Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 15C / b

Table of contents
- Solution b

Solution

For a time-independent function

F

, Hamilton's equations give

\frac{d F}{d t} = \sum_{k} (F_{q_{k}} H_{p_{k}} - F_{p_{k}} H_{q_{k}}) = [F, H] .

Thus a constant of motion has $[F, H] = 0$ . For

F = \sum_{k} p_{k}

and the given Hamiltonian,

[F, H] = - \sum_{k} V_{q_{k}}

. Simultaneous translation

q_{k} \mapsto q_{k} + s

leaves every difference

q_{i} - q_{j}

, and hence

V

, unchanged. Differentiation with respect to

s

gives

\sum_{k} V_{q_{k}} = 0

, so $[F, H] = 0$ also in this example. This is conservation of total momentum from translational symmetry.

 Ancestors (10)

 View article source

 Discussion (0) New discussion

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