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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 8D / Solution

The scalar and vector potentials satisfy

E = - \nabla ϕ - \partial_{t} A, B = \nabla \times A .

The canonical momentum and Hamiltonian are

p = m \dot{r} + q A, H = \frac{∣ p - q A ∣ ^{2}}{2 m} + qϕ .

For the uniform fields choose

A = (0, B x, 0), ϕ = - E x,

which produces the stated Hamiltonian. Since it is time independent and

y, z

are cyclic coordinates, three independent conserved quantities are

H, q u a d p_{y}, q u a d p_{z} .

Solved by gpt-5.6-sol high.

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