Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 34C / a

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- Solution a

Solution

For a particle of mass

m

and charge

q

, the Hamiltonian of a charged particle in an electromagnetic field is

H (x, p, t) = \frac{1}{2 m} [p - q A (x, t)]^{2} + qϕ (x, t) .

The canonical momentum is

p = m \dot{x} + q A

. Hamilton's equations are equivalent to the Lorentz force law

m \ddot{x} = q (E + \dot{x} \times B), E = - \nabla ϕ - \partial_{t} A, B = \nabla \times A .

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