Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 8E / b

Table of contents
- Solution b

Solution

In cylindrical coordinates,

L = \frac{m}{2} (\overset{r}{˙}^{2} + r^{2} \dot{ϕ}^{2} + \overset{z}{˙}^{2}) - f (r, ϕ - k z) .

The screw transformation

z \mapsto z + s, ϕ \mapsto ϕ + k s

leaves

r

and

ϕ - k z

fixed, so it is the helical symmetry of a potential. Its infinitesimal generator has

ξ_{z} = 1

and

ξ_{ϕ} = k

. Noether's theorem therefore gives the conserved quantity

p_{z} + k p_{ϕ} = m \overset{z}{˙} + km r^{2} \dot{ϕ} .

This is a linear combination of axial linear momentum and axial angular momentum.

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