum_{i}^{all}{p_{i}\dot{q}_{i}} - \mathcal{L}##, where it serves as a function of generalized momenta ##p## and generalized coordinates ##q##. To express the Hamiltonian in terms of ##p## and ##q## rather than ##p## and ##\dot{q}##, one must perform a change of variables to derive ##\dot{q}## in terms of ##p## and ##q##. This transformation is essential for transitioning from Lagrangian to Hamiltonian mechanics.
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