Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 8E / c

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- Solution c

Solution

With

H = p_{j} p_{j} / (2 m)

, the stated brackets give

\overset{q}{˙}_{i} = {q_{i}, H} = \frac{p _{i}}{m},

(1)

and

\overset{p}{˙}_{i} = {p_{i}, H} = \frac{e}{m} ϵ_{ijk} p_{j} B_{k} = e (\dot{q} \times B)_{i} .

(2)

Thus $m \ddot{q} = e \dot{q} \times B$ , the Lorentz-force equation with zero electric field. The magnetic interaction has been moved from the Hamiltonian into the noncanonical symplectic structure.

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8E
Paper 4
ii
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 8E / c

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