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

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- Solution a

Solution

Let

ξ_{i} (t) = \frac{\partial Q _{i} ( s , t )}{\partial s}_{s = 0}

be the infinitesimal generator of the continuous symmetry, and let

p_{i} = \frac{\partial L}{\partial q ˙ _{i}}

be the canonical momentum. Noether theorem states that along every trajectory satisfying the Euler-Lagrange equations,

J = i = 1 \sum n p_{i} ξ_{i}

is conserved.

Indeed, differentiate the invariance of

L

with respect to

s

at zero:

0 = \sum_{i} (\frac{\partial L}{\partial q _{i}} ξ_{i} + \frac{\partial L}{\partial q ˙ _{i}} \dot{ξ}_{i}) .

On an Euler-Lagrange trajectory,

\partial L / \partial q_{i} = d p_{i} / d t

, so

0 = \sum_{i} (\overset{p}{˙}_{i} ξ_{i} + p_{i} \dot{ξ}_{i}) = \frac{d}{d t} \sum_{i} p_{i} ξ_{i} .

This proves the Noether conserved quantity for a mechanical point symmetry.

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