Recent content by jack476

Divide through by four and use the double-angle formula ##\sin(2x) = 2\sin(x)\cos(x)## to turn the equation into ##\cos(2x) = \sin(2x)##, then divide through by ##\cos(2x)## and use the identity ##\sin(2x)/\cos(2x) = \tan(2x)## to get ##1 =\tan(2x)##; this is the simplified form of the...

Excellent, thanks for the suggestions, I'll look at Weinberg and those two papers right now.

Although strictly speaking this is motivated by a classical field theory problem, I felt that the quantum physics subforum would be more appropriate because it pertains to topics which to my understanding are almost entirely dealt with in the context of QFT. The most commonly stated form of...

Maybe the shark actually does know. Could it be part of their hunting instinct for species that prey on animals that swim right at the surface?

It means to identify the particles predicted by the theory and their masses.

I figured it out. By Noether's theorem, there is a conserved current ##j^a_{\mu}## associated with this global ##SO(3)## symmetry, and there is an energy associated with this current that is proportional to ##j^a_{\mu}A^a_{\mu}##. Let ##t^{ij}_a## be component ##i, j## of the ##a##th generator...

1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##. The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...

That's overcomplicating it just a little, although it is correct. Here's what I would say: Hamilton's equations of motion for a system define a vector field that gives a vector for every value of (q,p) in phase space. When supplied with initial conditions, Hamilton's equations can be solved for...

Sorry if this comes too close to the line for self-promotion, but a few months ago I wrote a Medium article about the derivation of the Navier-Stokes equation that mostly followed the approach in Grainger's textbook: (Yes, I know that it says "part 1" in the subtitle, implying the existence...

Can you think of a way to write ##L(q+\epsilon \psi)## in terms of derivatives of ##L## with respect to ##q##, given that ##\epsilon \psi## is very small?

Okay, I think I understand now. It makes no difference whether we integrate the distribution ##\vec{J}^\prime## over ##V^\prime## or if we integrate the following current distribution over a finite region ##V^{\prime \prime}## in which ##V^\prime## is properly contained: $$ \vec{J}^{\prime...

I don't know if I'm allowed to assume that ##V^\prime## has a very large surface.

Wangsness uses ##V^\prime## to represent the region occupied by the field sources (charges or currents). The problem doesn't specify any particular region (sphere, cylinder, torus, etc), just an arbitrary finite region bounded by a closed surface. But either way, the current source is...

The third identity in the list can be used to turn the expression into a surface integral. The current has to be tangent at the surface so at the surface the current be written as ##K\hat{t}_1## where ##\hat{t}_1## is a one of the two tangent vectors at the surface, and since ##d\vec{a}## is...

Since ##\vec{R}/R^3 = -\nabla(\frac{1}{R}) = \nabla^\prime(\frac{1}{R})##, the standard form of the Biot-Savart law for volume currents can be re-written as: $$\frac{\mu_0}{4\pi}\int\limits_{V^\prime}\frac{\vec{J}^\prime (\vec{r}^\prime)\times\vec{R}}{R^3}d\tau^\prime =...