Calculus of Variations - Noether's Theorem for fields
I am working on the attached problem from the book "Emmy Noether's Wonderful Theorem". I believe I can answer parts a,b,c and d, but I would like help with part e.
Now if my answer's to parts a-d are not correct then that might explain why I can't do part e. So I will put some of my working below.
A Lagrangian density function that gives the Poisson equation is:
for i = 1,2,3
Notice that this is slightly different to what I was asked to show. I think I am right and the text has a typo. Partly because my version looks more like an energy term.
The generators are [0 y -x 0]'. But I choose to make the problem a little more general. I choose to make the rotation about an axis , then the rotation around z is a special case and the generator becomes
Now the Rund Trautman identity is:
Taking care of the zero generators this simplifies to:
Now unless rho=1 and v=2 or rho = 2 and v=1 and tau is equal to 1 and -1 respectively. So the Hamiltonian term disappears and Rund Trautman becomes
because derivatives of a constant vector are zero
Noether's Theorem with (zeta = 0) simplifies to:
Cancelling the last two terms:
From here I have a few ideas but have not been able to make anything work. My ideas include:
1. Take the divergence of j then get all of the terms into a form of a vector dotproduct with n. The n can then be dropped as it is an arbitrary direction.
2. Integrate the divergence of j over a sphere with a radius that approaches infinity this integral will be a constant or zero. Taking the time derivative of this integral will get rid of any constant, then I can use the divergence theorem to write:
But I haven't been able to make either of these work.