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<p>Suppose I have a weightless spring connected perpendicularly to the ground, and it has on top of it some weightless surface. Now, I release some sticky object from height $h$ above the system of light spring-surface. The object eventually hits the surface and the spring is starting to contract till all the kinetic energy of the object is transformed to elastic potential energy. And the system continues to oscillate harmonically. I want to find the maximal contraction of the spring, therefore, I did:</p>
<p>(we assume that no energy is lost during the collision)</p>
<p>$U_{GR}=E_{k,max}=U_{SP, max}$</p>
<p>$mgh=\frac{1}{2}kx_{max}^2$</p>
<p>where $k$ is the stiffness coefficient.</p>
<p>Therefore:</p>
<p>$x_{max}=\sqrt{\frac{2mgh}{k}}$</p>
<p>I've chosen my 0 level for gravitational potential energy to be on the same level as 0 level of potential elastic energy. Suppose I want to choose another 0 level for GPE, e.g. at maximum contraction. Then my equation would be:</p>
<p>$mg(h+x_{max})=\frac{1}{2}kx_{max}^2$</p>
<p>It obviously leads us to a slightly different $x_{max}$ value.</p>
<p>According to the numbers in the answer in my book it seems like the second equation is correct.</p>
<p>My question is - how then correct equations should look like if I want to arbitrary choose my PE 0 level at any point? Or why the first one is wrong?</p>
<p>Also, I'm wondering why the amplitude of the oscillations is not $x_{max}$ (according to the book it is slightly less the correct $x_{max}$ value, even though we assumed that no kinetic energy was lost. Interesting coincidence: the amplitude on the contrary, according to the book, equals to the $x_{max}$ value of my first equation).</p> | 6,192 | [
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<p>(I found this related Phys.SE post: <a href="http://physics.stackexchange.com/q/30948/2451">Why is GR renormalizable to one loop?</a>)</p>
<p>I want to know explicitly how it comes that Einstein-Hilbert action in 3+1 dimensions is not renormalizable at two loops or more from a QFT point of view, i.e., by counting the power of perturbation terms. I tried to find notes on this, but yet not anything constructive. Could anybody give an explanation with some details, or a link to some paper or notes on it?</p> | 6,193 | [
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<p>I find <a href="http://en.wikipedia.org/wiki/Fredholm_theory" rel="nofollow">Fredholm theory</a> beautiful, especially the <a href="http://en.wikipedia.org/wiki/Liouville-Neumann_series" rel="nofollow">Liouville-Neumann series</a> for solving <a href="http://en.wikipedia.org/wiki/Fredholm_integral_equation" rel="nofollow">Fredholm integral equations of the second kind.</a> There seems to be a consensus that these equations are quite useful for physicists, but all I have read is "these are useful in physics but we don't have time for that here . . . blah, blah, blah compact operators blah blah blah existence and uniqueness blah blah blah Banach spaces blah blah blah". (I love all these topics, but the point stands.) </p>
<p>So my question is: Is there a simple problem in physics, perhaps suitable for advanced undergraduates, that contorts itself into a Fredholm integral equation? I reject the Poisson equation as an example, because it is too simple and there is no reason to learn Fredholm theory to solve it. </p> | 6,194 | [
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<p>My question relates to the third MIT's video lecture about Electricity
and Magnetism, specifically from $21:18-22:00$ : <a href="http://youtu.be/XaaP1bWFjDA?t=21m18s" rel="nofollow">http://youtu.be/XaaP1bWFjDA?t=21m18s</a></p>
<p>I have watched the development of Gauss's law, but I still don't quite
understand the link between Gauss's law and Coulomb law: How does
Gauss's law change if Coulomb law would of been a different one.</p>
<p>I also don't understand why is it so important for Gauss's law that
the electric field decrease proportionally to $\frac{1}{r^{2}}$ ?</p>
<p>For example, what would of happened if the electric field decrease
proportionally to $\frac{1}{r}$ , or $\frac{1}{r^{3}}$ ?</p> | 6,195 | [
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<p>While working on physics simulation software, I noticed that I had implemented discrete time (the only type possible on computers).</p>
<p>By that I mean that I had an update mechanism that advanced the simulation for a fixed amount of time repeatedly, emulating a changing system.</p>
<p>I was a bit intrigued by the concept. Is the real world advancing continuously, or in really tiny, but discrete time intervals?</p> | 114 | [
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<p>A solid disk of mass m is rolling along a surface its center has velocity v what is the kinetic energy of disk?I cannot solve the problem,</p> | 6,196 | [
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<p>a deSitter space is a maximally symmetric solution of Einstein equations, I have some problem picturing one thing: this space is past and future (time) infinite but spatial slices have finite size, how can we have finite size in slicing at constant time, and infinite time size when slicing at constant radial coordinate if the solution is supposed to be symmetric in the plane (t,r).</p> | 6,197 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/31247/why-do-we-need-higgs-field-to-re-explain-mass-but-not-charge">Why do we need Higgs field to re-explain mass, but not charge?</a> </p>
</blockquote>
<p>Why have scientists been looking for something to give mass to particles, but not looking for something to give charge (or other properties of particles)? </p>
<p>Or is mass not just another property of particles? </p>
<p>Also if the Higgs boson gives particles mass, what defines how much mass each particle is given? Why do some particles interact more with the Higgs boson (or do I mean Higgs field here?) than others? </p> | 217 | [
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<p>The Helmholtz equation is expressed as $$\nabla^2 \psi + \lambda \psi = 0$$. This equation occurs, for eg., after taking the Fourier transform (with respect to the time coordinate) of the wave equation in free space. While this equation for the case of $\lambda$ real is reasonably well discussed in the literature, the case where the coefficient $\lambda$ is complex-valued is not much discussed in the literature. However, such an equation does come up for the components of the electric and magnetic fields when we try to solve Maxwell's equations within a conductor. So my question is, what is the general solution of the equation $$\nabla^2 \psi + \lambda \psi = 0$$, where $\lambda$ is complex-valued(having both real and imaginary parts), in cylinderical coordinates? Also, can you quote a good reference from where I can get some good information about this?</p> | 6,198 | [
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<p>I am learning about electricity and magnetism by watching MIT video
lectures.</p>
<p>In the lecture about Gauss's law, while trying to calculate the flux through a sphere with charge in it, the lecturer states that the direction of the electrical field is radial, since it is the only preferred direction that there is (since the problem have spherical symmetry).</p>
<p>But why is this the only preferred direction?</p>
<p>I think that there is also the direction perpendicular to the sphere, but since this is actually a two dimensional subspace of $\mathbb{R}^{3}$ I think that I can rule this out.</p>
<p>Is my conclusion, that since I can't determine any direction (which corresponds to a one dimensional subspace of $\mathbb{R}^{3}$) in any other way, then indeed the redial direction is the only preferred direction?</p> | 6,199 | [
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<p>The usual definition given for a greenhouse gas is that it absorbs infrared radiation. Of course, then the gas emits its own thermal radiation, and it does so without preference for direction (assuming homogeneity).</p>
<p>I was looking at the <a href="http://en.wikipedia.org/wiki/File%3aGreenhouse_Effect.svg">greenhouse effect</a> heat balance and noticed a larger downward flow from the atmosphere than upward.</p>
<p><img src="http://i.stack.imgur.com/RvKA4.png" alt="Wikipedia greenhouse effect"></p>
<p>I can think of two ways you could explain this:</p>
<ul>
<li>the top of the atmosphere is colder than the bottom</li>
<li>the atmosphere reflects IR from the surface</li>
</ul>
<p>While the first point is trivially true, that doesn't rule out IR reflection. This isn't a completely pointless distinction. Since the atmosphere is cooler than the surface, the temperature of the down-coming IR radiation would be higher if there was reflection going on.</p>
<p>So is there some reflection? Or is the downward flow of IR radiation from the atmosphere completely from absorption and remission?</p> | 6,200 | [
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<p>A two-qubit system was originally in the state $ \frac{3}{4}|00\rangle-\frac{\sqrt{5}}{4}|01\rangle+\frac{1}{4}|10\rangle-\frac{1}{4}|11\rangle $ , and then we measured the first qubit to be 0 . Now, if we measure the second qubit in the standard basis, what is the probability that the outcome is 0 ? </p> | 6,201 | [
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<p>Due to some recent "discoveries" (yet to be completely confirmed), we have hinted the existence of tetraquark particles. </p>
<p><strong>Why QCD exotics like multiquark states beyond n=2,3 (mesons, protons,neutrons,...) like tetraquarks, pentaquarks,...or even glueballs could be useful for new physics? And more generally: Quark spectroscopy and "molecular" QCD/quark chemistry...What could they be useful for?</strong></p> | 6,202 | [
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<p>Suppose that I put in the outer space (where gravity from other bodies is negligible) a large, perfectly round sphere totally filled with water. At the bottom (even though "bottom" doesn't make much sense in the deep space) of the sphere there are several marbles, all touching each other.</p>
<p>If I make rotate the sphere, all marbles should stay exactly where they are. Right?
Is that true regardless of the force that I apply?
And what happens, instead, if I strongly shake the sphere?</p> | 6,203 | [
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] |
<p>The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall) and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc length)
.</p>
<p><strong>What shape does it take?</strong></p>
<p>My try so far: minimise potential energy of the whole string,
$$J(x,h, \dot{h})=\int_0^l gh(x) \rho \frac{ds}{l}=\frac{g \rho }{l}\int_0^l h(x) \sqrt{1+\dot{h}^2} dx$$</p>
<p>With the constraint
$$\int_0^l \sqrt{1+\dot{h}^2} dx- l=0$$
If it helps, it's evident that $\dot{h}(\frac{l}{2})=0$.</p>
<p>Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in
$$\int_0^l \frac{g \rho }{l}h(x) \sqrt{1+\dot{h}^2} +\lambda(\int_0^l \sqrt{1+\dot{h}^2} dx- l)dx$$</p>
<p>Must satisfy the Euler Lagrange equation. The constraint must also be satisfied.</p>
<p>But, in truth, by this point I am clueless. $\lambda$ is worked through $\nabla J = \lambda \nabla(\int_0^l \sqrt{1+\dot{h}^2} dx- l)$. I have tried this , but get nonsensical answers.</p>
<p>Is this method the best? If so, in what ways am I going about it wrongly thusfar?</p> | 6,204 | [
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<p><img src="http://i.stack.imgur.com/R70Fu.png" alt="enter image description here"></p>
<p>The question then asks for the potential difference between $X$ and $Y$, which is claimed to be $3.6\text{ volts}%$.</p>
<p>Why would there be a potential difference in this case? If I connect a lightbulb on $X$, and another on $Y$, it seems very obvious that the brightness would be the same.</p>
<p>Or is the standard direction of current important? So the current "encounters" the 60 ohm "first" when flowing to $X$, and the 30 ohm "second" when flowing to $Y$? That seems a little crackpot to me. So if we change the resistor to the left of $X$ to 100 ohms, and the resistor to the right of $Y$ to be 0 ohms, the differing definitions of current flow would actually cause a difference in measurement of potential?!?!</p>
<p>The correct answer is worded:
<img src="http://i.stack.imgur.com/kpT1i.png" alt="enter image description here"></p> | 6,205 | [
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<p>How much harder would it have been for Felix to use some powered sled and head for the ISS when he stepped out of his capsule? He was already above most of the atmosphere.
BTW, Is that capsule still up there? Why don't more missions use balloons for the first stage?</p> | 6,206 | [
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<p>I have two arbitrary vectors $\vec{x}$ and $\vec{x}'$ given in spherical coordinates $(|\vec{x}|=x,\theta,\phi)$ (as convention I take the "physics notation" given on Wikipedia <a href="http://en.wikipedia.org/wiki/Spherical_coordinate_system" rel="nofollow">http://en.wikipedia.org/wiki/Spherical_coordinate_system</a>). I now want to rotate the coordinate system so that it's $z$-direction points along $\vec{x}$. That means, $\vec{x}$ would have the values $(0, 0, x)$. I now need to compute the angles of $\vec{x}'$. The absolute value does not change, but the angles do. I need to figure out how the angles are in the new coordinate system. With the help of rotation matrices, one is able to get:</p>
<p>$\vec{x}' = x' (\sin(\theta')\cos(\phi'-\phi)\cos(\theta)-\sin(\theta)\cos(\theta'),\sin(\theta')\sin(\phi'-\phi),\sin(\theta')\cos(\phi'-\phi)\sin(\theta)+\cos(\theta)\cos(\theta')) \equiv x' (\sin(\alpha')\cos(\beta'),\sin(\alpha')\sin(\beta'),\cos(\beta')) $</p>
<p>Now $\alpha', \beta'$ are the angles in the normal sense but in the new coordinate system. I need a converting rule $\theta', \phi' \to \alpha', \beta'$. Anyone a hint?</p>
<p><strong>Edit (some further explanations):</strong> I need this to compute an integral of the form $\int \mathrm{d}^3x' g(\theta',\phi')f(|\vec{x}-\vec{x}'|)$ and I converted $\mathrm{d}^3x'=x'^2\mathrm{d}x\mathrm{d}\phi' \sin(\theta')\mathrm{d}\theta'$ to spherical coordinates. The problem is that $|\vec{x}-\vec{x}'|=x^2+x'^2-2xx'[\sin(\theta')\cos(\phi'-\phi)\sin(\theta)+\cos(\theta)\cos(\theta')]$ contains angles of both vectors and I need to get rid of the unprimed angles (which is possible in transforming the coordinate system under the integral to point with it's z-direcion along $\vec{x}$).</p> | 6,207 | [
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<p>To illustrate discord and its use, <a href="http://arxiv.org/pdf/quant-ph/0105072.pdf" rel="nofollow">Zurek in his paper on discord</a> (NB pdf) gives example of a partially decohered bell state i.e.
$$\rho_{AB}=\frac{1}{2}(|00\rangle\langle 00|+|11\rangle\langle 11|) + \frac{z}{2}(|00\rangle \langle 11|+|11\rangle \langle 00|)$$
using the measurement basis for Alice's side.
$$\{\cos\theta |0\rangle + e^{i\phi} \sin\theta |1\rangle,\ e^{-i\phi} \sin\theta |0\rangle + \cos\theta |1\rangle\}$$
He plotted it in the <a href="http://i.imgur.com/68vIYwJ.png" rel="nofollow">Fig.1</a>.</p>
<p>Now my problem is that I have not been able to reproduce this result. <a href="http://i.imgur.com/lUSsGXQ.jpg" rel="nofollow">Here</a> is what I got. I can not seem to locate what could be the source problem. The Mathematica code which I used is <a href="https://drive.google.com/file/d/0B39rxnxwFcLiZUZoSkdKNFJabFE/edit?usp=sharing" rel="nofollow">here</a>. Can anyone see what might be going wrong here.</p> | 6,208 | [
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<p>In the book Young and Freedman 13th edition, the wave equation is <br/> <br/> $y(x, t) = A\,\text{cos}(kx-wt)$ <br/> <br/> The problem is, I find it hard to console with the fact that <br/> <br/> $y(x, t) = A\,\text{sin}(wt-kx)$. <br/> <br/> How to derive $A\,\text{sin}(wt-kx)$ from $A\,\text{cos}(kx-wt)$?</p> | 6,209 | [
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<p>In particular I'm interested in any algorithms that can separate multiple tracks from one another reliably.</p> | 6,210 | [
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<p>From what I understand space itself is expanding, and the Big Bang attempts to describe this expansion at the very early stages of the universe.</p>
<p>This is usually described in a visual way as 2 dots on the surface of a balloon as the balloon is being inflated.</p>
<p>We exist in space, so as space expands we are expanding too aren't we? If this is correct how do we know space is expanding at all?</p>
<p>I'll attempt to explain further... 2 dots on a balloon with a ruler drawn between them measuring 1 cm. As the balloon expands, so do the dots, and the ruler, and it always says that the distance is 1 cm. From an observers point of view (somewhere away from the balloon) the dots, ruler, and ruler are all getting larger - but from the point of view on the balloons surface nothing has changed.</p>
<p>Another way to describe this would be a man in a room, the room and everything in it (including the man) are getting bigger at the same rate. From the man's perspective nothing is changing but from outside the room - you can see everything is getting bigger.</p>
<p>I guess what I am trying to say is that we are in the universe, and expanding at the same rate as it (aren't we?) so how do we know it's getting bigger?</p> | 56 | [
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<p>If I consider one single Dirac electron in momentum representation, I use the wavefunction $u(p)e^{-ipx}$, however if I consider an one-particle state in the Fock space I use $|p\rangle$. Should it not be same? </p>
<p>Obviously the Dirac 1-particle wavefunction is a bispinor, and probably $|p\rangle$ is not a spinor. But could it not be spinor? </p>
<p>For a 2-particle wavefunction $|p,k\rangle$, I would use
$$\frac{1}{\sqrt2}(u_1(p)u_2(k)e^{ipx_1+ikx_2} - u_2(k)u_1(p)e^{ipx_2 + ikx_1})$$
or something similar. I regret my limited way of expressing correctly.
Certainly there is the problem if I consider instead of a half-spin particle a scalar particle, then I would have to build my multi-particles state out scalar wave function instead of spinor wave functions. May be my understanding of the Fock space is incomplete.</p> | 6,211 | [
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<p>I would like to show that a particle orbiting another will follow the trajectory </p>
<p>\begin{equation}
r = \frac{a(1-e^2)}{1 + e \cos(\theta)}.
\end{equation}</p>
<p>I would like to do this with minimal assumptions. Any pointers? I've read that this is nothing but the equation for an ellipse, as defined from the focus of the ellipse. But I'm not satisfied in just taking the result, I mean why should I assume that orbits are elliptical? I guess it should be derivable from energy and angular momentum considerations...</p> | 311 | [
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<p>So <a href="https://www.youtube.com/watch?v=OTcdutIcEJ4" rel="nofollow">this guy</a> was the first to run a loop and in this (german) <a href="http://www.welt.de/vermischtes/article125146560/Erster-Mensch-der-Welt-rennt-einen-Looping.html" rel="nofollow">article</a> (and also in the video) a certain speed (13.8km/h) is mentioned.</p>
<p>Why must he run at this speed and not just "as fast as possible"?</p>
<p>My intuition says it has to do with to much centrifugal force and him not being used to it $\rightarrow$ trip hazard. </p> | 6,212 | [
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<p>any text on photonic crystals will highlight the almost perfect analogy between electrons in a periodic potential and photons in a periodic dielectric. The analogies are:</p>
<p>$$V(\vec r + \vec R) = V(\vec r) \Leftrightarrow \epsilon(\vec r + \vec R) = \epsilon(\vec r)$$</p>
<p>$$\psi(\vec r)=e^{i\vec k\cdot \vec r}u_n(\vec r) \Leftrightarrow \vec H(\vec r)=e^{i\vec k\cdot \vec r}\vec u_n(\vec r)$$</p>
<p>But they have a slight difference in their respective operators in their eigenvalue equations:</p>
<p>$$(\frac{-\hbar^2}{2m}\nabla^2 + V(\vec r))\psi = E\psi \Leftrightarrow \nabla^2\vec H= \epsilon(\vec r) (\frac{\omega}{c})^2\vec H$$</p>
<p>That is, the QM version <em>adds</em> the periodic part, whereas the EM version <em>multiplies</em> it. Why is this, and is it significant?</p>
<p>I don't know enough about differential equations, so maybe it's very insignificant and you could represent either in either way. If I had to guess, I'd say it's related to the fact that electrons and photons have different $k$ dependencies in their dispersion relations ($k^2$ and $k$), but I don't see how the two are related.</p> | 6,213 | [
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] |
<p>I am a little confused as to what the magnitude of <a href="http://en.wikipedia.org/wiki/Acceleration" rel="nofollow">acceleration</a> is and what it means.</p> | 6,214 | [
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<blockquote>
<p>The voltage of the battery signifies the difference in voltage between the positive and negative terminal</p>
</blockquote>
<p>What does this mean?</p>
<p>The definition of voltage difference I'm familiar with is the amount of potential energy charge one coloumb of charge would undergo from point A to point B.</p>
<p>At first I thought the quote was saying the voltage of the battery signifies the difference in potential energy one 1 C of charge would have if it were moved from the positive to the negative terminal. </p>
<p>But my interpretation of batteries is that they "push" electrons into the wire which push the electrons in front of them ahead which push the electrons in front of them ahead and so on.</p>
<p>How can I connect the two concepts of voltage being the difference in potential energy a charge would have if it moved from the positive to the negative terminal with the "pushing" power of a battery?</p> | 6,215 | [
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<p>I'm a high school student. The head is hydrophilic, the tail is fatty acid,in other words hydrophobic. Here is the thing I don't understand, all textbooks state that water is repelled by hydrophobic tail, why? The hydrophilic head is composed of polar molecule,and water is polar,so they will attract each other. But why hydrophobic tail will repel polar molecule?Isn't hydrophobic tail is just composed of nonpolar molecules?</p> | 6,216 | [
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<p>In the Bose-Hubbard model of cold atoms in an optical lattice, we consider only the short-range interaction or on-site interaction. Is it possible to realize long-range interaction similar to Coulomb interaction in the cold atom experiment? </p> | 6,217 | [
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<p>The increasing water pressure as you go deeper is generally explained in terms of the weight of the water column above the observation point pressing down. The question, then, is what would happen if you had a big blob of water in free fall, say 100m in diameter-- not big enough to produce large gravitational forces on its own-- and swam to the center of it. I'm imagining some sort of contained environment, here-- a giant space station, or the planet-sized envelope of air in Karl Schroeder's Virga novels, that sort of thing-- so you don't worry about the water boiling off into vacuum or freezing solid, or whatever. </p>
<p>Would there be any pressure differential between the surface of the blob and the center of the blob? At first glance, it would seem not, or at least not beyond whatever fairly trivial difference you would get from the self-gravity of the water. But it's conceivable that there might be some other fluid dynamics thing going on that would give you a difference.</p>
<p>For that matter, what difference would you expect between the pressure inside the water and outside the water? We know from shots of astronauts goofing around that water in free fall tends to stay together in discrete blobs. This is presumably some sort of surface tension effect. Does that lead to a higher pressure inside the water than out? How much of a difference would that be? Or would you not particularly notice a change from sticking your head into a blob of free-falling water? (Other than, you know, being wet...)</p>
<p>(This is just an idle question, brought on by thinking about the equivalence principle, and thus free-falling frames. It occurred to me that somebody here might know something about this kind of scenario, so why not post it?)</p> | 6,218 | [
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<p>This question pertains to EM signals exchanged between an inertial central clock and an orbiting clock i.e. non-inertial frame.
(I edited out the variables connected with gravity and separation between bodies)</p>
<p>This is how I found the frequency and wavelength received by clock (orbiting) as sent from clock (central). Hereafter let $$c=1$$ and angular speed as the proportion of c $$rω/c=β$$ so that the length contraction factor $$\gamma$$ for each instant in a rotating frame is given as $$γ=\sqrt{1-β^2}$$</p>
<p>Clock 2 is situated at the centre of clock 1’s uniform rotation. We can use the simplified formula $$f_1=f_2/\sqrt{1-β^2}$$ to determine the frequency of a signal that rotating clock 1 receives from central clock 2's emitted frequency.</p>
<p>Let central clock 2’s frequency=1, so $$f_2=1$$ Therefore, a familiar relation appears $$f_1 = 1/\sqrt{1 - β^2}$$ or just $$f_1=1/γ$$ </p>
<p>The frequency of a signal and its wavelength in relation to the speed of light is given by $$c=fλ$$ Let's view this relation from rotating clock 1's frame. So, $$c_1=(1/γ*f_1)(γλ_1)$$ but from central clock 2's frame, we get the reciprocal $$c_2=(γf_2)(1/γ*λ_2)$$</p>
<p>Is this correct?</p> | 6,219 | [
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<p>Padmanabhan's discussion of dynamics mentions that in general the two dimensional harmonic oscillator fills the surface of a two torus.</p>
<p>He further notes that there will be an extra isolating integral of motion provided that the ratio of frequencies is a rational number.</p>
<blockquote>
<p>$ -\frac{\omega_{x}}{\omega_{y}}\cos^{-1}\left(\frac{y}{B}\right)+\cos^{-1}\left(\frac{x}{A}\right)=c$</p>
<p>This quantity $c$ is clearly another integral of motion. But- in general - this does not isolate the region where the motion takes place any further, because $\cos^{-1}z$ is a multiple-valued function. To see this more clearly, let us write </p>
<p>$ x=Acos\left\{c+\frac{\omega_{x}}{\omega_{y}}\Big[Cos^{-1}\left(\frac{y}{B}\right)+2\pi n \Big]\right\}$</p>
<p>Where $Cos^{-1}z$ (with an uppercase C) denotes the principal value. For a given value of $y$ we will get an infinite number of $x$'s as we take $n=0, \pm 1, \pm 2, \dots $</p>
<p>Thus, in general, the curve will fill a region in the $(x,y)$ plane. </p>
<p>A special situation arises if $(\omega_{x}/\omega_{y})$ is a rational number. In that case, the curve closes on itself after a finite number of cycles. Then $c$ is also an isolating integral and we have three isolating integrals: $(E_{x}, E_{y}, c)$. The motion is confined to closed (one-dimensional) curve on the surface of the torus. </p>
</blockquote>
<p>This last part is not still clear to me.</p>
<p>Can someone please explain why a rational ratio of frequencies make a candidate integral of motion single valued and therefore the motion takes place on a closed (one dimensional) curve on the surface of the two torus? </p> | 6,220 | [
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] |
<p>Someone please tell (advice) me some research study problems on Classical Mechanics that can be tackled with Undergraduate level knowledge. </p> | 12 | [
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] |
<p>As far as I know, a black body is an ideal emitter. So how can it be that a non-ideal emitter emits more radiation than a black body?</p>
<p>This happens only in a very limited area at <strong>around 500nm</strong>, but it still happens: it looks like at the maximum it is around 15% above black body.</p>
<p>This seems impossibile for my understand of a black body. Especially because just for that there is the <strong>emissivity value ε</strong>, or better <strong>ε(λ)</strong></p>
<blockquote>
<p><strong><a href="http://en.wikipedia.org/wiki/Emissivity">Wikipedia Emissivity:</a></strong> Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature. The ratio varies from 0 to <strong>1</strong> </p>
</blockquote>
<p>Means <strong>0 ≤ ε(λ) ≤ 1</strong> </p>
<p>What is the right interpretation? What is the sun doing there, seems like ε(500nm)=1.15?</p>
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<p>It is always told as a fact without explaining the reason. Why do two objects get charged by rubbing? Why one object get negative charge and other get positive <a href="http://en.wikipedia.org/wiki/Electric_charge" rel="nofollow">charge</a>?</p> | 363 | [
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<p>If the fields in QFT are representations of the Poincare group (or generally speaking the symmetry group of interest), then I think it's a straight forward consequence that the matrix elements and therefore the observables, are also invariant.</p>
<p>What about the converse:</p>
<blockquote>
<p>If I want the matrix elements of my field theory to be invariant scalars, how do I show that this implies that my fields must be corresponding representations? </p>
</blockquote>
<p>How does this relate to S-matrix theory?</p> | 6,222 | [
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<p>Can anybody elaborate on the implications of the BICEP2 result for string theory?</p>
<p>The discussion here <a href="http://physics.stackexchange.com/questions/15/what-experiment-would-disprove-string-theory">What experiment would disprove string theory?</a> suggests that refuting string theory is rather difficult. It did not mention that planned/ongoing experiments in the near future could exclude many string models by measuring the polarization of the CMB.</p>
<p>I now read that the BICEP2 result might rule out lots of string theory models. Is that true? If so, why? Is this result in the so-called "swampland"? How big a blow is this for string theory?</p> | 6,223 | [
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<p>given the density of states according to Gutzwiller's trace formula</p>
<p>$ g(E)= g_{smooth}(E)+ g_{osc}(E) $</p>
<p>i know that the 'smooth' part comes from $ g_{smooth}(E)= \iint dxdp \delta(E-p^{2}-V(x)) $ for one dimensional system</p>
<p>however how it is the oscillating part of the trace obtained ?? :D i mean the sum over lenghts of the orbit (in the phase space)</p>
<p>also how does the condition for WKB energies appear ?? $ \oint _{C} p.dq= 2\pi \hbar (n+ \alpha) $ from the Gutzwiller trace ??</p> | 6,224 | [
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<p>(Background) In the scenario where a gauge symmetry is spontaneously broken and the gauge field eats the Goldstone boson to acquire a mass, the massive gauge field acquires a longitudinal polarization. I have seen the argument that the massless field has only two physical polarizations, and I have heard the hand-waving argument that now we can boost into the frame of the massive gauge boson, so I think I understand conceptually what has happened. However, I still don't understand how to show mathematically and rigorously that the new polarization has appeared, and I can't find any really decent explanation. This raises the following question</p>
<blockquote>
<p>Given only the Lagrangian of some QFT, and assuming that one does not already know anything about the answer from more elementary arguments, how does one (mathematically, rigorously) find the polarizations of a given field appearing therein?</p>
</blockquote>
<p>The proof that a massless vector field has only two physical polarizations (at least the one from my QFT I class) assumes the Lorentz gauge condition; this suggests the question</p>
<blockquote>
<p>Can one find the polarizations of a field in general gauge?</p>
</blockquote>
<p>Thank you in advance.</p> | 6,225 | [
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<p>I understand that it is more common in GR for the metric to be given a $(-,+,+,+)$ signature and more common in particle physics (or field theory, as Peskin & Schroeder tells me) to use the $(+,-,-,-)$ metric signature. I'm wondering why. Is there any advantage in these disciplines in terms of actual physics to using one over the other, or it simply an entrenched arbitrary convention?</p> | 6,226 | [
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<p>Right now I'm simply looking for an intuitive explaination of actions that integrate over a 4-volume element, $d^4x$ rather than a parameter say $\lambda$. More specifically I'm well versed in action principles that say have an action as $\int L d\lambda$. But in trying to understand actions such as the Einstein Hilbert action for example that are an integral over a volume element, in the form $\int(whatever)d^4x$. I'm not exactly seeing from an intuitive stance how these are pieced together, and how/if they relate to the extremization methods I am more familiar with. I do understand the notation of metrics, and GR and all that, that's not the problem. I just don't have any experience working with actions that are integrals over volumes. If anyone can try to point me in the right direction here, or recommend a good text perhaps it would help.</p> | 6,227 | [
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<p>What happens to the energy when waves perfectly cancel each other (destructive interference)? It appears that the energy "disappear" but the law of conservation of energy states that it can't be destructed. My guess is that the kinetic energy is transformed into potential energy. Or maybe it depends on the context of the waves where do the energy goes? Can someone elaborate on that or correct me if I'm wrong? </p> | 531 | [
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<p>How accurately can our current technological tools measure the human bio-electromagnetic field emitted by a person? Or, to put it differently, does each person have a different electromagnetic field signature, and can we measure that with enough accuracy to tell two people apart by their frequencies? </p> | 6,228 | [
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<p>Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. Penrose, 2004: Road to Reality. Vintage Books, 1136 pp.), the Ashtekar connection, in a way, represents the graviton. Let $\mathcal{Riem}$ denote the configuration space of GR. </p>
<p>If one managed to quantize the metric to yield the graviton, it would make sense to say that the $\mathcal{Riem}$ is also quantized, i.e., Wheeler's superspace becomes a Hilbert space with elements as gravitons. There is an complex inner product defined between the ``de-projections'' (the inverse projection mapping) of gravitons $\mathcal{A,A}_1$, defined as $\langle\mathcal{A}|\mathcal{A}_1\rangle$. An algebra of self-adjoint unbounded operators (of which the Hamiltonian is an element) on this quantum superspace corresponds to a set of all observables. The set of all possible states corresponds to the projective superspace.</p>
<p>Thus, is it possible to interpret the superspace as a Hilbert space (as mentioned above)and provide a rigorous mathematical formulation of quantum gravity?</p> | 6,229 | [
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<p>In <a href="http://www.het.brown.edu/people/danieldf/literary/eric-KKtheories.pdf" rel="nofollow">this</a> review about Kaluza-Klein theories, (page 1115) in order to determine the mass spectrum of the 5 dimensional theory the metric is expanded to first order.</p>
<p>Why this? Why not retain the full metric? Why does it consider only small perturbations? Is it because doing considering the full metric is just too complicated or is something else behind this?</p> | 6,230 | [
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<p>Using a laser setup, I was asked to determine the aperture of a given lens and then use some geometrical arguments and compare the theoretical value from the manufacturer and the experimental value. However, when I did this, the values <em>differed by orders of magnitude</em>.
The way I got the aperture was to let the laser light pass through the lens onto a white screen and then keep the screen at a fixed distance from the lens and measure the radius of the light spot obtained. Then I used geometry to determine the angle and thus its sine.
Could diffraction be the reason of such a large error? </p> | 6,231 | [
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<p>All the examples of plasma I have come across are visible. Is there any plasma which is not visible? For example, during a dark lightening we don't see the radiation because its gamma radiation. Is plasma formed during a dark lightening?</p> | 6,232 | [
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<p>When we calculate electric field intensity for a point charge at any point inside electric field the field intensity is $E = F/q$ where $F$ is the force acting on charge $q$. In this case, the charge $q$ should be very small. The practical value of $q$ cannot be so small as needed. In defining electric field by measuring value $F/q$ is smaller than the actual value.</p>
<p><strong>My question:</strong> How can the accurate value of electric field intensity be calculated?<br>
I'm quite confused. Can someone point me out?</p> | 6,233 | [
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<p>I've been trying to derive the relation</p>
<p>$$[\hat L_i,\hat L_j] = i\hbar\epsilon_{ijk} \hat L_k $$</p>
<p>without doing each permutation of ${x,y,z}$ individually, but I'm not really getting anywhere. Can someone help me out please?</p>
<p>I've tried expanding the $\hat L_i = \epsilon_{nmi} \hat x_n \hat p_m$ and using some identities for the $\epsilon_{ijk} \epsilon_{nmi}$ which gives me the LHS as something like $-\hbar^2\delta_{ij}$ but I've got no further than this.</p> | 6,234 | [
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<p>For demonstrating basic probability concepts, it would be nice to have a coin-like object that lands heads/tails not in 50/50% ratio, but biased in a way that can be revealed in a short experiment. What I'd like is to make an object satisfying:</p>
<ul>
<li>Thin disk shape, say thickness around 1/10 to 1/20 of diameter.</li>
<li>Lands heads/tails with some given lopsided ratio such as 60/40.</li>
<li>Feels evenly balanced to the students' hands.</li>
<li>Can be made by the average machinist, hobbyist woodworker, or 3D printing designer.</li>
<li>Size not important, but maybe 10-20cm diameter. It's meant to be a theatrical prop visible to a small audience, not an actual coin.</li>
</ul>
<p>Would a disk with an interior hollow zone closer to one side than the other do the job? I doubt it, since the coin will still rotate uniformly in the air, exposing both sides equally to any direction, including the floor. Making both sides "heads" is too obvious a cheat, and I don't want 100/0% probabilities anyway.</p>
<p>Note I'm not asking for any practical how-to workshop details, just the physical principle for designing such an object.</p> | 6,235 | [
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<p>This is in reference to the argument given towards the end of page $61$ of this <a href="http://arxiv.org/abs/hep-th/9702155" rel="nofollow">review paper</a>. There for the path-integral argument to work the author clearly needed some argument to be able to ignore the kinetic term(Kahler potential). But I can't get the argument fully. </p>
<p>Just above equation 5.5 in that review the author says, <em>"..since the Kahler potential is irrelevant we may choose it at will,and in particular we may choose it very small. To a first approximation, in fact, we may ignore it..."</em> And on Page 62 in his point 1, he says, <em>"..Nothing in our argument assures that the resulting kinetic term in the Calabi-Yau sigma model will be sufficiently “large” to ensure that sigma model perturbation theory will be valid.."</em></p>
<ul>
<li>It would be great if someone can help make precise as to what the above argument means (..espeically the second part about "alrge"..) and flesh out its contents in may be an equation form or something..</li>
</ul>
<p>Conceptually I would think that trying to ignore the kinetic term of the chiral fields is <em>different</em> than trying to ignore the kinetic term of the gauge fields. like say as written in equation 2.22 on page 11 of <a href="http://arxiv.org/pdf/hep-th/9301042v3.pdf" rel="nofollow">this paper</a>. </p>
<ul>
<li>Now one can see that in the above paper (unlike in the initially linked review) the author is trying to ignore the gauge kinetic term by trying to take the gauge coupling $e \rightarrow \infty$ limit as in Page 30 of the above paper by Witten. </li>
</ul>
<p>..but naively one might think that large gauge coupling is precisely the region where perturbative arguments would have started to fail and the entire idea of trying to minimize the classical potential would have started to become bad..</p>
<ul>
<li>But unlike in the review as linked to earlier, Witten's argument crucially needs the kinetic terms of the scalar chiral superfields (..from where the $F^2$ and the Yukawa terms come..). And Witten's argument also crucially needs the gauge kinetic term (from there the $D^2$ terms come..) Then why is he trying to justify the $e \mapsto \infty$ limit? Like he comments towards the end of Page 60, <em>``..the effects of finite e, are believed to be “irrelevant” in the technical sense of the renormalization group.."</em></li>
</ul>
<p>It would be great if someone could explain these two arguments about ignoring the kinetic term of the gauge fields and the chiral fields. </p> | 6,236 | [
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<p>The Heterotic (HO and HE) string is found by tensoring the left movers of the bosonic string theory state and the right movers of the Type II string theory state:
$$|\psi_{\operatorname{H}}\rangle=|\psi_{\operatorname{B}}\rangle\otimes|\psi_{\operatorname {II}}\rangle$$ </p>
<p>The bosonic state is of course based on the Polyakov Action. Now, what if we had another state (I'll call it the k-heterotic state) which is formed by tensoring the fermionic string theory state and the type II state.
$$|\psi_{\operatorname{H}}\rangle=|\psi_{\operatorname{F}}\rangle\otimes|\psi_{\operatorname {II}}\rangle$$
The fermionic string theory state is based on the Dirac Action:
$$S_F=S_{RNS}-S_B$$
Why is this theory not consistent? Whether the bosonic state is chosen or the fermionic state, how does that matter? Is it because the Dirac Action is imaginary (since it has an i outside the integral if one expands out the Dirac matrix fermionic field product)? Does it have anything to do with that?</p> | 6,237 | [
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<p>This question disusses the same concepts as <a href="http://theoreticalphysics.stackexchange.com/questions/848/relativistic-center-of-mass">that question</a> (this time in quantum context). Consider a relativistic system in spacetime dimension $D$. Poincare symmetry yields the conserved charges $M$ (a 2-form associated with Lorentz symmetry) and $P$ (a 1-form associated with translation symmetry). The center-of-mass trajectory $x$ is defined by the equations</p>
<p>$$x \wedge P + s = M$$
$${i_P}s = 0$$</p>
<p>I'm implicitely identifying vectors and 1-forms using the spacetime metric $\eta$</p>
<p>Define $X$ to be the point on the center-of-mass trajectory for which the spacelike interval to the origin is maximal. Substituting $X$ into the 1st equation, applying $i_P$ and using the 2nd equation and $(P, X) = 0$ yields</p>
<p>$$X = -\frac{{i_P}M}{m^2} $$</p>
<p>Passing to quantum mechanics, this equation defines $X$ as vector of self-adjoint operators, provided we use symmetric operator ordering the resolve the operator ordering ambiguity between $P$ and $M$. In particular these operators are defined on the Hilbert space of a quantum mechanical particle of mass $m$ and spin $s$.</p>
<blockquote>
<p>What is the spectrum of the operator $X^2$, for given $m$, $s$ and $D$?</p>
</blockquote> | 6,238 | [
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<p>The answers in this question: <a href="http://physics.stackexchange.com/questions/1/what-is-spin-as-it-relates-to-subatomic-particles">What is spin as it relates to subatomic particles?</a>
do not address some particular questions regarding the concept of spin:</p>
<p>How are some useful ways to <em>imagine</em> a particle without dimensions - like an electron - to spin?<br>
How are some useful ways to <em>imagine</em> a particle with spin 1/2 to make a 360° turn without returning to it's original position (the wave function transforms as: $\Psi \rightarrow -\Psi$).<br>
When spin is not a classical property of elementary particles, is it a purely relativistic property, a purely quantum-mechanical property or a mixture of both?</p> | 383 | [
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<p>I was wondering if anyone put together a law to describe the rising temperature of the water coming out of a tap.</p>
<p>The setup is fairly simple: there's a water tank at temperature T, a metal tube of length L connected to it and a tap at the end where temperature is measured. The water flows at P l/s.</p>
<p>Given that the metal tube is at room temperature initially, what law describes the temperature of the water at any instant? What is the limit temperature of the water?</p>
<p>Thanks.</p> | 6,239 | [
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<p>I would like to have a good understanding of what is happening when you add salt to boiling water.</p>
<p>My understanding is that the boiling point will be higher, thus lengthening the process (obtaining boiling water), but at the same time, the dissolved salt reduce the polarization effect of the water molecules on the heat capacity, thus shortening the process.</p>
<p>Is this competition between these two effects real ? Is it something else ?</p> | 6,240 | [
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<p>I'm interested in the process that goes from, for example, huge data sets of temperature, pressure, precipitation etc readings to a model of the atmosphere. I'd like to know about how much relies on a priori knowledge of Navier-Stokes, and how much curve fitting is done.</p>
<p>Some sort of general overview of the process (ideally a fairly accessible one) would be perfect.</p>
<p><strong>Edit</strong></p>
<p>I'm looking for a reference, specifically. I know more or less the science from talking to people who do this stuff, but I would like to have some summary article or book chapter overview I can refer to in written work.</p> | 6,241 | [
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<p>The following situation interests me and I was wondering if there is software to model it? </p>
<p>A large set of n spheres of uniform density and discrete sizes, mass proportional to volume, are dropped together on a narrow inclined plane under low gravity (so they are free to bounce and sort themselves) . The spheres are assigned mutual repulsion and attraction functions according to size, such that balls of a certain size are at equilibrium a certain distance apart. </p>
<p>This doesn't reflect any physical phenomenon I am aware of, but it seems like the sort of thing a physicist might model. I have seen physical contraptions that drop balls into slots so they sort themselves into a more or less "normal" distribution, and this is not unlike what I have in mind. But the situation is a little more complicated and I don't really have any preconceived idea about the results. </p>
<p>It may be there is no simple program that does this sort of thing, and that too would be an answer.</p>
<p>Suggestions appreciated. Thanks. </p> | 6,242 | [
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<p>Magnetic monopole predicted by Dirac nearly a century ago was found in spin ice as quasi-particle(2). My question is Why magnetic monopole found in spin ice don't modify the Maxwell's Equations? (I know they are not elementary particles but quasi-particles.)</p>
<p>(1) Dirac, P. A. M. Quantised singularities in the electromagnetic field. Proc. R. Soc. A 133, 60–72 (1931)</p>
<p>(2) Castelnovo, C., Moessner, R. & Sondhi, S. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).</p> | 6,243 | [
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<p>Why don't the stars in a star cluster attract each other gravitationally, forming one big star? What causes a cluster to disperse the stars in it? </p> | 6,244 | [
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<p>Today in my Physics lecture I suddenly thought of something. We all know that gravitational force is proportional to the two masses and inversely proportional to the square of the distance between them. So, naturally, there exists a system of natural units which equates the proportionality by setting $G=1$. Here I'm referring to Planck Units. But I realized that even though numerically using this system of units $F=\frac{m_1m_2}{r^2}$, dimension-wise that is wrong because the units do not correspond to the unit of force as defined by Newton's Second Law. Similarly, the electrostatic force, when making $k=1$, has the wrong unit as well. Why do we have to throw in those proportionality constants to convert the units? Shouldn't the units work out to be the same?</p> | 6,245 | [
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<p>I'm really stuck. I need to figure out the mass/light ratio of a galaxy in solar units. I know its mass is 5.7 x 10<sup>10</sup> solar masses. I know its absolute magnitude (-17.3) and distance (29 Mpc). I'm feeling dumb here because I think this ought to be clear to me, but I'm getting hung up at the magnitude - luminosity - light part. </p>
<p>Here's what I think:
mass/light = 10<sup>0.4*(Magnitude galaxy-magnitude Sun)</sup> x (mass galaxy/mass Sun)</p>
<p>I get a mass to light ratio of around 90. Am I doing this right?</p> | 6,246 | [
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<p>If you put a object in contact with a heat reservoir that is infinitesimally higher in temperature than the object and allow equilibrium to be reached the entropy change is zero right?</p> | 6,247 | [
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<p>Can someone send me pointers to work (either theoretical or simulations)
showing (in)stability of satellite orbits around tidally locked exoplanets?
I want to know firstly if satellite orbits can survive the inward migration
and secondly if once the planet becomes tidally locked if that has any
implications for the orbits of its satellites - do they too become tidally
locked? How do the tidal interactions from the star impact their orbits?
In particular, I'm interested in what happens to orbits of earth-sized moons around gas giants that are in the HZ.</p> | 6,248 | [
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<p>The twinkling of stars, or scintillation, occurs because the optical path length of the atmosphere varies in both space and time due to turbulence.</p>
<p>This means that when the wavefront from a distant star enters a telescope, it is distorted from the flat wavefront expected for an object at infinity.</p>
<p>Because the fundamental Fourier mode of this wavefront varies in time, the average position of the PSF on the retina of a human observer also changes in time; the star appears to shake back and forth. (By this I mean that the wavefront at the telescope may be considered a superposition of plane waves. One particular plane wave is the strongest, or is the median; that is the one I mean by "fundamental Fourier mode". I'm not sure if I'm using exactly the right term. When I worked with an adaptive optics team one summer, we had a "tip-tilt mirror" specifically to correct for this mode, and then the deformable mirror corrected for higher-order aberrations. )</p>
<p>That much makes sense to me. However, I have read <a href="http://en.wikipedia.org/wiki/Scintillation_%28astronomy%29">on Wikipedia</a> that this is not the dominant effect in what we observe as scintillation. Instead, the overall apparent brightness of the star varies.</p>
<p>I presume this has essentially the same physical origin. The wavefront is complicated and dynamic, and so interference effects sometimes reduce and sometimes increase the brightness. What are the details of how this works? How can one work out that this effect should occur? Perhaps a specific example of what a wavefront might do and how this would result in varying brightness would be helpful.</p> | 6,249 | [
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<p>A question came up on Outdoors StackExchange, <a href="http://outdoors.stackexchange.com/questions/1093/how-to-tell-the-time-at-night">How to tell time at night</a>. I wrote an answer to that question, and in my answer I said that a standard sundial wouldn't work without some fiddling, but that an astronomer would know what fiddling was required. (I invite/encourage any qualified astronomers to go there and answer the question, BTW.)</p>
<p>So I did some searching around, and some thinking about it. Here's the question. Suppose we have an ordinary horizontal sundial, at a fixed latitude (in the Northern hemisphere), which has been calibrated to correctly show solar time at that latitude. The sundial <em>works</em>. </p>
<p>Now, suppose there's a Full moon. What adjustments would be necessary, what additional information is needed, what would it take to transform/convert the ordinary sundial into a moondial? The definition of "success" would be that we could look at the shadow and get the "correct" time, where correct might mean up to the precision of an ordinary sundial which might neglect the <a href="http://en.wikipedia.org/wiki/Equation_of_time" rel="nofollow">Equation of Time</a> (is a different equation needed?). One website I read said that no adjustment was necessary, but I'm not sure I agree.</p>
<p>If that's not too bad, how would the answer change if we were some days <em>off</em> of a full moon? (Obviously, if we are sufficiently far off then there will not be enough light to make a discernible shadow.)</p>
<p><strong>EDIT:</strong>
I've been thinking about it a lot, and thinking about the E.o.T.. The EoT has two (dominant) parts: the Kepler eccentricity part, and the oblique axis part. At first I thought the Kepler part wouldn't matter, but then I learned that the Moon/Earth eccentricity (0.055) is even bigger than the Earth/Sun eccentricity (0.017). So it would seem any reliable moondial would need to be corrected for this problem.</p>
<p>I'm still wrapping my head around the obliqueness part. I understand that the style of the gnomon must be angled from the horizontal (equal to the latitude), but that's to make sure the style points toward true north. In fact, even if a horizontal sundial is used at an incorrect latitude, we can fix it by angling the whole sundial so the style points north. Clearly the Moon is in another orbital plane (apparently by about 5 degrees, I'm guessing that's an average), but if we change the angle of the style to compensate for the Moon then it wouldn't be pointing north anymore. So, at the moment, I'm thinking to leave the sundial as-is and then figure (somehow) the sine-wave that describes the Moon's obliqueness.</p>
<p>The further this goes the more it looks like I'm talking about figuring an Equation of Moon Time or similar.</p> | 6,250 | [
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<p>Josh Hill, 9, Oakdale Elementary has always talked Theory of Relativity and Astrophysics etc., I can answer most but lately he has stumped me and has been begging me to ask a pro, so here it is....</p>
<p>"What happens when Dark Matter comes in contact with the event horizon of a large Black Hole?”</p>
<p>Josh Hill
& Mike Hill (DAD)</p>
<p>P.S. we should probably anticipate “ What happens when Dark Energy comes in contact with a Black Hole?” as the follow up question , just so we kill 2 birds with one stone , Thanks. </p> | 6,251 | [
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<p>Every object at a non-zero temperature radiates light, i.e. it glows.
(Is that called <em>blackbody radiation</em>?)</p>
<p>What is the physical reason to this?</p>
<p>Is it because more heat implies that the atoms vibrate, and vibrating charges (the electrons or the nuclei) generate electromagnetic radiation?</p> | 218 | [
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<p>The question is available <a href="http://puu.sh/42HC2.png" rel="nofollow">here</a>:</p>
<p><img src="http://i.stack.imgur.com/MLall.png" alt="enter image description here"></p>
<p>I've modeled the building as a rod on a torsional spring (with a pendulum hanging from the top).</p>
<p>$\phi$ is the angle from the centre for the pendulum and $\theta$ is the angle from the centre for the building.</p>
<p>Taking the Lagrangian,</p>
<p>$$L = T - V,$$</p>
<p>we have to find $T$ and $V$.</p>
<p>Inertia of the pendulum was given as $mL^2$ and the building (rod) was given as $\frac{mh^2}{3}$, therefore</p>
<p>$$T = \frac{1}{2}(mL^2)\dot \phi^{2} + \frac{1}{2}(\frac{mL^2}{3})\dot \theta^2.$$</p>
<p>Potential energy of the pendulum is $-mgL \cos\phi$ and $\frac{1}{2}k\theta^2$, therefore</p>
<p>$$V = -mgL \cos\phi + \frac{1}{2}k\theta^2.$$</p>
<p>So,</p>
<p>$$L = \frac{1}{2}(mL^2)\dot \phi^{2} + \frac{1}{2}(\frac{mL^2}{3})\dot \theta^2 + mgL \cos\phi - \frac{1}{2}k\theta^2.$$</p>
<p>Evaluating:</p>
<p>$$\frac{\partial d}{\partial dt}(\frac{\partial L}{\partial \phi}) = \frac{\partial dl}{\partial d\phi}$$</p>
<p>and</p>
<p>$$\frac{\partial d}{\partial dt}(\frac{\partial L}{\partial \theta}) = \frac{\partial dl}{\partial d\theta},$$</p>
<p>I get</p>
<p>$$mL^2 \ddot{\phi} = mgL\sin \phi - r_{\phi}\dot{\phi}$$</p>
<p>and</p>
<p>$$\frac{mh^2}{3}\ddot{\theta} = k\theta - r_{\theta}\dot{\theta},$$</p>
<p>where $r_{\phi}$ and $r_{\theta}$ are the damping forces on the pendulum and torsional coil respectively.</p>
<p>I'm reasonably certain that this is incorrect? This is a coupled system meaning that there should be variables from one in the equation of the other. I'm not entirely sure how to get that. Any pointers in the correct direction would be appreciated.</p>
<p>If anything is unclear in my post (such as what variables are or something messed up in latex), please let me know and I will try to correct/clarify.</p> | 6,252 | [
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<p>Why <a href="http://en.wikipedia.org/wiki/Gauss%27s_law" rel="nofollow">Gauss' law</a> is applied? Why is there a need of finding electric field by Gauss' law if we can find the electric field through Coulomb's law?</p>
<p>or has it got more applications than Coulomb's law?</p> | 6,253 | [
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<p>Who proposed the <a href="http://www.google.de/search?q=bulk+edge+correspondence" rel="nofollow">bulk-edge correspondence principle</a>?</p>
<p>The principle is often quoted in counting the number of zero energy states localized on the interface between two insulators with distinct band topology. However, I could not retrieve who was the first to say that.</p> | 6,254 | [
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<p>It seems to me that diffraction gratings are completely described by the double slit experiment-why then is it called a diffraction grating?</p> | 6,255 | [
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<p>Since the medium in which light propagates is spacetime, would light be able to exist if spacetime did not exist? Is this like one of those chicken/egg problems, or can light be thought of as a legitimately independent entity? This might be bordering on an philosophical question, so if it is, let me know and I'll delete it.</p> | 6,256 | [
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<p>I am trying to follow <a href="http://arxiv.org/abs/cond-mat/0503717" rel="nofollow">this paper</a> and track the dynamics of vortex motion on a discrete (square) lattice. The idea is to simulate the time evolution of the Gross-Pitaevskii (GP) equation, which reads (in rescaled units)</p>
<p>$$i \partial_t \psi = - \nabla^2 \psi + |\psi|^2 \psi.$$</p>
<p>As initial condition one takes a wavefunction of the form of Eq. (10) in the above-mentioned paper, </p>
<p>$$\psi(z) = \prod_{z_+ \in [Z_+]} \frac{(z-z_+)}{|z-z_+|}
\prod_{z_- \in [Z_-]} \frac{(z^*-z_-^*)}{|z-z_-|}$$</p>
<p>where $z=x+iy$ are complex coordinates. This wavefunction describes a set of vortices (V) with coordinates $z_+$ in $Z_+$ and a set of anti-vortices (AV) with coordinates $z_-$ in $Z_-$. This is equivalent to using a wavefunction of the type $e^{i \phi}$ (polar coords.) for vortices and $e^{-i \phi}$ for anti-vortices, with $\phi$ the polar angle. One can calculate the velocity as gradient of the phase of the complex wavefunction. An example of the associated velocity field, for a V at (x,y)=(0,2.5) and AV at (0,-2.5) is shown below.</p>
<p><img src="http://i.stack.imgur.com/nLZqZ.png" alt="velocity-lines"></p>
<p>This should correspond to Fig. 1 in the paper. However, there are several things I do not understand.</p>
<p>First, why do the authors say that they cannot look at the dynamics of just one vortex, but must include an anti-vortex in the initial conditions to satisfy the periodic boundary conditions?</p>
<p>Second, apart from the initial V-AV pair, they also seem to include "a few mirrow images with respect to the boundary". However, they do not specify how many nor where they are positioned. (I assume that the mirrow image of a V is and AV and vice-versa, please correct me if I'm wrong).</p>
<p>Lastly, they mention "evolving the system in imaginary time" in order to "cool it down" (although no temperature is included in the model!) and then "turning on a superflow" and continuing the evolution in real-time. I though all was needed was the discretization of the GP equation and following its evolution given the initial wavefunction and periodic boundary conditions ($\psi(\text{left margin})=\psi(\text{right margin})$ and $\psi(\text{top margin})=\psi(\text{bottom margin})$).</p>
<p><strong>Edit</strong></p>
<p>Following the suggestion of @BebopButUnsteady, I repeated the 'unit' of my system, the V-AV pair, until obtaining the following tiling pattern (vortices are circles, AVs are squares)</p>
<p><img src="http://i.stack.imgur.com/OQSXa.png" alt="tiles"></p>
<p>Let us first look at the real (left) and imaginary (right) parts of the wavefunctions for the initial V-AV pair:</p>
<p><img src="http://i.stack.imgur.com/ECgmG.png" alt="initial-boundaries"></p>
<p>One can see that, especially in the imaginary part, the values on the boundaries are not equal. Now we start adding copies of the "unit cell" along the x axis, and plot a cut though the imaginary part of the wf:</p>
<p><img src="http://i.stack.imgur.com/VVDSm.png" alt="boundaries-tiled"></p>
<p>It appears that, as we increase the number of copies, the values at the ends get closer and closer to 0, which would be the ideal periodic case. Of course this is just numerics, I have no formal proof yet that this procedure actually converges.</p> | 6,257 | [
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<p>The intensity of particles come to sea level depend to the zenith angle $I=Iv (\cos(z))^n$, where $z$ is the zenith angle and $n$ depend to the particle.
I know the vertical intensity depend to the energy or momentum of the particle, for exemple, to muons, $n$ is about 2 and vertical intensity is $0.0094\, \mathrm{cm}^{-1}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ to $>0.35 \mathrm{GeV/c}$.
I would like to know what is the vertical intensity and the $n$ for electrons, positrons, protons and hadrons. </p> | 6,258 | [
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<p>A simple question about notation of <a href="http://arxiv.org/abs/hep-th/9803265" rel="nofollow">Moore Nekrasov and Shatashvili</a> which makes me confused. Page 3, the authors rearranged the action into a novel form.</p>
<p>For D=3+1,5+1,9+1 respectively, the path integral is
$$
I_D=(\frac{\pi}{g})^{\frac{(N^2-1)(D-3)}{2}} \frac{1}{{\rm{Vol(G)}}}\int d^DXd^{\frac{D}{2}-1}\Psi e^{-S} $$
where $S=\frac{1}{g}(\frac{1}{4}\Sigma {\rm{Tr}}[X_\mu,X_\nu]^2+\frac{i}{2}\Sigma {\rm{Tr}}(\bar{\Psi}\Gamma^\mu[X_\mu,\Psi]))$</p>
<p>The authors rearrange the fields into the following form:
$$ \phi=X_D+iX_{D-1} \\
B_j=X_{2j-1}+iX_{2j} ~~~~~~(\rm{for ~~~j=1,2...D/2-1})\\
\Psi \rightarrow \Psi_a=(\psi_j,\psi_j^{\dagger}), \vec{\chi
},\eta$$
$B_j$ are often denoted as $\mathbf{X}=\{X_a, a=1,....D-2\}$</p>
<p>My question is what the rearrangement of fermion field means? For D=4, a=2, j=1 ,$\chi$ has one component, before arrangement, we have a Dirac spinor $\Psi$, and what do we have after arrangement? are $\chi$ and $\eta$ Weyl spinors?</p>
<p>If we expand formula (2.4) using nilpotent symmetry (2.3), why no such terms like $\chi^{\dagger}[\bullet,\bullet]$?</p>
<p><strong>Edit:</strong> From some related paper <a href="http://arxiv.org/abs/hep-th/9810035v2" rel="nofollow">Kazakov Kostov and Nekrasov</a>, the rearrangement is clear while there are some other puzzles. From KKN, first rewrite matrix model into complex fermion form, which is the formula (2.1) of KKN
$$\frac{1}{\rm{Vol(G)}}\int dXd\lambda \exp(\frac{1}{2}\Sigma _{\mu<\nu}\rm{Tr}[X_\mu,X_\nu]^2+ )+\rm{Tr}\bar{\lambda}_\dot{\alpha}\sigma_\mu^{\alpha \dot{\alpha}}[X_\mu,\lambda_\alpha]$$
Two complex fermions $\lambda$ can be written as four real fermions $\chi$, $\eta$,$\psi_\alpha$, $\alpha=1,2$,
$$ \lambda_1=\frac{1}{2}(\eta-i\chi),\\
\lambda_2=\frac{1}{2}(\psi_1+\psi_2),\\
\bar{\lambda}_{\dot{a}} =\sigma_2^{a\dot{a}}\lambda_a^*,~~~s=[X_1,X_2]$$
Using the following nilpotent symmetry:
$$ \delta X_\alpha=\psi_\alpha,~~~\delta\psi_\alpha=[\phi,X_\alpha] ,\\
\delta\bar{\phi}=\eta,~~~\delta \eta [\phi,\bar{\phi}],\\
\delta\chi=H,~~~\delta H=[\phi,\chi],\\
\delta \phi=0$$ </p>
<p>KKN claims that the action can be written as formula (2.5),
$$ S=\delta \left( -i\rm{Tr} \chi s -\frac{1}{2} \rm{Tr}\chi H - \Sigma_a\psi_a[X_a,\bar{\phi}] -\frac{1}{2} \eta[\phi,\bar{\phi}]\right) $$</p>
<p>I found that there is some inconsistence between formula (2.1) and formula (2.5).
Look at fermionic part that consist $X_2$ in (2.5) which is proportional to $i\rm{Tr} \chi[\psi_1,X_2] + \rm{Tr}\psi_2[X_2,\eta] $. If we start from (2.1), fermionic part that consist $X_2$ should be proportional to $ \bar{\lambda}_\dot{1}\sigma_2^{2\dot{1}}[X_2,\lambda_2] +\bar{\lambda}_\dot{2}\sigma_2^{1\dot{2}}[X_2,\lambda_1] $. Further using definition of $\bar{\lambda}_\dot{a}=\sigma_2^{a\dot{a}}\lambda_a^*$, we always get terms of form $\lambda_1^*[X_2,\lambda_1]$ and $\lambda_2^*[X_2,\lambda_2]$ which can not consist $i\rm{Tr} \chi[\psi_1,X_2] + \rm{Tr}\psi_2[X_2,\eta] $.</p>
<p>Anything wrong here? Some comments and references are welcoming.</p> | 6,259 | [
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<p>What is the importance of deriving the results of perturbation theory in condensed matter physics in terms of spectral functions ?</p> | 6,260 | [
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<p>I would like to understand better the role of Higgs mass in the gravitational interaction. I read here in some posts, that the Higgs mass has nothing to do with gravitation. Is it justified to say that? Gravitation acts on "energy" rather than mass but it is only mass, among the physical observables, which provides an equivalent to energy in contrast to electrical charge for example. </p>
<p>Hadron masses are generated by QCD but doesn`t one need still Higgs-massive quarks for this? If so, planet masses are connected to the Higgs mass.</p>
<p>In the picture of general relativity space-time curvature causes gravitation but remembering Kepler`s laws, why should be "Higgs-based" mass unessential to gravitation, where it is the correct input, describing the orbits of planets etc.?</p>
<p>Sorry, that I mix a few questions and it is a bit unstructured.</p> | 6,261 | [
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<p>As a physics bsc student, I have a very limited knowledge of QM:</p>
<p>Dirac formalism, Schrodinger equation and simple solutions (oscillators, particle in a given potential, hydrogen-like atom etc). There are more advanced QM courses I can take, but I find more interest in QFT.</p>
<p>What are the prerequisite subjects of non-QFT QM courses I should learn before starting with QFT?</p> | 443 | [
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<p>It is well known that the negative cosmological constant of AdS spacetime can act like a confining potential. That is, in contrast to asymptotically flat spacetime, in an asymptotically AdS spacetime <em>massive particles cannot escape to infinity</em>. However, massless particles can escape to infinity and actually do so in a finite time.</p>
<p><strong>As tachyons travel faster than massless particles, is it true that all tachyons can escape to infinity as well?</strong></p>
<p>If the answer is yes, then I have some trouble understanding the following argument in a paper by Horowitz on holographic superconductivity (see <a href="http://dx.doi.org/10.1088/1742-6596/484/1/012002" rel="nofollow">here</a>). Here, the considered action of the holographic dual to the superconductor (the bulk action) is</p>
<p>$S=\int d^4x\sqrt{-g}\left(R+\frac{6}{L^2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-|\nabla\Psi-iqA\Psi|^2-m^2|\Psi|^2\right)$,</p>
<p>i.e., a complex scalar field $\Psi$ and a Maxwell field $A_t$ (electric) coupled to gravity. The effective mass for $\Psi$ following from this action is
$m_{eff}^2=m^2+q^2g^{tt}A_t^2$.</p>
<p>In constructing this dual theory, Horowitz argues that "<em>In AdS, the charged particles cannot escape, since the negative
cosmological constant acts like a confining box, and they settle outside the horizon.</em>" (Of course, only for particles for which the sign of the charge is the same as that of the black hole.)</p>
<p>However, the case considered subsequently is $m^2=-\frac{2}{L^2}$, which implies that $m_{eff}^2<0$ since also $g^{tt}<0$. Hence they consider tachyons!</p>
<p><strong>Being tachyons, how can these particles settle outside the horizon? Why would they be confined by the cosmological constant rather than escape to infinity?</strong></p>
<p>EDIT: right now I'm actually questioning my claim that all tachyons automatically travel faster than light...</p> | 6,262 | [
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<ol>
<li><p>Recently I am studying electromagnetic waves, and I am wondering why it is formed by acceleration of electric charges? </p></li>
<li><p>Can the EM waves be formed by other movements of electric charges, such as at a constant velocity? </p></li>
<li><p>And also, is there a way to visualize the formation of EM waves?</p></li>
</ol> | 219 | [
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<p>Special relativity was well established by the time the schrodinger equation came out. Using the correspondence of classical energy with frequency and momentum with wave number, this is the equation that comes out, and looks sensible because this is of the form of a wave equation, like the one for sound etc, except with an inhomogeneous term</p>
<p>$$\nabla^2 \psi - 1/c^2 \partial_{tt} \psi = (\frac{mc}{\hbar})^2\psi$$</p>
<p>Instead, we have schrodinger's equation which reminds of the heat equation as it is first order in time. </p>
<p><strong>EDIT</strong> Some thoughts after seeing the first posted answer</p>
<p>What is wrong with negative energy solutions? When we do freshman physics and solve some equation quadratic in, say, time... we reject the negative time solution as unphysical for our case. We do have a reason why we get them, like, for the same initial conditions and acceleration, this equation tells usthat given these iniial velocity and acceleration, the particle would have been at the place before we started our clock, since we're interested only in what happens <em>after</em> the negative times don't concern us. Another example is if we have areflected wave on a rope, we get these solutions of plane progressive waves travelling at opposite velocities unbounded by our wall and rope extent. We say there is a wall, an infinite energy barrier and whatever progressive wave is travelling beyond that is unphysical, not to be considered, etc.</p>
<p>Now the same thing could be said about $E=\pm\sqrt{p^2c^2+(mc^2)^2}$ that negative solution can't be true because a particle can't have an energy lower than its rest mass energy $(mc^2)$ and reject that. </p>
<p>If we have a divergent series, and because answers must be finite, we say.. ahh! These are not real numbers in the ordinary sense, they are <em>p-adics</em>! And in this interpretation we get rid of divergence. IIRC Casimirt effect was the relevant phenomenon here.</p>
<p><strong>My question boils down to this</strong>. I guess the general perception is that mathematics is only as convenient as long as it gives us the answers for physical phenomenon. I feel this is sensible because nature can possibly be more absolute than any formal framework we can construct to analyze it. How and when is it OK to sidestep maths and not miss out a crucial mystery in physics. </p> | 6,263 | [
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<p>I would like to clarify something that mixes cosmology and relativistic effects. Maybe I'm not understanding something or maybe there a difference of vocabulary between the cosmological and the relativistic people. </p>
<p>If you ask a cosmologist at which scale the relativistic effects appears (for example in N-body simulations), he will probably answer you that a problem may occur at the scale of the horizon (so at the scale of the observable Universe). If you ask the same question to a specialist of the backreaction or a specialist of general relativity I suspect that he will answer you that a problem may occur at the scale of matter inhomogeneities (so at the scale of the cosmic web, filaments and voids). </p>
<p>My question is : why are there these 2 point of views ?</p>
<p>I suspect that is because a cosmologist hypothesizes an homogeneous space-time background and apply a perturbation theory on this background whereas for a people from general relativity there is no such background. Is it the answer or I don't understand something ?</p> | 6,264 | [
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<p>I couldn't find anything about the parity of an electron. Neither in the german, nor in the spanish and nor in the english version of Wikipedia. </p>
<p>I only found one sentence in the parity article of Wiki:</p>
<blockquote>
<p>One way to fix a standard parity operator is to assign the parities of three particles with linearly independent charges B, L and Q. In general one assigns the parity of the most common massive particles, the proton, the neutron and the electron, to be +1. - <a href="https://en.wikipedia.org/wiki/Parity_%28physics" rel="nofollow" title="it's under 'Fixing the global symmetries'">Wiki</a></p>
</blockquote>
<p>But I cannot find other sources to confirm it.</p>
<p>I do not need it for a special exercise, it's just to understand the whole thing...</p> | 6,265 | [
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<p>I want to transmit sunlight over a distance of approx 4 m via optical fiber. Ideally I want to light up the whole house with sunlight.</p>
<p>I was reading that the largest diameter for an optical fiber may not be the right way to solve this issue. I was wondering if there some sort of equation where I can insert the length that I would like to carry light over (length is a variable) and maybe the number of cores of optical fiber (also a variable) with the output of this equation being the efficiency or loss in transmission.</p>
<p>This way I can integrate all variables in one equation and optimize against the cost to procure.</p> | 6,266 | [
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<p>I was reading through the first chapter of Polyakov's book "Gauge-fields and Strings" and couldn't understand a hand-wavy argument he makes to explain why in systems with discrete gauge-symmetry only gauge-invariant quantities can have finite expectation value. This is known as <a href="http://en.wikipedia.org/wiki/Elitzur%27s_theorem" rel="nofollow">Elitzur's theorem</a> (which holds for continuous gauge-symmetry). </p>
<blockquote>
<p>Polyakov says : <em>[...] there could be no order parameter in such systems (in
discrete gauge-invariant system) [...], only gauge invariant quantities are nonzero. This follows from the fact, that by fixing the values of $\sigma_{\mathbf{x},\mathbf{\alpha}}$ at the boundary of our system we do not
spoil the gauge invariance inside it.</em></p>
</blockquote>
<p>Here $\sigma_{x,\alpha}$ are the "spin" variables that decorate the links of a $\mathbb{Z}_2$ lattice gauge theory. I would like to understand the last sentence of this statement. Could anyone clarify what he means and why this implies no gauge-symmetry breaking ?</p> | 6,267 | [
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<p>In the derivation of <a href="http://en.wikipedia.org/wiki/Lorentz_transformation" rel="nofollow">Lorentz transformations</a>, the <a href="http://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations" rel="nofollow">Wikipedia article</a> mentions a couple of times that the linearity comes from the <a href="http://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#Galilean_and_Einstein.27s_relativity" rel="nofollow">homogeneity of space</a>. I am looking for a thorough explanation on this. </p> | 6,268 | [
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<p>In my readings of Mirman (1995), "Group Theory: An Intuitive Approach", on p.35 he asks me to consider a so-called "water group" that has 4 transformations. I'll list them for completeness, but I'm only concerned about two of them. </p>
<ol>
<li>The identity element</li>
<li>Reflection in the plane element</li>
<li>Rotation by $\pi/2$ about a line through "O" perpendicular to the line joining the two "H"s</li>
<li>Reflection through the line described in 3. </li>
</ol>
<p>Now for some context. This book is primarily a book about group theory, but the author likes to use examples from physics in order to provide examples that yield a concrete picture of what the formalism can and can't describe. </p>
<p>Leading up to the example of the water group, I calculated the Cayley table for the symmetry group $S_3$; the dihedral group $D_3$; and then the table for an isosceles triangle as an example of symmetry breaking from the equilateral triangle in the prior example. All of that I get. </p>
<p>What I don't understand is how a quarter-turn rotation leaves the figure invariant in the water example. I tried it from two different angles: </p>
<ol>
<li><p>using cylindrical coordinates, there is only one possibility: namely a 1/4 of a full turn in the plane of the figure which yields a picture that is different than what you started with. </p></li>
<li><p>or, using spherical coordinates, you can perform the rotation out of the plane but that leaves you with a top-down view and the three "atoms" following in a straight line which isn't what you started with, either. </p></li>
</ol>
<p>So, I'll ask again: is there a way to do a 1/4 turn of an isosceles triangle that leaves the original shape unchanged? </p> | 6,269 | [
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<p>This is a follow up of this <a href="http://physics.stackexchange.com/questions/119886/how-to-calculate-roll-yaw-and-pitch-angles-from-3d-co-ordinates-euler-angles">question</a> :</p>
<p>I have the rotation matrix </p>
<p>$$
\left( \begin{matrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}\\
\end{matrix}\right)
$$</p>
<p>I'm using pre-multiplying rotation matrix (that operates on column vectors) for <a href="http://en.wikipedia.org/wiki/Tait-Bryan_angles#Intrinsic_rotations" rel="nofollow">intrinsic rotations</a> (i.e. I make rotations about the axes of the plane that rotates). And since the fixed frame is my reference frame ---</p>
<p>$$
\left( \begin{matrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{matrix}\right)
$$</p>
<p>My rotation matrix is nothing but the column unit-vectors of the axes of the rotated frame, i.e.</p>
<p>$$
\left( \begin{matrix}
x_{1} & x_{2} & x_{3}\\
y_{1} & y_{2} & y_{3}\\
z_{1} & z_{2} & z_{3}\\
\end{matrix}\right)
$$</p>
<p>So therefore I have the values of a11,a12,a13,a21,a22,a23,a31,a32,a33 as x1, x2, x3, y1, y2, y3, z1, z2, z3.</p>
<p>Now if I consider a particular set of rotation (say X first, then Y , then Z), with the corresponding <a href="http://en.wikipedia.org/wiki/Tait-Bryan_angles#Tait.E2.80.93Bryan_angles" rel="nofollow">Tait-Bryan angles</a> --- a,b and c. My rotation matrix will be the following. <a href="http://en.wikipedia.org/wiki/Tait-Bryan_angles#Rotation_matrix" rel="nofollow">Rx(a)*Ry(b)*Rz(c)</a> ---</p>
<p>$$
\left(\small{ \begin{matrix}
\cos(b)\cos(c) & -\cos(b)\sin(c) & \sin(b)\\
\cos(a)\sin(c) + \cos(c)\sin(a)\sin(b) & \cos(a)\cos(c) - \sin(a)\sin(b)\sin(c) & -\cos(b)\sin(a)\\
\sin(a)\sin(c) - \cos(a)\cos(c)\sin(b) & \cos(c)\sin(a) + \cos(a)\sin(b)\sin(c) & \cos(a)\cos(b)\\
\end{matrix}} \right)
$$</p>
<p>Now if I have to solve for the above angles a, b & c (pitch, yaw and roll), I basically have nine equations but three unknowns. Following are the equations ---</p>
<pre><code>a11 = cos(b)∗cos(c)
a12 = −cos(b)∗sin(c)
a13 = sin(b)
a21 = cos(a)∗sin(c)+cos(c)∗sin(a)∗sin(b)
a22 = cos(a)∗cos(c)−sin(a)∗sin(b)∗sin(c)
a23 = −cos(b)∗sin(a)
a31 = sin(a)∗sin(c)−cos(a)∗cos(c)∗sin(b)
a32 = cos(c)∗sin(a)+cos(a)∗sin(b)∗sin(c)
a33 = cos(a)∗cos(b)
</code></pre>
<p><a href="http://mathworld.wolfram.com/EulerAngles.html" rel="nofollow">This</a> is where I learnt about it.</p>
<p>Now I am using a non-linear least squares curve fitting method to solve the above set of over-determined equations. There are two major problems that I am encountering</p>
<ol>
<li>The final values of a,b, c change as I change the initial values in the iterative algorithm. I get different results if I start from [50 50 50] and different results with [0 0 0].</li>
<li>Secondly I don't think that the values obtained are correct; since the angles of pitch, yaw & roll seem pretty much different in the video. I'm using the <a href="http://stackoverflow.com/questions/24382626/how-to-solve-an-overdetermined-set-of-equations-using-non-linear-lest-squares-in">lsqcurvefit</a>(click for the question I asked on stackoverflow) command in <a href="http://www.mathworks.in/help/optim/ug/lsqcurvefit.html" rel="nofollow">Matlab</a>. (click for documentation)</li>
</ol>
<p>I have been on this problem of how to calculate pitch, yaw & roll for quite some time now. I think this post will give you all the details of what I have tried. I need your help to know if what I am doing is the best approach to tackle my problem. If so, please point out what is wrong in my method. If you think there are other simpler methods, please let me know about them. I'm sure there has to be a better method, since this seems like a pretty simple thing to do.</p>
<p>Should I change my Matlab algorithm? Anyone know any special Matlab/Mathematica toolbox that calculates the yaw, pitch, roll?</p>
<p>Thanks!</p> | 6,270 | [
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<p>This is a question with a philosophical, as well as physical, flavor.</p>
<p>Why should a physical principle (or a description of one), be applicable to different systems that can be in different positions in space and time?</p>
<p>In other words why should there be such (<em>modulo</em>) equivalence classes (with respect to a physical principle)? Why not a "<em>custom (physical) principle per case</em>"?</p>
<p>Let me give one simple example. Newton's laws are applicable to my desk, to my table, to your desk, to your table and to someone's table, say 1 year from now (or 1 year ago).</p>
<p>Why is this? Why should the same principle be aplicable to such different systems (different in the sense stated above)?</p>
<p>Anyone know how to approach such a question?</p>
<p><strong>Note in order to avoid misunderstandings</strong>:</p>
<p>regardless of the tags of this question, this is <strong>not</strong> specifically about <em>covariance</em>, the tag was used, by an editor, because the original tag <em>laws-of-physics</em> was removed for other reasons</p> | 179 | [
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<p>Given a hot air balloon of radius 10 meters and negligible mass, calculate the maximum weight it can carry if the density of outside air is 1.2 $\frac{kg}{m^3}$ and the density of inside air is 0.9 $\frac{kg}{m^3}$. </p>
<p>Edit:
My attempt:
Since the maximum weight the balloon can carry is determined by the relation :
$Q=mg$
I somehow need to set bouyancy force equal to $Q$.
Is this correct and if not, how to do it correctly?</p> | 6,271 | [
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<p>If the moon was alone in space and covered in X feet of water and it began to rotate would it displace to a bias along the equator? </p> | 6,272 | [
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<p>Let's have <a href="https://en.wikipedia.org/wiki/K%C3%A4ll%C3%A9n%E2%80%93Lehmann_spectral_representation" rel="nofollow">Kallen–Lehmann spectral representation</a> for the scalar theory:
$$
\tag 1 D(p) = \int \limits_{0}^{\infty} d(\mu^{2})\frac{\rho (\mu^{2})}{p^{2} - \mu^{2} + i\varepsilon}.
$$
We can represent $(1)$ in a form
$$
D (p) = D_{free}(p, m)Z + \int \limits_{m^{2}}^{\infty}d(\mu^{2})\rho (\mu^{2})D(p, \mu),
$$
where $\mu$ refers to the mass of the multiparticle states and $m$ refers to the mass of one-particle free state. From here there is interesting result: it can be showed that $0 \leqslant Z \leqslant 1$.</p>
<p>How to get analogous result for the arbitrary spin field (or at least for fermions with spin $\frac{1}{2}$ and for photons)?</p>
<p>I know the general structure of the propagator of the free theory (for simplicity I assume massive case): for field $\Psi $ with spin $s$
$$
\hat {\Psi}_{l} (x)= \sum_{\sigma = -s}^{s}\int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2E_{\mathbf p}}}\left( u^{\sigma}_{l}(\mathbf p )e^{-ipx}\hat {a}_{\sigma}(\mathbf p ) + v^{\sigma}_{l}(\mathbf p )e^{ipx}\hat {b}^{\dagger}_{\sigma}(\mathbf p ) \right)
$$
it will be</p>
<p>$$
D_{lm}(p) = \frac{F_{lm}(p)}{p^{2} - m^{2} + i\varepsilon}, \quad F_{lm} = \sum_{\sigma}u^{\sigma}_{l}(\mathbf p )\left(u^{\sigma}_{m}(\mathbf p) \right)^{\dagger}.
$$ </p>
<p>For the derivation of spectral representation for the scalar field see the reference above. Here also you can see the problem with this derivation which appears for the case of field with nonzero spin: it is impossible to introduce the scalar density function $\rho (\mu^{2})$. </p>
<p>Also there is bigger problem: there was using relation $\langle |[\hat {\Psi}(\mathbf x , t), \frac{d}{dt}\hat {\Psi}^{\dagger}(\mathbf y, t)]|\rangle = i\delta (\mathbf x - \mathbf y)$ for getting relation $0 \leqslant Z \leqslant 1$ (look to the Weinberg's "QFT" (chapter "Kallen–Lehmann spectral representation" of the section "Nonperturbative methods") or to Greiner's "Field Quantization" (pp. 278-282)). But this relation is correct only if one field is canonical conjugated to another (one field is canonical coordinate while the another one is canonical momentum). But for the arbitrary spins the canonical momentums must be chosen "individually".</p> | 6,273 | [
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<p>Which mass of the particle is the source of gravitational field? If we define mass as a pole of the propagator, and calculate loop corrections to the pole we get infinities. Now the way we get rid of these infinities defines what is our renormalized propagator and what is its pole. What is the mass of the particle is dependent on which renormalization scheme we choose. Now, gravity field knows about everything that happens to the particle (as any energy gravitates), all contributions of whatever virtual particles contribute to the mass of particle (up to very short distances). </p> | 6,274 | [
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<p>Not sure if this is a physics or chemistry question. But if the motion of atoms and it's particles can be described by quantum mechanics, then is there a software that simulate full atoms and it's boundings, in a way you can visualize them, and that can be used, for instance, to throw 2 molecules together and watch them reacting?</p> | 6,275 | [
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<p>This is thought experiment. I couldn't get a good answer because I keep getting negative mass.</p>
<p>Gauss's Law say that eletric field is proportional to charge, how much charged is enclosed. Newton's gravitational pretty much says the same thing. More mass, strong gravitational strength.</p>
<p>So to put it mathematically</p>
<p>$$\vec{E} \propto \sum Q_{en}$$</p>
<p>But charge can be negative, a negative sum of charge means the electric field is going <em>inwards</em>. </p>
<p>$$\vec{g} \propto \sum M_{en}$$</p>
<p>But mass can only be positive, but g is then propotional to the mass enclosed, which means it will go radially outwards. That doesn't make sense because we know gravity is radially inward for a spherical shape (like earth)</p> | 6,276 | [
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<p><strong>Problem/Solution</strong></p>
<p><img src="http://img221.imageshack.us/img221/4285/20841428g.jpg" alt="">!</p>
<p>I am deeply confused. </p>
<p>B) We know that </p>
<p>$x = 2\sin(3\pi t)$.</p>
<p>$x' = 6\pi\cos(3\pi t)$</p>
<p>So max <em>speed</em> is $6\pi$</p>
<p>$6\pi = 6\pi \cos(3\pi t)$</p>
<p>$\cos(3\pi t) = 1$</p>
<p>$3\pi t = 2\pi$ (reject 0 since t >0)</p>
<p>$t = 2/3$</p>
<p>But it isn't 2/3s. I understand where the solution is coming from, but I don't understand why I am wrong.</p> | 6,277 | [
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<p>I've seen the post on mathoverflow.SE asking almost the same question, and I have indeed flipped through said answers, but most are in a more general context ie quantum mechanics and do not provide a conceptual answer with physical interpretation. Anyone able to offer any insight, or even an example in the aforementioned context? ( More particularly the theory of representations for symmetry groups. )</p> | 6,278 | [
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<blockquote>
<p>A ball is thrown vertically upwards at $5\text{ m/s}$ from a roof top of $100\text{ m}$. The ball B is thrown down from the same point $2\text{ s}$ later at $20\text{ m/s}$. Where and when will they meet?</p>
</blockquote>
<p>Well at first I separated the fact that while ball B falls at say $t$ seconds, ball A's time is given by $t+2$ (due to it being thrown 2 seconds prior to ball B).</p>
<p>Given what's given above (the velocity and height)...I wrote down the following equations to find TIME, where lower a and b denote ball A and B:</p>
<p>$$\begin{align}s_a &= s_{0a} + u_a(t+2) + \frac{1}{2}a(t+2)^2 \\
s_b &= s_{0b} + u_bt + \frac{1}{2}at^2\end{align}$$</p>
<p>$S_a(0) = 100$ m, $S_b(0) = 100$ m, $u_a = 5$ m/s, $u_b = 20$ m/s, $a=-9,8$ m/s$^2$ </p>
<p>Now plugging those in gives:
$S_a = 100 + 5(t+2) + \frac12(-9.8)(t+2)^2$ and
$S_b = 100 + 20t + \frac12(-9.8)t^2$ </p>
<p>but i tried equalizing the above two equations with all the proper variables replaced by the values provided in the problem, but unfortunately i don't get the right answer.eventually knowing time i can answer to the "where" part of the question.</p>
<p>I'm lost, I'm not even sure if I tackled the problem correctly.</p> | 6,279 | [
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-0.01920895278453827,
-0.005926385521888733,
0.001991699216887355,
-0.03415742516517639
] |
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