Question About the Hamiltonian

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The Hamiltonian is defined as the sum of the product of momenta and velocities minus the Lagrangian, expressed as ##\mathcal{H} \equiv \sum_{i}^{all} p_{i} \dot{q}_{i} - \mathcal{L}##. It is considered a function of the generalized coordinates ##q## and momenta ##p## rather than velocities ##\dot{q}##. To achieve this representation, a change of variables is necessary to express ##\dot{q}## in terms of ##p## and ##q##. This transformation is crucial for the transition from Lagrangian to Hamiltonian mechanics. Understanding this relationship is essential for grasping the fundamentals of Hamiltonian dynamics.
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If the hamiltonian is defined as ##\mathcal{H}\equiv \sum _{ i }^{ all }{ p_{ i }\dot { q } _{ i } } - \mathcal{L} ##, how is it considered a function of ##p## and ##q## instead of ##p## and ##\dot{q}##?

Chris
 
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You should make a change of variables at the end to find ##\dot{q}## in terms of ##p,q##.
 
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