By the way Cleonis, I really like your Site. I had visited it before for the sections on the Coriolis effect and Foucault's pendulum, but I didn't know you are on this forum as well.
I'm wondering, are there cases where the action has no local minimum? With respect to small s, that is. Because if {\partial \over \partial s}S(\mathbf{q}(t))|_{s=0} were not 0, then it would no longer be equivalent to \mathbf{F} = m\mathbf{a}.
On a related note, \mathbf{F} = m\mathbf{a} plus...
Anyone knows? ... Btw I left out the imaginary number in the OP; corrected below.
Say we have f(x, y). We can Fourier decompose it in terms of f1(y, v) and e^{\ i\ x\ v}, f2(x, u) and e^{\ i\ u\ y}, or both variables simultaneously f3(u, v) and e^{\ i\ (x\ v\ +\ u\ y)}. Similarly for any...
So, for any particular operator, a change of basis would give different mathematical forms to its eigenstates (I assume physically they don't change)? But the eigenvalues would still be the same, right, since they have to be the possible outcomes of observation?
Say we have the same state \boldsymbol{\psi_r} in momentum basis, or \boldsymbol{\phi_p} in position basis. I want to make either a position observation or a momentum observation. How do I write the operation and the result mathematically, \mathbf{r} \boldsymbol{\psi}, \mathbf{r}...
Multi-Variable / Dimension Fourier Decomposition
Say we have f(x, y). We can Fourier decompose it in terms of f1(y, v) and e^{\ x\ v}, f2(x, u) and e^{\ u\ y}, or both variables simultaneously f3(u, v) and e^{\ x\ v\ +\ u\ y}. Similarly for any greater number of variables or dimensions. Now, is...
Actually, they had all assumed constancy of c before even getting into the point on causality.
I only meant in terms of freedom of travel. Space is not actually turning into time or vice versa.
Well, check these out.
(1) http://en.wikipedia.org/wiki/Lorentz_transformation#Derivation
(2) http://arxiv.org/pdf/gr-qc/0107091v2.pdf
Note especially the discussion on whether FTL violates causality, benign versus paradoxical tachyons, and the candidate definition of "cause and effect" using...
SR is the derivation, validation, and application of Lorentz transformations. The derivation of Lorentz transformations requires causality (one-way past-->present-->future) not be violated in order to dismiss the faster-than-c solutions and leave only SR (& its limiting case where v << c)...
It's not that hard to explain. When the rear tires (friction forces) push on the ground, they also push on the front tires (rigid body constraint through the chassis, axles, etc.). The front tires are turned at an angle, however; so unless you're doing something unsafe, the friction forces on...
You can have static friction force while nothing is moving relative to one another. The tire is not moving relatively to the ground at the contact point, but the rest of it (& the car) is.
In most derivations I've seen of the Lorentz transformation (the basis of special relativity effects), faster-than-light solutions (in some cases, c being a lower limit) are thrown out simply because they are not observed; i.e. causality is assumed to be in place and trumps over mathematical...
Propagation of energy-momentum has an upper speed limit locally. Even locally, you don't need to exceed any speed limit to get sucked into something, as long as you have some non-zero velocity along that direction. I think black holes simply distort space-time while locally, c remains the same...