- #1
niko_.97
- 18
- 4
Hi, I hope this is in the right section. It's for EM which I guess is a relativistic theory but the question itself is not to do with any Lorentz transformations or anything similar.
I'm reading through Jackson with my course for EM and I'm on the section where he is generalising the Hamiltonian and is trying to justify why the Hamiltonian density is the canonical stress tensor. I've included a little extract for reference.
Firstly, why should the Hamiltonian be Lorentz invariant? Is it because, under certain conditions, it is equal to the energy, which we know to be Lorentz invariant? (if so what about when those conditions don't hold? I think the conditions are that the Lagrangian doesn't depend on the conjugate velocities, which it doesn't so that's good, and that the holonomic contrinsts aren't time dependent but I'm not sure what these would be aprt from the ME).
Secondly, when they generalise it to the canonical stress tensor, is it only the 00 component that is the Hamiltonian density? Since that is the component that would have time derivatives and also the component that would give you the energy if integrated wrt d^(3)x.
I'm reading through Jackson with my course for EM and I'm on the section where he is generalising the Hamiltonian and is trying to justify why the Hamiltonian density is the canonical stress tensor. I've included a little extract for reference.
Firstly, why should the Hamiltonian be Lorentz invariant? Is it because, under certain conditions, it is equal to the energy, which we know to be Lorentz invariant? (if so what about when those conditions don't hold? I think the conditions are that the Lagrangian doesn't depend on the conjugate velocities, which it doesn't so that's good, and that the holonomic contrinsts aren't time dependent but I'm not sure what these would be aprt from the ME).
Secondly, when they generalise it to the canonical stress tensor, is it only the 00 component that is the Hamiltonian density? Since that is the component that would have time derivatives and also the component that would give you the energy if integrated wrt d^(3)x.