Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 8E (Classical Dynamics)

Table of contents
- a 8E
  - Solution a
- b 8E
  - Solution b
- c 8E
  - Solution c
- d 8E
  - Solution d
- e 8E
  - Solution e

a

Solution

For canonical position and momentum coordinates

(q_{i}, p_{i})

, Hamilton's equations are

\overset{q}{˙}_{i} = \frac{\partial H}{\partial p _{i}}, \overset{p}{˙}_{i} = - \frac{\partial H}{\partial q _{i}} (i = 1, \dots, n) .

b

Solution

The Poisson bracket of two phase-space functions is

{f, g} = i = 1 \sum n (\frac{\partial f}{\partial q _{i}} \frac{\partial g}{\partial p _{i}} - \frac{\partial f}{\partial p _{i}} \frac{\partial g}{\partial q _{i}}) .

c

Solution

The canonical Poisson-bracket relations are

{q_{i}, q_{j}} = 0, {p_{i}, p_{j}} = 0, {q_{i}, p_{j}} = δ_{ij} .

d

Solution

By the multivariable chain rule and Hamilton's equations,

\frac{df}{d t} = \sum_{i} (\frac{\partial f}{\partial q _{i}} \overset{q}{˙}_{i} + \frac{\partial f}{\partial p _{i}} \overset{p}{˙}_{i}) + \frac{\partial f}{\partial t} = \sum_{i} (\frac{\partial f}{\partial q _{i}} \frac{\partial H}{\partial p _{i}} - \frac{\partial f}{\partial p _{i}} \frac{\partial H}{\partial q _{i}}) + \frac{\partial f}{\partial t} .

The sum is the Poisson bracket

{f, H}

\frac{df}{d t} = {f, H} + \frac{\partial f}{\partial t} .

e

Solution

Let

f

and

g

be conserved quantities, allowing explicit time dependence. Then

\partial_{t} f = - {f, H}, \partial_{t} g = - {g, H} .

Using bilinearity of the Poisson bracket, commutation of partial differentiation with it, and the Jacobi identity for the Poisson bracket,

\frac{d}{d t} {f, g} = {{f, g}, H} + {\partial_{t} f, g} + {f, \partial_{t} g} = {{f, g}, H} - {{f, H}, g} - {f, {g, H}} = 0.

Thus ${f, g}$ is also conserved.

 Ancestors (9)

 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