Recent content by -dove

Gah! Of course! Sorry for the stupidity. I'm just so use to dealing with Lagrangians with a single extremum for given endpoints that I entirely forgot about the Euler-Lagrange equations' ability to deal with local extrema. (I blame undergrad physics.) I see your point now, and agree that it...

Maybe I'm misunderstanding you. But consider the following situation: Imagine we were interested in the classic 'fastest path' problem between two points where we are given some 'speed limit' function (a function of position). In this case we are dealing with two dimensions. This is a classic...

So you mean try every value for n that seems reasonable, perform the optimization routine for each one, then calculate the resulting functional for each case, compare all of them by hand, and choose the one that is optimal? To be honest, the whole thing seems more convoluted than it ought to...

But what if it the optimum solution is one where it doesn't touch at all? More importantly, what if you don't know how many times it touches in the optimal solution? Remember, this approach has to handle general functionals that may not have anything to do with the arc length of the path...

EDIT: The end of that first paragraph should read "This makes sense for a DC current or a steadily increasing one. In the former case, the magnetic field will be constant inside and zero outside; the electric field outside will be zero as well. In the latter case, the magnetic field inside will...

150+ views and no leads? Is the question at least clear? As an addendum: I also wonder how restrictions of smoothness get involved. The Euler–Lagrange approach doesn't specify whether the function space it is optimizing over is the space of all continuous functions, or all differentiable...

It is said that toroidal inductors/coils (see http://fa.tu-sofia.bg/te/Brandisky/images/toroidal_coil.jpg ) that are perfectly axially symmetric will completely confine B fields. That is, the B field inside will be nonzero and will circle the toroid, but the B field outside the toroid will be...

All looks good to me! Now, with this field configuration a priori, all I am saying is that if you then wrap a wire around to make a circle centered about the z axis, and this wire loop is entirely in the region of vanishing B, you will still find an EMF and current induced in this wire. Of...

As a hint, if you try to apply the Parseval's identity to x and x^2, and do so correctly, you will find 0=0. If a_n are the x coefficients and b_n are the x^2 squared coefficients: a_0=0 a_n for n≠0 are imaginary b_n are all real. a_n*b_n is imaginary or zero (either way the real part is...

In the interest of full disclosure, I haven't been paying full attention to this thread. I just knew the answer to this last question off hand. I'm not going to swear on anything given any assumptions or workflows that have been presumed already, but I'll say this: There is an 'inner-product...

What they mean by dB/dt is the constant value it takes in the confined region. They do not mean the value it takes at the point at which you evaluate E. This is evident if you see how they derive that formula. E~1/r outside R and E~r inside R. The proportionality constant for the latter case is...

Parseval's theorem proves that the Fourier transform (discrete or continuous) is unitary. So yes. Proving that an operator preserves the norm of all vectors is equivalent to proving that the operator preserves all two vector inner products. Furthermore, knowing the quadratic form of any linear...

Yes. Like I said, I get your point. Maxwell's equations continue to be valid for the E and B (or ɸ and A), of course. My point was just that one should be careful not to then get the idea that classical electrodynamics gives the right answer. The classical failing is a result of the...

I am familiar with basic calculus of variations. For example, how to find a function that makes some integral functional stationary (Euler-Lagrange Equations). Or for example, how to perform that same problem but with some additional holonomic constraint or with some integral constraint. The...

Exactly. Well it turns out that Faraday's law does work here, because there is an EMF and it is given by the same old formula. I was merely looking for ways in which the situation could get away with not inducing an EMF even though Faraday's law says 'it should regardless of whether or not B=0...