 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2015/s7/2/1/solution

p_{i} = \partial L / \partial \overset{q}{˙}_{i}

. If the velocity-momentum relation can be locally inverted, the Legendre transform defines

H (q, p, t) = \sum_{i} p_{i} \overset{q}{˙}_{i} - L

, with the velocities expressed in terms of

q, p, t

. Its differential is

d H = \sum_{i} \overset{q}{˙}_{i} d p_{i} - \sum_{i} \frac{\partial L}{\partial q _{i}} d q_{i} - \frac{\partial L}{\partial t} d t .

\overset{q}{˙}_{i} = \frac{\partial H}{\partial p _{i}}, \overset{p}{˙}_{i} = - \frac{\partial H}{\partial q _{i}} .

A singular velocity Hessian instead requires a constrained-Hamiltonian treatment.

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