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

Table of contents
- Solution 1

Solution

The canonical momentum is

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 .

The Euler-Lagrange equations then yield Hamilton's equations,

\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.

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version