Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 2 / b

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- Solution b

Solution

The canonical momenta are

p_{ρ} = m \overset{ρ}{˙}, p_{θ} = m ρ^{2} \dot{θ} + \frac{qB}{2} ρ^{2} .

Taking the Legendre transform gives

H = \frac{p _{ρ}^{2}}{2 m} + \frac{1}{2 m} (\frac{p _{θ}}{ρ} - \frac{qBρ}{2})^{2} + \frac{k ρ ^{2}}{2} - qEρ cos θ .

Its four Hamilton equations, in a polar chart with

ρ > 0

, are

\overset{ρ}{˙} = \frac{p _{ρ}}{m}, \dot{θ} = \frac{p _{θ}}{m ρ ^{2}} - \frac{qB}{2 m},

\overset{p}{˙}_{ρ} = \frac{p _{θ}^{2}}{m ρ ^{3}} - \frac{q ^{2} B ^{2} ρ}{4 m} - k ρ + qE cos θ, \overset{p}{˙}_{θ} = - qEρ sin θ .

Canonical angular momentum includes the vector-potential contribution and is not simply mechanical angular momentum.

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