Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 301 / 2 / Solution

Suppose an infinitesimal continuous transformation has variations

δ ψ, δ \overset{ˉ}{ψ}

and changes the Lagrangian density by

δ L = \partial_{μ} K^{μ}

. Expanding by the chain rule and integrating derivatives by parts gives

δ L = [\frac{\partial L}{\partial ψ} - \partial_{μ} \frac{\partial L}{\partial ( \partial _{μ} ψ )}] δ ψ + [\frac{\partial L}{\partial ψ ˉ} - \partial_{μ} \frac{\partial L}{\partial ( \partial _{μ} ψ ˉ )}] δ \overset{ˉ}{ψ} + \partial_{μ} [\frac{\partial L}{\partial ( \partial _{μ} ψ )} δ ψ + δ \overset{ˉ}{ψ} \frac{\partial L}{\partial ( \partial _{μ} ψ ˉ )}] .

On solutions of the Euler-Lagrange equations, the first two brackets vanish. Hence Noether's theorem gives the conserved current

j^{μ} = \frac{\partial L}{\partial ( \partial _{μ} ψ )} δ ψ + δ \overset{ˉ}{ψ} \frac{\partial L}{\partial ( \partial _{μ} ψ ˉ )} - K^{μ}, \partial_{μ} j^{μ} = 0.

For spacetime transformations, the coordinate variation supplies the corresponding energy-momentum term.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version