 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s7/1/c/solution

a = q B_{0} / (2 L) > 0

. Differentiating the Lagrangian with respect to each generalized velocity gives

p_{x} = m \overset{x}{˙}, p_{y} = m \overset{y}{˙}, p_{z} = m \overset{z}{˙} - a x^{2} .

The coordinates

y

and

z

are cyclic, so

p_{y}

and

p_{z}

are conserved. The momentum

p_{x}

is generally not conserved: its equation is

\overset{p}{˙}_{x} = - 2 a x \overset{z}{˙}

. In particular, conserved canonical

p_{z}

is not the same as conserved mechanical

m \overset{z}{˙}

, which changes as the particle moves across the magnetic-field gradient.

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