# Potential/Potential Energy

1. Nov 23, 2004

### pmb_phy

I'm curious. How many of you know what potential is and potential energy is and what role it plays in relativistic electrodynamics. How many of you know the difference between 4-momentum and generalized 4-momentum. Do you know what the time component of each of these 4-vectors are?

Thanks

Pete

2. Nov 23, 2004

### pervect

Staff Emeritus
I'd answer a qualified yes to the first question - I would assume you're talking about the 4-version of the magnetic vector potential A if you started talking about 4-potentials in electrostatics, the 4-vector that generates the Faraday tensor when you take it's exterior derivative.

But it's possible that you'd be talking about something completely different the way things have been going :-)

As far as generalized momentum goes, I'd answer qualified no. Usually a generalized momentum is derived from a Lagrangian, so I'd guess that that's what you were doing, but I don't think I've seen people talk about generalized 4-momenta.

3. Nov 26, 2004

### pmb_phy

Yes and no. There is a relativistic generalized potential which appears in the Lagrangian. Off the top of my head (and check me on this) this is U = qA*u - q*Phi. The Phi is the Coulomb potential. V = q*Phi is the potential energy of the charged particle in an EM field. The total energy, W, is an integral of motion and has the value

W = K + E_o + V

where V = potential energy.

Its well worth your time to check into this. I recommend Goldstein's "Classical Mechanics - 3rd Ed."

I think this is an interesting subject myself. I've been thinking about it more lately and will be making new web pages to describe/discuss it (better than trying to post it all here in Latex ) But I have to wait until my modem comes. Its being shipped by UPS and will be here within a week. I'll be back more then and can get more detailed then too.

One interesting notion is the idea of gauge transformation. E.g. if you take the make the following gauge transformation A -> A + grad Psi, then you have to change Phi to Phi -> Phi - @Psi/@t (@ = partial derivative). Psi is a function of space and time. Then the electric field

E = -grad Phi - @A/@t

becomes

which simplifies to

E = -grad Phi - @A/@t

and therefore the field remains unchanged as it must. But if you make this gauge transformation in the Lagrangian then what you're really doing is adding d(Psi)/dt to the original Lagrangian. Since this is the time derivative of a function of space and time it can be deleted from the Lagrangian since both L and L' = L + d(Psi)/dt yield the exact same equations of motion. However the time component of the generalized 4-momentum changes from W to W + @Psi/@t. In such a case the time component of the generalized 4-momentum is no longer the total energy. Interesting stuff.

Happy Thanksgiving by the way!

Pete

Last edited: Nov 26, 2004
4. Nov 30, 2004

5. Dec 1, 2004

### speeding electron

Nice work there pmb phy! Thanks.

6. Dec 1, 2004

### pmb_phy

You're most welcome. Please call me Pete.

I plan on making many more. During the worst part of my convelesance (i.e. back surgery) I had plenty of time to think about some of the more tricky things.

That one topic was in response to someone (whom shall remain nameless since its from another forum/newsgroup) who claimed that potential and potential energy is meaningless in special relativity. I used the Coulomb potential as an example since its the time component of a 4-vector, i.e. the 4-potential. That person's response was to claim that since the time component can be transformed away by a gauge transformation then it was meaningless. I found that impossible to believe and it seemed like a vastly mislead comment. It seemed to me that such a claim was meant to imply that since the Phi can be transformed away that what it does also goes away by such a transformation. However as those calculations show - the gauge transformation can null out the time component of the 4-potential but it does nothing to the total energy and hence it does nothing to the potential energy.

I later realized that I ended up answering two homework questions from Goldstein's Classical Mechanics - 3rd Ed text. :rofl:

So it was a useful exercise. Glad you liked it. There's a bunch of stuff I want to write up that can't be found in texts so that's how I'll be spending my time during the rest of my convelesance.

Pete

Last edited: Dec 1, 2004