Lagrangian depends on second derivative of field

In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu a}+\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi-\frac{\xi}{2}B^{a}B^{a}+B^{a}\partial^{\mu}A_{\mu}\,^{a}+\bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} $$ (for example in Peskin and Schröder equation 16.44) If you expand the last term (for the ghost fields) you get: $$ \bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} = -\bar{c}^{a}\partial^{2}c^{a}-gf^{abc}\bar{c}^{a}\left(\partial^{\mu}A_{\mu}\,^{b}\right)c^{c}-gf^{abc}\bar{c}^{a}A_{\mu}\,^{b}\partial^{\mu}c^{c} $$ And so, the Lagrangian has a term proportional to the second derivative of $c^a$. In this case, how does one find the classical equations of motion for the various ghost fields and their adjoints? I found the following equations of motion so far: $$ D_{\beta}\,^{dc}F^{\beta\sigma}\,^{c}=-g\bar{\psi}\gamma^{\sigma}t^{d}\psi+\partial^{\sigma}B^{d}+gf^{dac}\left(\partial^{\sigma}\bar{c}^{a}\right)c^{c} = 0 $$ $$ \sum_{j}\partial_{\sigma}\bar{\psi}_{\alpha,\, j}i\gamma^{\sigma}\,_{ji}-\sum_{\beta}\sum_{j}\bar{\psi}_{\beta,\, j}\left(gA_{\mu}\,^{a}\gamma^{\mu}\,_{ji}t^{a}\,_{\beta\alpha}-m\delta_{ji}\delta_{\beta\alpha}\right)=0 $$ $$ \left(i\gamma^{\mu}D_{\mu}-m\right)\psi=0 $$ $$ B^{b}=\frac{1}{\xi}\partial^{\mu}A_{\mu}\,^{b} $$ $$ \partial^{\mu}\left(D_{\mu}\,^{dc}c^{c}\right)=0 $$ $$ f^{abd}\left(\partial_{\sigma}\bar{c}^{a}\right)A^{\sigma}\,^{b}=0 $$ But it is the last equation that I suspect is false (I saw the equation $ D_\mu\,^{ad} \partial^\mu \bar{c}^d = 0 $ in some exercise sheet (http://www.itp.phys.ethz.ch/education/fs14/qftII/Series7-3.pdf Exercise 3) and I also saw the equation $D^\mu\,^{ad}\partial_\mu B^d = igf^{dbc}(\partial^\mu\bar{c}^b)D_\mu\,^{dc} c^c$ which I don't understand how they were derived.) EDIT: Thanks to Qmechanic's answer I was able to derive the correct equations of motions (as noted in the comment to that answer) but I still don't know where to "obtain" the last equation I mentioned which connects the auxiliary field with the ghost fields.


Inelastic nucleon-nucleon cross section at LHC energies

I am trying to reproduce the number of participants $N_{part}$ in Pb-Pb Collisions at LHC using a Glauber Monte Carlo simulation, specifically aiming to reproduce the values given in Phys. Rev. Lett. 106, 032301 (2011) by K. Aamodt et al. (the ALICE Collaboration). To do this, I am looking for the inelastic nucleon-nucleon cross section. A paper on the software written for the PHOBOS experiment (available on arXiv here) suggest that it is (at LHC energies) " _expected to be around $\sigma_{NN} = 72$ mb (with large uncertainty from the unknown elastic cross section)_ ". I have been using this web-interface provided by the GSI to get an estimate of $N_{part}$. The website suggests to use $60$mb at $\sqrt{s_{NN}} = 5500 \text{GeV}$. However, even using the 72mb suggested above, for 0%-5% centrality I only get about $376$ participant nucleons, compared with $N_{part} = 382.8$ for $2.76$ TeV published in the ALICE paper. I looked at the PDG website as well, but since (as hinted above) the plots do not show data for the elastic part of the cross-section, no value is provided for the energy I'm looking at. The fit funciton provided does not seem to give a reasonable value for LHC energies. > Am I right in assuming that we simply don't know the value of $\sigma_{NN}$ > yet and that it must thus be estimated for such calculations? If that is the > case, is there a common way to do so?


Eigen modes in a room

In a rectangular room 10*8*5 how can the eigen modes belonging to octave and third octave bands centered at 50Hz be found ? I have the formula to calculate the number of modes but don't have the idea how to find the modes. Do I need to use Matlab for it ?


Capacitance and Light Bulb And Dielectric Slab

An electric lamp having coil of negligible inductance connected in series with a capacitor and an AC source is glowing with certain brightness. How does the brightness of the lamp change on reducing the (i) capacitance, and (ii) the frequency ? What if the diaelectric slab is introduced between the capacitor.


Virgo SuperCluster and space time

Is the Virgo Supercluster affected at all by the expansion of space? What about the larger filament that Virgo resides in?I don't quite understand how gravity affects space time locally in groups of galaxies, and am wondering if the expansion of space is lumpy. It would be nice to know we are not all being stretched... :)


How to find the electric field of a charged tube?

Let's say there is a charged tube(cylinder with no top or bottom) with radius $a$, length $l$ and charge $q$ and a point which is collinear with the centre of the charged tube. Anyway, since we can think the tube as infinite charged rings we first should calculate the electric field of a single ring.(the distance between ring and point charge is $r$ and the distance between the centre of the ring and point charge is $d$) $$ dE= (k.dq)/r^2 $$ where $dq$ is infinitesimal charge. $$ dE_x=(k.dq)/r^2cos(\theta) $$ $$ dE_x=(k.d.dq)/(a^2+d^2)^{(3⁄2)} $$ $$ E_x=∫(k.d.dq)/(a^2+d^2)^{(3⁄2)} =(k.d)/(a^2+d^2)^{(3⁄2)}∫dq=kdq/(a^2+d^2)^{(3⁄2)} $$ Finally, we found the electric field of a single ring. However, I am stuck at how to integrate the equation to find the total electric field of the charged tube. How should I integrate it? P.S: It may seem like the other question I asked but they are different in concept of what the question is.


Speed of light and distance

Our measure of distance (the meter) is defined in terms of how far light in a vacuum travels in a specific time. When light travels through another medium, we say it travels at a different speed. Why not say it travels at the same speed, and the distance in the medium is different that the distance outside? Is it only because the 2nd explanation feels less intuitive than the first, or is there more to it?


energy of electrons outside an atom

The higher the quantum number(energy levels)m the higher the energy. What does the energy refers to? Kinetic energy, potential energy, or the total mechanic energy?


Stresses on an inclined plane

# Background This issue has been bothering me for a few days now. It's actually part of my homework, but I've already finished them and in a sense they're not part of the question here. What I would like to know is more of a theoretical question about this formulation. Now my professor's book and many online resources (sample) maintain the same mathematical relation, hence my dilemma. # Problem Consider the following: ![Problem](http://i.imgur.com/bTaKei0.png) Under axial loading, the stresses can now be computed for any section. The question posed is what if we make an imaginary section at an _inclined_ , its normal defined by an angle $\theta$. I will not write every detail here, but its clear that we can use $A$ as a placeholder for the area and we can define $\sigma_x$ to be $\sigma_x = \frac{F}{A}$ and therefore in the diagram below we will use only stress. ![Problem](http://i.imgur.com/u4PU86j.png?1) Now, I will refrain working with vector quantities, I will only use scalar relations. If you take a look at this triangle, its easy to see that the area is projected to the section and we can get the relationship: $$ A' = \frac{A}{\cos(\theta)} $$ Now the rest of the equations go by the books: $$ T = \frac{F}{A'} = \frac{F}{A}\,\cos(\theta) = \sigma_x\,\cos(\theta) $$ $$ T_n = T\,\cos(\theta) $$ $$ T_s = T\,\sin(\theta) $$ My dilemma starts here. First of all _notice_ how in each of the above expressions the value for the length of the object is not taken into consideration. Also, we're working with areas and forces hence the length itself is not part of the equation at all even as a _contained_ information. Now that would be all fine, if it were not for the following section(to simplify, I've taken the origin to be at the center of the object). ![Section C](http://i.imgur.com/CZYfjZA.png) Now as you can see this section of the object gets very interesting. Clearly the formula for $A'$ does not hold anymore because the area itself is shrinking on the left side. To make things even worse, notice how the area on the right appears. The two areas are clearly linked, in this case, the sum of the two should be equal to the original area - from the fact that the height appearing on the right is equal to the height subtracting from the left. Calculating the angle at which the breaking point is made, that is when the section plane cuts the corner is easily calculated using: $$ \tan(\theta_k) = \frac{l}{h} $$ Where $ \theta_k $ stands for the critical angle, just so we can refer to it. Now the question that is on my mind is what is the distribution of the stresses on this section. Clearly there is stress on the left, there's stress on the right. Now I am confused as to the distribution of stress at this angle. # My Work Now, I've attempted a solution here. First, I am in a tight spot between assumptions. Clearly the original force $F$ must still be present here. But the force distribution is confusing me at this point. The best idea I could come up with, is the following. I would be grateful for your review. ## FBD ![Problem 3](http://i.imgur.com/0DIhQMU.png) I've only written here $F_1$ and $F_2$ for visualization purposes, because personally I don't think that we will need the $F_2$. Knowing that $F_1 = F$ it's obvious that they will cancel each other out, and they will also cancel the moments they create against the newly formed centroid on the trapezoid. Hence I think the only remaining task is to compute the areas of each individual piece and compute stresses. ## Areas Computing the individual areas for the cuts, since we won't need the area at where supposedly $F_2$ was located, we will ignore that for the time being. Though it will come naturally as we try to find the left area. The inclined plane area is the easiest to find, clearly, if we project the length $l$ to the top of the smaller rectangle we can see that using trigonometric relations we can find: $$l\,'=\frac{l}{\sin(\theta)}$$ Multiply that by the base and you've got the area $A'=b l\,'$. To find the areas of the left and right sections one must notice that the amount we deviate from the original height we add to the right side. So if we're to project the length like we did last time we get this: ![Area2](http://i.imgur.com/9m1MbQ0.png) From the figure above it's clear that the following relation holds true: $$ h = h' + 2\,h_2 $$ From trig we know $h'=l\cot(\theta)$. From this, $h_2$ is defined by: $$ h_2 = \frac{1}{2}(h - l\cot(\theta)) $$ Now $ h_1 = h - h_2 $ and after some mathmagic we should get the following: $$ h_{1/2} = \frac{1}{2}(h\pm l\,\cot(\theta)) $$ Now let's multiply this by the base and we've got ourselves $A_1 = h_1\,b$. ## Stress Now we have two stresses, namely $\sigma_{x_1}$ and $T$. Let's analyze the relations. The forces clearly have an eccentric deviation from the centerline. Now, is this going to be a problem? I seem to think that it won't be, and I base my logic on the fact that they will cancel the bending moments as we said above. The formulas are: $$\sigma_{x_1} = \frac{F}{A_1}$$ $$T = \frac{F}{A'} $$ Now obviously these are general equations, one could always dig deeper and I believe we could even express these stresses as a function of elementary stresses, namely $\sigma_x$ and some form of this other _tangential_ stress $F/(l\,b)$. ## Appendix To be completely accurate, the formula for $\theta_k$ seems to be a bit more complex, since there are four corners on the cut (in our case). Knowing that paired corners are symmetric to the x-axis we can use this equation: $$\theta_k(n) = |\arctan(\frac{l}{h}) + \pi\,n|$$ where $n\in(-2,-1,0,1)$. Is this ugly or is it just me? # Closing I hope to get some review on the ideas here, I'd really love to know how would this get solved correctly. And please someone, help me fix the appendix formula.


A Puzzle about $SO(3)$

Lie algebra of nonabelian group is $[T^a,T^b]=if^{abc}T^c$. For $SO(3)$ case, is the representation $T^a_{ij}=-i\epsilon^{aij}$ fundamental or adjoint? The fundamental representation is defined as identifying $T^a$ the original generator. The adjoint representation is defined as identifying $T^a$ as structure constant. For $SO(3)$ case, generators seem to be same as structure contant. Reference:   1. Srednicki QFT


what is the basic form of the 'fire'?

> **Possible Duplicate:**   >  Is fire matter or energy? What is the basic form of fire? physics defines every entity by a basic form either solid or liquid or as a gas, example: water is liquid, ice as solid, water vapor: gas so what is fire? when I queried is 'Plasma a fire?' I landed in below link, where it is said that fire isn't plasma, but not defined which basic form it belongs to! the link: Is fire plasma? What I understand is, it's a form of energy caused by reaction of gases(?) so it's basically a gas! is it true?


How does weight add up to press on things?

I think I understand how pressure works with gases. More molecules bouncing around -> more random impacts -> stronger force. But I realized to my embarrassment that I don't understand how solid things press on each other, on a molecular level. Say I put a block of iron on my head. If I put another one on top of it, I feel twice the weight. The two blocks together can tear through thin paper where one block can't. But the contact between my head (or paper) and the blocks is just a very thin layer of atoms of the lower block's structure. If the lower block doesn't move when I put the upper one on it, what causes this thin layer to "press" on my head (or paper) more? When the two blocks together tear through thin paper, where does the force come from that acts on the paper molecules - it can't be gravity from the upper block, right? And how come that whatever this source of pressure is only depends on the weight of the upper block, and not on what it's made out of, iron or wood?


What is the result of applying a fourier transform n times to a distribution?

For a function applying the fourier transform twice is equivalent to the parity transformation, applying it three times is the same as applying the inverse of the fourier transform, and applying four times, is the identity transformation. Wikipedia has a good description of this. What do we get when we apply the transform multiple times to a distribution? For example applying it twice would look like this: $ \mathscr F \mathscr F T(f) = T(\mathscr F \mathscr F f) = T( f_-) = ? $ Where $f_- = f(-x)$. Here $\mathscr F$ is the fourier transform $T$ is a distribution, and $f$ is a function.


Solution of multiple materials

## Introduction I'm trying to make some strawberry liqueur. The recipe I got says I should put alcohol, strawberries and sugar in a bowl and leave it soaking for 4-5 weeks. I read that making liqueur is not a chemical reaction, but only a solvent (mostly alcohol) acting on some fruit, herb or nut. So, intuitively, I would add the sugar only after the soaking time. I wonder if that makes a difference. ## Question If I dissolve material A in alcohol until it reaches saturation, how much of material B can I still dissolve in the solution?   * Nothing   * As much as with pure alcohol   * Something in between None of the materials are gaseous and both are soluble in alcohol.


Speed of air through valve

Right now I'm working on a hot air balloon in Physics formulas. At the moment it's flying up like it should, but once it is at a certain height we want it to go back down. After some research about hot air balloons we found out that they drop down by opening a valve at the top. So what we want to do is once the balloon gets at a certain height it will open the valve and drop down again. We can get the size of the valve and that it opens at a certain height. How can we calculate the speed of cold air going into the valve? We know the speed of the balloon, the size, temperature of the air in and outside of the balloon, air pressure etc. Is there some kind of formula we can use to calculate how fast the cold air will flow in and then cool of the air in the balloon making it drop down.


Earth's stationary iron core

Why is the earth's iron core stationary, while the liquid matal circles around it creating the magnetic shield. Don't understand how can the entire planet rotate where as the planet's center is stationary. Anyone could explain this?


Why are bubbles formed in the empty space of soda when turned upside down?

I know this may actually be a chemistry question, but I don't know where else to ask. When I have a soda plastic bottle, I turn it upside down so the soda fills the empty space, and bubbles start growing on the walls of the plastic. So, why does this happen? I can think of something like a film forming around the empty space, and it reacts with the liquid, if so, what is it? why does it react that way?


Sources to learn about Greens functions

For a physics major, what are the best books/references on Greens functions for self-studying? My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary Boas.


What physical forces pull/press water upwards in vegetation?

Each spring enormous amounts of water rise up in trees and other vegetation. What causes this stream upwards? Edit: I was under the impression that capillary action is a key factor: the original question therefore was: what are the fundamental forces involved in capillary action? Can perhaps anybody make an estimation of the amount of gravity energy that is involved in this process earth-wide each season?


Expressing a particle's matter wave in terms of its momentum

I'm following Zettili's QM book and on p. 39 the following manipulation is done, Given a localized wave function (called a wave packet), it can be expressed as $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi}} \int_{-\infty}^{\infty} \phi(k) e^{i(kx-\omega t)} dk$$ Now use the de broglie relations: $p = \hbar k$ and $E = \hbar \omega$ and define $\tilde{\phi}(p) = \phi(\frac{k}{\hbar})$. This should yield $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi \hbar}}\int_{-\infty}^{\infty} \tilde{\phi}(p) e^{i(px-E t)/ \hbar} dp$$ but I get $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi} \hbar}\int_{-\infty}^{\infty} \phi\biggl(\frac{p}{h}\biggr) e^{i(px-E t)/ \hbar} dp$$ when I make the change-of-variable $k=\frac{p}{\hbar}$. What am I missing?


What is the condition for accelerating charge to radiate?

I was always taught that any accelerating charge produces radiation, but I don't think this condition is sufficient condition. For instance, any free charge on Earth is accelerated due to Earth orbiting the Sun but it doesn't produce radiation.


A question about verifying the transverse of electric field

I came accross a question about verifying the transverse of electric field in Peskin and Schroeder's QFT p179. Given > $$ \mathcal{A}^{\mu}(\mathbf{k}) = \frac{ -e}{| \mathbf{k} | } \left( > \frac{p'^{\mu}}{k \cdot p'} - \frac{p^{\mu}}{k \cdot p} \right) \tag{6.7} $$ and > $$ \mathcal{E} = -i \mathbf{k} \mathcal{A}^0 (\mathbf{k}) + i k^0 > \mathcal{A}(\mathbf{k}) \tag{6.9} $$ it is said > you can easily check that the electric field is transverse: $\mathbf{k} > \cdot \mathcal{E}(\mathbf{k}) =0 $ I tried to vertify this almost trivial relation and I didn't get it... Here is my attempt $$ \mathbf{k} \cdot \mathcal{E}(\mathbf{k}) = \left( \frac{ -e}{| \mathbf{k} | } \right) \left[ - i \mathbf{k} \frac{ \mathbf{k} p'^0 - k^0 \mathbf{p}'}{ k \cdot p'} + i \mathbf{k} \frac{ \mathbf{k} p^0 - k^0 \mathbf{p}}{ k \cdot p} \right] \tag{1}$$ I may use $p^{\mu}=(p^0, \mathbf{0})$ (since the particle is initially at rest) and $k^0=|\mathbf{k}|$. It seems that Eq. (1) is still not necessarily zero. Would you provide, at least, any hint how to show Eq. (1) is zero?


Resonant inductive coupling and Schumann resonances

I was reading about WiTricity (http://en.wikipedia.org/wiki/WiTricity) a technology developed by MIT to wirelessly transmit electricity through resonance, and I have this question: Given the phenomenon of resonant inductive coupling which wikipedia defines as: > the near field wireless transmission of electrical energy between two coils > that are tuned to resonate at the same frequency. > http://en.wikipedia.org/wiki/Resonant_inductive_coupling And the Schumann resonances of the earth ( ~7.83Hz, see wikipedia), would it be theoretically possible to create a coil that resonates at the same frequency or one of it's harmonics (7.83, 14.3, 20.8, 27.3 and 33.8 Hz) to generate electricity? I have a feeling that these wavelengths may be too big to capture via resonance (they are as large as the circumference of the earth if I understand it correctly), so alternatively would it be possible to create a coil that resonates with one of the EM waves that the sun sends our way?


Why do we take the first derivative of EPR/ESR spectra?

Apologies if this question is a bit too chemistry-flavoured. In electron paramagnetic resonance spectroscopy, there's a practically ubiquitous convention of plotting the first derivative of the absorption with respect to field, rather than the simple absorption. Why is this?


Reversible Stirling Engine

So it has been asserted that a Stirling engine with proper regeneration can be made reversible. It will consist of two isothermal quasi-static processes connected by two constant-volume processes. The working gas is supposed to be heated/cooled through infinitesimal temperature differences (thus preventing the creation of entropy). I'd like to know an explanation on the working gas properties during each step of the cycle. What are the state conditions for the gas at each of the points in the cycle (including inside the regenerator)? I've been looking for a the past week and found nothing.


Additional mass of block on inclined plane

I would like some help in solving this problem. ![Height is 15 cm , base is 88 cm](http://i.stack.imgur.com/fjVLX.png) Height is 15 cm , base is 88 cm My attempt: $$(119.3 + m)(\sin(\theta)) = (88*9.8)$$ $$ 15^2 + 88^2 = 89.269^2 $$ $$ \sin^{-1} = 15/89.269 $$ $$ (119.3 + m)(9.6733) = (88*9.8) $$ $m = -30.14$ ....Am I missing something? I feel like it has something to do with the force of friction...


History of the use of the concept of phase space in engineering

Engineering textbooks constantly use the concept of 'phase space' (see e.g. http://www.cs.cmu.edu/~baraff/sigcourse/notesc.pdf). That is, they think of the state of a mechanical system as a high-dimensional geometric space. For example, for a system of particles, the phase space has $6n$ dimensions, where $n$ is the number of particles. My understanding is that the notion of phase space initially developed among theoretical physicists (possible in conjunction with quantum mechanics? or possibly before that point?). Therefore, I'm wondering when and how engineers began to adopt this terminology and concept from physicists. It might have something to do with differential equations, since the concept of representing the state of a mechanical system as a point in a space becomes obvious when you view ODE's in terms of flows on euclidean spaces. NOTE: The phase space concept was invented by theoretical physicist and mathematician Willard Gibbs.


Is a weak measurement the same as an unsharp measurement or POVM?

This is prompted by the strong claims made in Science 332, 1170 (2011) to have observed trajectories of photons, "something all of our textbooks and professors had always told us was impossible". I'm suspicious of this claim because they work entirely within quantum theory, and it seems to me that in making weak measurements at two points on what they say is a trajectory, they do nothing else than make unsharp measurements of incompatible observables, a concept that was first introduced, to my knowledge, by Busch and Lahti in Phys. Rev. D 29, 1634–1646 (1984), and that is now much used in quantum information. Certainly observables that are at time-like separation are in general incompatible, so an unsharp measurement approach would seem applicable. A brief account of the Science paper mentioned above can be found here, amongst other places, and a few quotes can be found here, but, this being Science, I couldn't find a preprint. I was first alerted to this paper by a page at the BBC. The claim highlighted by ZapperZ, "Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm–de Broglie interpretation of quantum mechanics (8), although the reconstruction is in no way dependent on a choice of interpretation", seems to me particularly egregious, insofar as both their choice of what experiment to do and of what to do with their raw data seem partly driven by the form they want to present their results in. It seems entirely possible to present the data in terms of unsharp measurements and POVMs, which would give relatively less support to the de Broglie-Bohm interpretation.


Car parking special-relavity puzzle

Hi I read the following puzzle from an old text book long time ago. However it doesn't provide the answer. So what is the solution? Let's suppose a car is going to park to a garage and the garage is exactly the same size of the the static size of the car. Now the car is running in a speed that in its point of view the garage is just 1/10 of its static size. So it will consider the garage is too small to fit. On the other hand, the garage manager consider the car is only 1/10 of the garage size so it definitely fit. The question is to explain what will ACTUALLY happen if the car try to park in the garage, and what the garage manager and the driver will see.


Are the hypotheses of the Bernoulli equation satisfied for a bird or airplane wing at low Mach number?

A previous question by David Zaslavsky was a request for a broad, "how things work" type of explanation of the lift of an airfoil. The answers given there are enlightening, but don't address a more specific question I have. First a summary of my current understanding. There is no logical reason to assume that parcels of air take equal time to travel above and below the wing. If we did make such an assumption, Bernoulli's principle would make all kinds of incorrect predictions. "How things work" explanations (e.g., this one) often speak as though Bernoulli's principle is one effect, and then there are some other effects, such as conservation of momentum; it's implied that these effects are additive, and it's often stated that the Bernoulli effect is negligible. But in fact Bernoulli's principle is, under certain hypotheses, equivalent to conservation of energy. Conservation of energy and conservation of momentum are not effects to be added. They're two physical laws, both of which need to be obeyed. It's only under certain hypotheses that Bernoulli's principle is equivalent to conservation of energy. It requires nonviscous flow, and the version given in freshman physics textbooks also assumes incompressible flow, although there is also a version for compressible flow. The WP article on Bernoulli's principle states that it's usually valid for Mach numbers lower than about 0.3, which I guess might be valid for birds and some small planes (cruise speed for a Cessna 172 is Mach 0.25), but not passenger jets. Based on this, I can envision two logical possibilities: (1) The hypotheses of Bernoulli's principle hold for birds and some small planes. The equation gives at least an approximately correct result for the net force on the wing, but there is no obvious, simple way to know what velocities to assume. (2) The hypotheses of Bernoulli's principle fail in these cases. Which of these is correct? If #2, which hypothesis is it that fails? Is the incompressible-flow version inapplicable but the compressible-flow version OK?


Quantum Cyclotron Frequency - Why is it off by a factor of 2?

Say you have a magnetic field $\vec{B}=(0,0,B_0)$. Then the Schrodinger Equation Hamiltonian for a spin-2 particle of charge $e$ moving in this field is: $$H = \frac{1}{2m}[\vec{p}-e\vec{A}]^2-\vec{\mu}\cdot\vec{B},$$ where $\vec{A}=(-\tfrac{1}{2}y,\tfrac{1}{2}x,0)$ is the magnetic vector potential. You can find the speed by finding this commutator: $\frac{d\vec{r}}{dt}=i[H,\vec{r}]$. When I did out this calculation, I ended up with $\frac{\vec{p}-e\vec{A}}{m}$. However, I'm told that I should have gotten $\frac{\vec{p}}{m}+\vec{\omega}_c\times\vec{r}$, where $\vec{\omega}_c$ is the cyclotron frequency. This has the value $\vec{\omega}_c=-\frac{e\vec{B}}{m}$, but if you plug that in, you get $\vec{\omega}_c\times\vec{r}=-\frac{2e\vec{A}}{m}$. Why is this off by a factor of 2?


Spring coupled platforms & conservation of momentum - can it be solved with freshman physics?

This question came up as an exercise in a first year undergraduate course I was a TA for. It turned out to be a lot more difficult (impossible?) than anticipated... > Two platforms of mass $M_1$ and $M_2$ ($M_1\neq M_2$) are connected by a > spring of constant $k$, and are initially at rest with the spring > unstretched, sitting on a frictionless surface. A man of mass $m$ stands on > one platform and begins to run, always with a constant speed $v$ measured > relative to the platform he is running on. What is the maximum speed reached > by the other platform, relative to the ground? My intuition is telling me that I need to know something about how the man gets from rest to his constant speed (instantaneously? very slowly? with some smooth acceleration?) to solve this, but I haven't been able to prove to myself that this is required. If this is a requirement, I think the most reasonable assumption would be that he reaches his full speed instantaneously. What I'm most interested in is whether this problem can be tackled with a typical freshman's toolbox - simple arguments around conservation of energy/momentum, no/very limited differential equations, basic calculus. I can see an easy way to get an upper bound on the maximum speed from energy/momentum considerations, but I don't see a way to check if this speed is ever reached.


What is the relation between the Higgs field and chirality?

Wikipedia states that the spontaneous breaking of chiral symmetry _"is responsible for the bulk of the mass (over 99%) of the nucleons"_. How do the nucleons gain mass from the spontaneous breaking of chiral symmetry? Why don't leptons gains mass from it? What is the role of the Higgs field in this all?


Finding the components of the tensor for potential and kinetic energy

I have a rather poor understanding of what a tensor is, but enough to apply it to the biggest part of the classical mechanics I'm studying.   However, I've run into a small problem while studying "Free vibrations of a linear triatomic molecule". (We're assuming all 3 atoms are in 1 straight line and the forces they exert on each other are represented by springs with constant values for $k$). My potential energy is described as: $V=\frac{k}{2}(\eta_1^2+2\eta_2^2+\eta_3^2-2\eta_1\eta_2-2\eta_2\eta_3)$   Where $\eta$ is a coordinate relative to the equilibrium position. And 'hence' the tensor has the form: $\vec V=\begin{bmatrix} k & -k & 0\\\ -k & 2k & -k\\\ 0 & -k & k \end{bmatrix}$ How do I go from 1 to the other and vice versa?   I've tried googling terms like "Equation to tensor", "Potential energy tensor", "Absolute value of tensor" etc. but they didn't yield anything usable for me.


Question on static friction

![enter image description here](http://i.stack.imgur.com/fzIUA.png) This is the question, and the answer is given as 2). Now my basic doubt is, are they asking us the direction of the total contact force on the object by the floor? Or, are they asking the direction of the static friction on the block by the floor? What is it? Because, the normal reaction between any two surfaces in contact is ALWAYS perpendicular to the two surfaces. Please help me, and please answer this before closing it as being too specific.


Collection of histories vs. collection of momentary configurations

For a given Hamiltonian, is the space of histories of a classical system the same as the symplectic manifold? Do I have to take care of gauge equivalences and if so, is this only an issue for fields (not for trajectories)?


Car on a frictionless banked curve

> _A car is on a banked curve, following a path which is part of a circle with > radius $R$. The curve is banked at angle $\theta$ with the horizontal, and > is a frictionless surface. What is the speed the car must go to accomplish > this?_ What I don't understand about this problem is why we assume there is only the normal force and the gravitational force on the vehicle. From that point onwards, I have no trouble following the solution. The way I see it, if we're considering only that there exists a normal force and a gravitational force, something the car is doing (accelerating?) must be "adding" to the normal force. Or is it possible that the car could just be coasting, and it could keep a constant height along the banked curve?


The expansion of a function in powers of a parameter

In the perturbation theory for non-degenerate levels, the energy $E_n(\lambda)$ of an eigenstate $|\psi_n(\lambda)\rangle$ of the hamiltonian $\mathcal{H}=\mathcal{H}_0+\lambda \mathcal{H}_1$ (where $\mathcal{H}_0$ is the unperturbed hamiltonian with eigenstates $E_n^0$ and $\lambda\mathcal{H}_1$ is the perturbation) is described by the equation $$E_n(\lambda)=E_n^0+\dfrac{\lambda\langle\phi_n|\mathcal{H}_1|\phi_n\rangle+\lambda^2\langle\phi_n|\mathcal{H}_1|\psi_n^1\rangle+\lambda^3\langle\phi_n|\mathcal{H}_1|\psi_n^2\rangle+\dots}{1+\lambda a_n^{(1)}+\lambda^2 a_n^{(2)}+\dots}.$$ Here, $|\phi_n\rangle$ is the $n$'th eigenstate of the unperturbed hamiltonian, and $a_n^{(p)}=\langle\phi_n|\psi_n^p\rangle$, with $|\psi_n^p\rangle$ being the $p$'th correction to the $n$'th eigenstate. The zeroth correction of course equals the eigenstate of the unperturbed hamiltonian: $|\psi_n^0\rangle=|\phi_n\rangle$. **My question is the following** : how does one go from the above equation to this? $$\begin{array}{r l} E_n(\lambda)&=E_n^0+\lambda\langle\phi_n|\mathcal{H}_1|\phi_n\rangle\\\ &+\lambda^2\left[\langle\phi_n|\mathcal{H}_1|\psi_n^1\rangle-\langle\phi_n|\mathcal{H}_1|\phi_n\rangle a_n^{(1)}\right]\\\ &+\lambda^3\left[\langle\phi_n|\mathcal{H}_1|\psi_n^2\rangle-\langle\phi_n|\mathcal{H}_1|\psi_n^1\rangle a_n^{(1)}-\langle\phi_n|\mathcal{H}_1|\phi_n\rangle a_n^{(2)}\right]+\dots \end{array}$$ The book I'm working with tells me this is done by _"Expanding this expression in powers of $\lambda$"_ , but I can't recognize the procedure. Of course the first term $E_n^0$ is of order $\lambda^0$, and the second term $\lambda\langle\phi_n|\mathcal{H}_1|\phi_n\rangle$ is of order $\lambda^1$, and these can be taken from the original equation right away, but I can't find out how the subtractions in the third and fourth term got there.


Hamiltonian formulation of single particle in an electromagnetic field

Consider a charged particle in a static electromagnetic field. Suppose that the domain is simply connected so that the second law of Newton's dynamics reads: $$ m\frac{\mathrm{d}^2\vec{x}}{\mathrm{d}t^2}=-e\left(\vec{\nabla}\Phi+\frac{\mathrm{d}\vec{x}}{\mathrm{d}t}\times(\vec{\nabla}\times\vec{A})\right) $$ where:   * $m$ is the mass of the particle,   * $\vec{x}$ is position vector,   * $\Phi$ the (scalar) electric potential,   * $\vec{A}$ the (vector) magnetic potential. My question is how to derive the Hamiltonian formulation from here. The Hamiltonian is: $$ H(\vec{x},\vec{p})=\frac{1}{2m}\left\|\vec{p}-e\vec{A}\right\|+e\Phi(\vec{x}) $$ and the Hamilton equations look like: $$ \begin{cases} \frac{\mathrm{d}x_i}{\mathrm{d}t}=\frac{1}{m}(p_i-eA_i)\\\ \frac{\mathrm{d}p_i}{\mathrm{d}t}=\frac{e}{m}\sum_{j=1}^3(p_j- eA_j)\frac{\partial A_j}{\partial x^i}-e\frac{\partial\Phi}{\partial x^i} \end{cases} $$ Another question related to this one is why the magnetic potential is "included" in the "generalized" moment $\vec{p}-e\vec{A}$. I undertand why from the Lagrangian of the system but perhaps there is a physical explanation. **EDIT:** Consider a single particle in a conservative force field $F=-\vec{\nabla}V$. Then the Newton's equations are: $$ m\frac{\mathrm{d}^2\vec{x}}{\mathrm{d}t^2}=-\vec{\nabla}V $$ This is a second order differential equation, which we can transform into a system of first order differential equations, by adding a new variable which is usually the speed $v$ in mathematics but here we will take the momentum $p=mv$. The system I am talking about is: $$ \begin{cases} \frac{\mathrm{d}\vec{x}}{\mathrm{d}t}=\frac{\vec{p}}{m}\\\ \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}=-\vec{\nabla}V(x) \end{cases} $$ which is just the hamiltonian form of the Newton's equations. Can we do the same kind of trick in the case of the system above?


Help wanted achieving cryogenic temperatures

My son's science project involves liquifying air. I found the Linde process which is the way it was done first. We are trying and not getting very far so I am looking for advice. I have an air compressor good to 150psi. We are slowly ramping up the pressure in the system. We are currently testing at 80psi. We pressurize 20 feet of copper tube bent in a coil running through an ice bath. The tube exits and goes into the bottom of a thermos. The end of the tube is capped, and we allow a small leak so that pressure drops from 80psi to 1atm. We insulated the few inches between the ice bath and the thermos. We do not yet have a double walled tube, but we taped the thermos so all the air coming out is exiting through the insulation. When we run, the thermos drops from room temperature (about 68F in the basement) to our best of 45F. I'm quoting in Fahrenheit right now because the boys are very obnoxious about not wanting to use a temperature system they aren't comfortable with, and I just haven't been able to convince them that they are being unreasonable. The point is, we are not even achieving the temperature of the ice bath. Incidentally, as measured with my multimeter, the ice bath is always 3C. I'm curious about that too, but that will be a separate question. Needless to say, we are not achieving any refrigeration. The next step we will increase the leak so that the flow through the system is much higher. And we will start trying to use a Peltier cooler so we can have an intermediate stage that is 20C or so cooler than the ice bath. Obviously we can up the pressure, but I would think we should see something more than we are getting. Any suggestions? Should I take a picture of our setup?


Ice bath is always 3C, why?

I've always learned that a mixture of ice and water should reach equilibrium at approximately 0C. I've actually tried to create that a number of times in different contexts and always fail. First, an ice bath in a plastic container. I get 3C, occasionally as low as 2C with a multimeter thermocouple. Granted those are inaccurate, so I've tried a digital thermometer from Vernier with claimed accuracy of 0.5C. I confirmed with an IR thermometer. Same story. Thinking that perhaps it is due to warming from the outside, I have put the ice bath in a thermos and chilled it a while. Same story. I am not using distilled water, so there are some minerals in the water, but if anything I thought that would lower the temperature of the ice bath.


Adding matter to make galaxies bigger

What are the actual factors that play a role in the accretion of matter into galaxies? I read about Accretion Disks but I don't quite understand how they work yet.


Theoretic Mass Energy Conversion?

say you had a large quantity of a heavy element such as uranium... AND you put a massive amount of energy into it, so that it began to undergo nuclear fission and transformed to a significantly smaller element(s). Would it be possible to use this energy to now begin a process of fusion, to output energy? And then once it's fused beyond a certain size, back to fission? Basically the idea is to extract back and forth as much energy as possible from the original sample.


Is there a better approximation of drag (other than quadratic drag)?

This is a student inquiry sparked by sheer curiosity. Wikipedia states the drag equation, $F = 1/2v^2pC_dA_c$. (p = mass density of fluid/gas, v = velocity, c_d = drag coefficient, a_c = cross sectional area perpendicular to velocity.) Wikipedia claims that this equation is only accurate under certain conditions: > _The objects must have a blunt form factor and the fluid must have a large > enough Reynolds number to produce turbulence behind the object._ **Does there exist a more general equation that can accurately measure the drag of all objects, at all speeds/Reynolds numbers and considering all physical properties affecting the drag?**


What happens when electron degeneracy pressure is being overcome (right before an object turns into a black hole)?

I was wondering about hydrostatic equilibrium (the balance of radially inward and outward forces) in large (i.e. above the Chandrasekhar limit) stars. It is often said that when the gravitational force exceeds any outward forces or pressures, mainly the electron degeneracy pressure I'm thinking, the star collapses into a black hole. But how can this happen without the Pauli exclusion principle being violated? A similar question: Does black hole formation contradict the Pauli exclusion principle? The answer by @Siva in that question makes some sense to me, but I don't understand at what point we start our counting of states from the object being a star to a black hole. What happens in the middle? Is there a sharp change?


Escape Velocity of Asteriod 243 Ida

I was reading about this asteroid (apparently, it has a moon, isn't that awesome?) and I started thinking about if I was on this asteroid, and I jumped, would I fall off? It's been a while since I did something like this, so I went to Wikipedia to get some formulas. There I found $$v_e = \sqrt{\frac{2GM}{r}}$$ Where   * $G$ is the gravitational constant ($6.673 × 10^{-11}\ \mathrm{m}^3\mathrm{kg}^{-1}\mathrm{s}^{-2}$)   * $M$ is mass of the body (found to be $4.2 × 10^{16}\ \mathrm{kg}$)   * $r$ distance from center of mass (found to be $53.6\ \mathrm{km}$) $$v_e = \sqrt{\frac{2(6.673\times 10^{-11})(4.2\times 10^{16})}{53600}} = \sqrt{104} = 10.22\,\frac{\mathrm{m}}{\mathrm{s}}$$ I'm a fairly fit guy, about 73kg and .6m vertical leap here on earth. I found a formula for the speed of a free falling object to be $$v = \sqrt{2gd}$$ If I assume that my speed when I hit the ground is the same as when I jump up, that means my jump speed is about 3.4 m/s, which is significantly less than the required 10.4 m/s. So it would seem I would not be able to jump off of this asteroid. Are my formulas, assumptions, and calculations correct here? Also, how would I use this information to determine how high I would jump (would it be enough to freak me out? Well I'd be jumping off an asteroid, so it would freak me out anyway... although I imagine I would be going slow enough for (my ship?) to pick me back up.)


Is there a maximum number of types of elementary particles?

Doing a Google search i found a paper called The maximum number of elementary particles in a super symmetric extension of the standard model. It claims in the abstract that the upper bound is 84 (i don't have access to the article) My question is: Is there a max number of types of elementary particles predicted in advanced physics theories such as string theory? What are the reasons for this?Are the arguments purely mathematical?


Earnshaw's theorem for extended conducting bodies

I read on Wiki that Earnshaw's theorem has been proven for extended conducting bodies. If we consider the case of a positive charge at the centre of a symmetric metal cavity- the positive charge and the induced negative charge constitute a system which violates Earnshaw's law. Where am I wrong in thinking this way?


Consideration of static atomic displacements in electronic structure calculations

I am hoping to discuss some details of electronic structure calculations. I am not an expert on this topic, so please forgive any abuse of terminology. It is my understanding that first principles electronic structure calculations are based on perfect translational symmetry of a crystalline lattice. Further, that the first round of calculations traditionally treat the atomic nuclei as fixed points due to the large disparity in nuclei and electron velocities, and after the initial electronic structure is solved, one can then turn on the nuclear motions and understand the interaction between the nuclear and electronic motions. The heart of my question is: is it possible to investigate local distortions of structures using existing techniques? By a local distortion, I mean a distortion which does not disrupt the long range crystalline periodicity, but results in a very different local atomic environment?


Chiral edge state as topological properity of bulk state

As far as I know, quantum hall effect and quantum spin hall effect has chiral edge state. Chiral edge state is usually closely related with delocalization, since back scattering is forbidden. However, some topological nontrivial state does not have chiral state, such as majorana bound state in topological superconductor. I am quite interested whether the existance of chiral edge state is determined by bulk topological properity? Thanks in advance


Indicators on how even the heat is distributed?

I'm wondering if there are any good indicators on how even the heat is distributed on an object (for simplicity, a flat object maybe)? What are the possibly reasonable ways to maximize the evenness if varying the shape of the object is allowed.


What are the physycal meaning of universal constants such as the magnetic, electric and gravitational constants?

I know that: $G=6.67300 × 10^{-11}\dfrac{Nm^2}{kg^2}$(Gravitational constant) $K_e=9×10^9\dfrac{Nm^2}{C^2}$(Electric constant or Coloumb's constant) $k_m=1×10^{-7}\dfrac{Ns^2}{C^2}$(Magnetic constant) * * * But what are the meaning of this values? Would the universe be much different with a little change on this values? What if we hadn't known them?


Question about entangled states

I have a question about entangled state. Suppose I consider the entangled state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. I saw an argument for how measurement of the first bit is affected by whether or not the second bit is measured because this is an entangled state in an article I am reading. The argument is clear to me and I describe it below, so please keep reading: If I measure the first bit without first measuring the 2nd bit, then the probability that I get $|0\rangle$ is 1/2. This is clear to me. If I measure the 1st bit after measuring the 2nd bit, the probability that it is $|0\rangle$ is 0 or 1 depending on whether the measured value of the 2nd bit was $|0\rangle$ or $|1\rangle$. This is clear to me too. But what I am confused about is this: If I measure the first bit after the second bit, this is what I would calculate the probability of measuring the first bit being $|0\rangle$ to be: $\frac{1}{2}\times 1 + \frac{1}{2}\times 0 = \frac{1}{2}$. The first term is the prob. of measuring the second bit as $|0\rangle$ and then the 1st bit as 0 and the 2nd term is the prob of measuring the 2nd bit as 1 and then measuring the 1st bit as zero. If the probability of measuring the first bit as 0 is to same regardless of whether the 2nd bit was measured or not is unchanged, what is the effect of the entangled state? Or am I missing something?


Is spacetime an illusion?

In consistent histories, for gauge theories, can the projection operators used in the chains be not gauge invariant? In quantum gravity, for a projection operator to be gauge invariant means it has to be diffeomorphism invariant. There are no localized diffeomorphism invariant operators, not even localized in time. No localized observables. So, what are the permissible chains in quantum gravity? Do they have to be delocalized, or localized on the boundaries of spacetime? Are space and time illusions, as Andrew Strominger claimed? A Cosmic Hologram projecting out the illusion that is spacetime from the future boundary of time?


Can Jupiter be ignited?

Our solar system itself contains two candidate "Earths" One is Jupiter's moon Europa and another is Saturn's moon Titan. Both of them have the problem of having at low temperature as Sun's heat cannot reach them. Jupiter is made up mostly of Hydrogen so is our Sun. Can we ignite Jupiter so that it will produce enough heat to warm these two earth like planets/satellites? What can be consequences of any of such attempt?


Why aren't there more natural satellites orbiting the Earth than just the Moon?

Considering that there's a lot of debris in space and that impacts fling out rocks into space all the time, why do we only have one large natural satellite - the Moon? Shouldn't there be all kinds of rocks in all shapes and sizes orbiting the Earth?


Is spectrum of Discrete-Time Fourier Transform (DTFT) periodic or not

I can't think of any better title. Here is the content that I got question http://cnx.org/content/m10247/2.31/ As it state the nature of DTFT's spectrum is periodic as it show in figure 1 ![enter image description here](http://i.stack.imgur.com/RHr9S.png) ![enter image description here](http://i.stack.imgur.com/892ct.png) However, in figure 3, spectrum is non-periodic ![enter image description here](http://i.stack.imgur.com/lO3dP.png) So, the spectrum of DTFT is periodic or not or it is dependent on each case of $s(t)$


"Consciousness causes collapse"-interpretation or free will excluded by quantum zeno effect?

This question is if the von Neumann–Wigner interpretation, aka "consciousness cause collapse" (see e.g. John von Neumann's 1932 book The Mathematical Foundations of Quantum Mechanics) is compatible with the quantum Zeno effect (see Sudarshan, E. C. G.; Misra, B. (1977) "The Zeno's paradox in quantum theory", Journal of Mathematical Physics 18 (4): 756–763) respectively its empirical observation (such as by Wineland 1989 or by Raizen 2002). Or if the Quantum Zeno effect falsies or created the possibility to empirically verify the von Neumann-Wigner interpretation. Assume measuring the state of a system every second causes a zeno effect and measuring every three seconds causes an anti-zeno effect. Now let's assume the state is measured every second and recorded. A proponent of the "consciousness causes collapse" interpretation would argue that the wave function only collapses when a conscious observer actually looks at the measurement results. But what if the conscious observer first looks at only every third result and decides only afterwards on whether to look at the other results too. Will she observe the zeno effect depending on her future decision? Wouldn't such an experiment show that either (1) the consciousness of the observer has no impact and there is a zeno effect even if two out of three measurement results are destroyed before a conscious observer looks at them or that (2) the observer has no free will and whenever she observes a zeno effect looking at every third measurement she can't help but looking at the other results too, and conversely if the first set of results she looks at shows an anti-zeno effect she has an irresistible urge to destroy the other results? Well the first option looks more plausible to me, but it seems like something that could be experimentally tested.


Can an object reverse its direction of acceleration even though it continues to move in the same direction?

Can anyone please explain me on this matter along with day to day examples?


Scalar-fermion bound state

Is it possible to have a bound state between a scalar and a fermion? For example, a squark--anti-squark bound state, provided that the decay width is sufficiently small compared to the binding energy?


what is the mechanism of water conductivity?

I'm trying to wrap my head around the kelvin water drop generator and more specifically what happens in the water before it drops. If on one side the drops of water are positive and on the other side negative, that means there is current flowing through the water _before it drops_. What I dont understand even more specifically is how electricity is carried by water, and why water conductivity increases with ions content (and other impurities perhaps). Are there electrons flowing ? Or are there ions flowing ? Let's imagine that we are below the potentials at whitch electrolysis occurs (about 1 volt). What carries the charges ? I first thought that ions were responsible for the charges in the kelvin generator, but when a spark occurs, because the water has dropped into separate boxes it's impossible for them to move, so there must be some kind of redox reaction occuring. But if table salt is in the water to raise it's conductivity, would that mean that Na+ is reduced to sodium and Cl- oxydized to chlorine ?


plasma pressure initial condition

The authors in the paper below did a simulation of plasma plume evolution from a laser heated metal target. They used compressible transport equations and ideal gas EOS for Ar. I am having difficulty understanding their initial condition. They say ", the initial internal pressure of plasma is assumed constant, 210 MPa.". How is this value determined or assumed. It is based on the temperature of the surface? So, im thinking about the problem some more, and it seems the density is what should be known because when ablation occurs, phase transition occurs in the metal surface which generates a vapor. Thanks S. Harilal, G. V. Miloshevsky, P. K. Diwakar, N. L. LaHaye, and A. Hassanein, Phys. Plasmas 19(8), 083504 (2012).


calculating electrodynamic momentum of a dumbbell (consisting of two point charges) in longitudinal motion

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure how. $$ \int_{0}^{\infty}\int_{0}^{\pi} {r^{4}\sin^{3}\left(\theta\right) \over \left\\{(r^{2} + d^{2}/4)^2 - \left[rd\cos\left(\theta\right)\right]^{2}\right\\}^{3/2}} \,{\rm d}r\,{\rm d}\theta \space-[1] $$ If I focus on the integration w.r.t. r first, I get [2]. If I focus on the integration w.r.t. $\theta$ first, I get [3]. A,B,C are just constants w.r.t. the variable of integration. I believe I can make a series of substitutions but they don't seem to get me closer to an answer. $$ \int_{0}^{\infty} {Ar^{4} \over \left[(r^{2} + B)^2 -\left(rC\right)^{2}\right]^{3/2}}\,{\rm d}r \space-[2] $$ $$ \int_{0}^{\pi} {A\sin^{3}\left(\theta\right) \over \left\\{(C)^2 - \left[B\cos\left(\theta\right)\right]^{2}\right\\}^{3/2}} \,{\rm \rm d}\theta \space-[3] $$


Does measuring the exact position of a ball destroy the ball

If you have a macroscopic ball (say, a tennis ball) and you (hypothetically) try to measure the exact position of the center of that ball by measuring the exact positions of the atoms making up the ball, will this measurement destroy the ball (i.e. is there a point where, as a result of the uncertainty principle, the more accurately you measure the position of each of the atoms, the more likely the atoms are to scatter off in some random direction, destroying the ball)?


What is the capacitance of parallel plate capacitor with different electrode material? The capacitance of parallel plate capacitor depends on dielectric material, surface area, and separation between two plates. $$C = ε_r * ε_0 * \frac{A}{D} $$ But how about electrode materials between dielectric, will the capacitance be different? Thanks.


Signal strength drop-off from a geosynchronous satellite as a function of angular pointing error

If we have a compact antenna used to communicate with a satellite in geosynchronous orbit somewhere between the $K_u$ and $K_a$ bands of the spectrum (for high broadband applications where we can't use something like the Iridium network), we presumably need to properly orient the dipole of the antenna with respect to the position of the geosynchronous satellite. Let $\alpha$ be the angular pointing error of the satellite (some fraction of a degree). How does the signal strength fall off as a function of $\alpha$? Or, if it makes a difference, we can uniquely define an azimuth angle $\theta$ and elevation angle $\phi$. Now, I'm guessing that the answer to this question has something to do with the satellite beam configuration / geometry. So is it possible for there to be a "typical" answer to this question? Also, to be clear, here I'm asking about the "form" of the expression for the drop-off in signal intensity as a function of $\alpha$ (or azimuth and elevation angles $\theta$ and $\phi$), as the exact parameters will likely depend on the precise antenna being used. Where I'm coming from with respect to this question: I heard recently that is indeed possible possible to send data packets back and forth to a geosynchronous satellite from a sufficiently large boat (or aircraft) with proper antenna equipment (indeed to establish a continuous connection), and I was wondering how in the world this worked? In other words, in the context of something like a small vessel or an airplane, to what extent is sophisticated inertial navigation equipment required for real-time compensation of rotational (pitch / roll / etc.) and translational transformations on the orientation and position of the antenna? [3-11-2014] Is it safe to assume that these vessel-to-GEO antennas use circular polarization to avoid issues associated with the relative orientation of the transmitting and (primarily) receiving antenna? It seems then safe to assume that only the angular error $\alpha$ should be sufficient to estimate signal strength?


Shear Flow corresponding to Eccentric Shear Force of a Closed-section Beam (Structural Analysis - Mechanics)

Been stumped with this question for way too long... its a beam with a thin- walled rectangular cross-section, and a shear force is acting at a distance from the shear center. I know my decomposition of that force into the torsional moment and shear-center acting shear force is correct, but for some reason the shear flows don't add up.... rather than causing a higher shear to the left of the cross-section like I'd expect from my diagram, they cause a higher shear on the right side of the cross-section, sort of the mirror image of what I would expect. I've tried finding my mistake looking at books, websites, notes but none of them address this topic directly, leaving me with no other choice. :( Could anyone please tell me what I'm doing wrong? Also sorry for the blurry image and messy diagrams... I'll draw a new one if it's useless. ![enter image description here](http://i.stack.imgur.com/U31Qw.jpg) ![enter image description here](http://i.stack.imgur.com/zxXJH.jpg)


total intensity measurement (of the whole visible light spectrum; from 400 - 800 nm) with a powermeter PM100?

I was using a power meter PM100, with a Si-photodiode sensor head, to measure the intensity of the white light source (250 W quartz halogen lamp). I have recorded the power, shown by the sensor, by setting the wavelength equal to 633 nm (i.e. the maximum intensity is supposed to be at this wavelength, in the spectrum); however, this value for the intensity (power divided by area of the sensor) is only true for the monochromatic light (supposed to use for laser light?) and want to estimate the intensity (of the white light source) from the measured value. I have the responsivity curve for the senor, I used, and the intensity at a wavelength. How can I estimate the total power/intensity of the light source? Your help will be much appreciated.


Explain Momentum

It's a shame for me to be in Year 12 and still haven't understood the concept of Momentum. This is what I think, and I know I'm wrong. But, it is a good place to start for you to explain: "Since the unit of the momentum is kg/ms^-1, it is a measure how much mass is being moved at the speed of 1ms^-1. By the way, this is not a homework or anything. I'm really eager to learn about this concept. Thanks In Advance.


what is the proper way to connect two light bulbs in a circuit? in series or parallel?

What is the proper way to connect two light bulbs in a circuit? in series or in parallel and why? My thought is that it's better to hook them in parallel, since if we take into account Ohm's Law, the sum of their resistance will be less than when connected in series therefore the battery will last longer. What do you think?


Hawking radiation: direct matter -> energy conversion?

When a black hole evaporates, does it turn all the matter that has fallen in directly to energy, or will it somehow throw back out the same kind of matter (normal or anti) that went in?


If the universe could emerge from nothing, what about physical laws?

I've watched Lawrence Krauss's lecture and read his book. I think I got what he was saying, and I don't have any problems with that; however, what I can't get is how the laws of physics that makes this phenomenon possible came to be in the first place?


From where randomness comes from and why it exists?

I recently began to study statistics and probability and I have two questions:   1. Where does randomness come from? What is the source of randomness?    2. Why does the randomness exist? Is it possible to answer my question that non mathematician or non physicist can understand it? I searched Internet for the answer but there is very little information that non mathematician or the information is for professional physicists and mathematicians. Also please suggest books concerning my questions that is written so that non mathematician or non physicist can understand it.


can radiocarbon dating be used on living things?

I have been going through a wiki article about worlds oldest living creature. As a matter of fact its a plant, a shrub to be precise. Wiki says that the plant age was determined by carbon dating. But what i know is that carbon dating can only be done for dead tissues i.e which no longer assimilates atmospheric carbon onto itself. Can someone clarify this?? regards,


Optics: finding total power of a syste, including wavelength

Specifically I need to create an ABCD matrix given 2 radii of curvatures, the thickness of the lens, type of glass (BK7) and a given wavelength. I will have to figure this out eventually, so help would be appreciated with this as well, but my main question is, how do I correlate the wavelength of the incident light to the total power? If I have the wavelength of light, how do I decide what the power of the lens is if nothing in the lens changes, just the wavelength of the beam of light (for example, how would the power differ if a yellow helium line was shone, or a blue hydrogen line)? Thank you!


Quantum entropy in term of density matrix

Why in von Neumann expression of quantum entropy we have trace of density matrix expression? Why don't off diagonal term play a role?


Simple question regarding the Green's function for the diffusion equation

The differential operator for diffusion in three dimensions is given by $\partial_t - k \nabla^2$ where $k$ is a constant. The Green's function is (according to Wikipedia) $$\theta(t)\left( \frac{1}{4\pi k t} \right)^{3/2} e^{-r^2/4kt}$$ where $\theta$ is the Heaviside step function and $r = |\mathbf{r}|$. Hence, we should have $$ (\partial_t - k \nabla^2)\theta(t)\left( \frac{1}{4\pi k t} \right)^{3/2} e^{-r^2/4kt} = \delta(t)\delta^3(\mathbf{r}).$$ I've played around with this for a while, but I simply can't see how evaluating the left-hand side in the above expression gives the right-hand side. Where do the delta functions even come from?


Describing a motion of gyroscope with gimbal

Can you tell be how to set the equations to describe the motion of this machine in movie "Contact": https://www.youtube.com/watch?v=TSaO9VGjLXc This is gyroscope with gimbal, am I right?


Infinitely charged wire and Differential form of Gauss' Law

I have tried calculating the potential of a charged wire the direct way. If lambda is the charge density of the wire, then I get $$\phi(r) = \frac{\lambda}{4 \pi \epsilon_0 r} \int_{-\infty}^\infty \frac{1}{(1+z^2/r^2)^{1/2}} dz.$$ But this comes to $+\infty$ unless I am doing the calculation wrong. Why doesn't this work the direct way? Also, is it possible to calculate the potential of a charged wire using Gauss' differential law? What about in the case of an infinite charged sheet? Or does Gauss' differential law only apply to charged volumes?


As the universe expands, why do some things stretch but not others?

I got into watching a video on Olbers' Paradox a few days ago, and from there read about the origins of the universe, its expansion, and so on... it's always fascinated me, but now something about it bothers me. I've heard many analogies about it (the dot-balloon, the raisin bread loaf, and others), but none really seem to explain this question. (This comes close, but dances around the answer more than explain it.) At the beginning of the universe as we know it, the universe itself was very small, so all the stars giving off light would have made it very bright (16:29 in this video). Since that time, the wavelength of that light has been stretched (17:01, same video). I found a few explanations saying that space itself stretched (here; described as "ether" in the article), which would stretch out the wavelengths. But here's what bothers me: If space is stretching out, redshifting all the light soaring around our universe, why are we not stretching? Theoretically, the universe is expanding an incredible amount faster than the speed of light, and the edge of the universe is an unimaginably large number of megaparsecs away from us. But should we not notice some of the stretch here, too? That is to say, if the light in space (the "ether", though I'm not fond of that term) is stretching out, why is everything on Earth still the same size as it was a hundred years ago? Is it stretching uniformly, but we are just unable to notice such a small stretch? Or does mass have some property that space and light do not, that prevents it from stretching out? I've also heard about time stretching, too; does this have an impact on it?


Flux through a conduting cylinder?

> _A point charge of magnitude $Q$ is placed inside a conducting cylinder of > length $L$ and radius $R$ at its centre. What is the flux through the > cylinder?_ I know that I have to use Gauss Law here but my concern is: > _Since it is a conducting cylinder, there would be equal and opposite > charges induced on its surface. How do I account for these charges?_ Help would be appreciated.


Which limit for matsubara frequency sum?

in the context of a simple toy problem for Feynman path integrals, I consider a two-site Hubbard model for spinless fermions. I expand the path integral to first order in the interaction $V$, which means that I have to compute an "average" of the type $$\left\langle \bar \psi_1(\omega_n) \bar \psi_2(\omega_m) \psi_2(\omega_p) \psi_1(\omega_q) \right\rangle$$ with respect to the non-interacting action. Since the Grassmann numbers for sites $1$ and $2$ are different numbers, this average immediately factors, and I'm left with two-variable averages, which give me just the non-interacting Matsubara Green's functions. But now I have to calculate a Matsubara sum of the type $$\frac{1}{\beta} \sum_n \frac{1}{i\omega_n - \epsilon}$$ where $\epsilon$ is just a constant. From searching in the literature, I know that contour integration is the way to go, but unfortunately this sum is ambiguous and I have to introduce a factor $e^{-i\omega_n \tau}$ with $\tau \rightarrow 0^+$ or $\tau \rightarrow 0^-$, and the result of the contour integration will be quite different depending on which limit I take. My question now is, how do I know what limit to take? And is there an easier way of carrying out the sum over frequency?


What's best shape for linear concentrated solar panel?

I found 3 shapes which are suitable for concentrated solar panel.      1\. **Semicircle** \- first thing that comes to ones mind.   It's a simple shape, the position of the focal point is obvious.      2\. **Parabolic mirror**   A bit more difficult shape.   ![Parabolic mirror](http://i.stack.imgur.com/qWq30.png)      3\. **Compound parabolic mirror**   ![Compound Parabolic mirror](http://i.stack.imgur.com/VUupy.jpg)   Compound parabolic mirror has an advantage over the parabolic mirror in acceptance angle. Collector doesn't necessarily need to track sun, because it's able to receive sunlight from a range of angles.   **Is there any other shape, that might have interesting properties for the purpose of solar collector?**   I found many DIY designs with semicircle mirrors. And many scientific papers with parabolic mirrors.   **What is the advantage of parabolic mirrors are over semicircle mirrors?**


POVM advantage in state discrimination

Suppose we are given the task of discriminating, with minimum error, between a set of states $\\{|\psi_1\rangle,|\psi_2\rangle,\ldots,|\psi_N\rangle\\}$. In other words, we are given an unknown state form this set and our task is to output which of the states we were given. Let $P_{err}$ be the probability of incorrectly identifying the state. Under what circumstances is this probability minimized by using a projective measurement?


The most general procedure for quantization

I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not gotten too far into reading the book, so bear with me): > _A physical system is usually described in terms of states and observable. > In the Hamiltonian framework of classical mechanics, the states form a > symplectic manifold $(M,\omega)$ and the observables are functions on $M$. > The dynamics of a (time invariant) system is a one parameter group of > symplectic diffeomorphisms; the generating function is the energy or > Hamiltonian. The system is said to be free if $(M,\omega)$ is an affine > symplectic space and the motion is by a one-parameter group of symplectic > transformations. This general descriptions applies to any system that > includes classical particles, fields, strings and other types of objects._ The last sentence, in particular, has really intrigued me. It implies a most general procedure for quantizing all systems encountered in physics. I haven't understood the part on symplectic diffeomorphisms or free systems. Here are my questions:   1. Given a constraint-free phase-space, equipped with the symplectic 2-form, we can construct a Hilbert space of states and a set of observables and start calculating expectation values and probability amplitudes. Since the passage says that this applies to point particles, fields and strings, I assume this is all there is to quantization of any system. Is this true?    2. What is the general procedure for such a construction, given $M$ and $\omega$?    3. For classical fields and strings what does this symplectic 2-form look like? (isn't it of infinite dimension?)   4. Also I assume for constrained systems like in loop quantum gravity, one needs to solve for the constraints and cast the system as a constraint-free before constructing the phase, am I correct?   5. I don't know what 'the one-parameter group of symplectic diffeomorphisms' are. How are the different from ordinary diffeomorphisms on a manifold? Since diffeomorphisms may be looked at as a tiny co-ordinate changes, are these diffeomorphisms canonical transformations? (is time or its equivalent the parameter mentioned above?)   6. What is meant by a 'free' system as given above?   7. By 'affine' I assume they mean that the connection on $M$ is flat and torsion free, what would this physically mean in the case of a one dimensional-oscillator or in the case of systems with strings and fields?   8. In systems that do not permit a Lagrangian description, how exactly do we define the cotangent bundle necessary for the conjugate momenta? If we can't, then how do we construct the symplectic 2-form? If we can't construct the symplectic 2-form, then how do we quantize the system? I have asked a lot of long questions, so please answer as many as you can and link relevant articles.


Potentials in Helmoholtz Theorem

I read about Helmholtz theorem on the Griffiths (Introduction to electrodynamics) and in the appendix B it shows how to use the Helmholtz theorem to determine the field **F** starting from the potential. The book says that: Suppose we are told that the divergence of a vector function **F** (r) is a specified scalar function D(r): $\nabla \cdot \textbf{F}=D(r)$ and the curl of **F** (r) is a specified vector function **C** (r): $\nabla \times \textbf{F}=\textbf{C}$ For consistency, **C** must be divergenceless, $\nabla \cdot \textbf{C}=0$ because the divergence of a curl is always zero. Question: can we, on the basis of this information, determine the function **F**? If D(r) and **C** (r) go to zero sufficiently rapidly at infinity, the answer is yes, as I will show by explicit construction. I claim that $\textbf{F}=-\nabla U + \nabla \times \textbf{W}$ where $U(\textbf{r})=\frac{1}{4\pi}\int \frac{D(\textbf{r'})}{\textbf{r-r'}}d\tau'$ and $\textbf{W(r)}=\frac{1}{4\pi}\int \frac{\textbf{C(r'})}{\textbf{r-r'}}d\tau'$ Can someone show me how to obtain the potentials formulas written above?


Correlation between speed, distance, mass, etc between a planet and a star (or moon and planet)?

While playing around with some little scripts, generating solar systems, I started to wonder about the correlation between the different objects. Up until now, distance, mass, size and speed are randomly generated. Are there any 'common rules' on how some of theese objects correlate? Are planets close to a star always smaller and faster, and planets far away usually larger and slower? Is the rotation speed affected? And how about the correlations between moons and their planet? At the end of the day, I'm triyng to make the the generation a bit more accurate / realistic. Thanks in advance! Sorry for the english, it's a second language for me.


Very short decay: A bare quark?

I keep reading the same phrase about the very short life time of the top quark: > > Because the t-quark decays on a shorter than the characteristic QCD > interaction-time it cannot hadronize. Therefore it give to possibility to be > seen as a bare quark. I cannot find any more information beyond this simple phrase. There are however a few things that I would like to seen cleared out (sorry for the naive questions) :   * The first question is simple: I don't understand why this offers to opportunity to see a bare quark, since this quark decays that fast it must be very difficult te see it...   * A bare quark?? I thought this was ruled to see colorful particles?


How do projectors work?

So, objects are certain colors because they are absorbing every color except for that one. So why is it that if I take a projector and project a blue image on a red wall, the red wall still reflects the blue image rather than absorbing the blue light? I don't know if this is considered a dumb question, but I was thinking about it and it got me curious (also, I haven't taken any kinds of physics).


What can I say about compatibility between predictions and results?

If I have these theoretical predictions: \begin{align} \omega_{p_1} = 4.5132 \pm 0.0003~\text{rad/s} && \omega_{p_2} = 4.5145 \pm 0.0002~\text{rad/s}\\\ \omega_{b_1} = 0.0707 \pm 0.0003~\text{rad/s} && \omega_{b_2} = 0.0700 \pm 0.0002~\text{rad/s} \end{align} And I got these experimental results: \begin{align} \omega_{p_1} = 4.5148 \pm 0.0001~\text{rad/s} && \omega_{p_2} = 4.5147 \pm 0.0001~\text{rad/s}\\\ \omega_{b_1} = 0.0702 \pm 0.0001~\text{rad/s} && \omega_{b_2} = 0.0707 \pm 0.0001~\text{rad/s} \end{align} Can I only say that the experimental measurements are incompatible with the theoretical predictions even if the discrepancy is in the order of $\sim 10^{-3}\text{ s}$? Or can I say something else more?


What happens in a car crash?

My mom and I were in a car accident. We are ok, but I want to know how fast the car that hit us was going. We were stopped at a light. The car that hit us from behind was a big GMC SUV. Our car was a Cadillac srx and mom looked it up it weighs 4,277 pounds. Then we hit a little car in front of us at 12mph and the air bag went off! Mom says the car behind us must have going faster than that because she says our car absorbed some kinetic energy. I don't know what that means but I think it was going slower and the hit pushed us to go faster. How can we figure out what actually happened?


Universe Expansion as an absolute time reference

Why we call "constant" to the Hubble constant?, if the universe were really expanding then the Hubble "constant" should change, being variable, smaller and smaller..with "time". Other example/view of this question: if Hubble constant were constant then the universe "age" would be constant too, so let's say 13.7 billion years, and stop growing. That's not an expanding universe. On the other hand: If the Hubble constant and universe age were not so constant then: There is some property that is changing like a "time" that -everyone in universe- could get same number, so there would be an absolute time.. I see the Hubble Constant, and even the age of the universe, a really too speculative theory (perhaps it is as speculative as the posibility of measuring), having more accuracy will lead to better theories... Is it because some theoric impossibility in measuring that kind of variation? I see a lot of inconsistency. Any light at this respect is welcome, thanks


Calculating the age of the Universe now and 30 billion years from now

I am confused about the age of the Universe. If you calculate it by $1 / H_0$, won't the answer be roughly the same today as it will be 30 billion years from now? I know $H_0$ is a parameter, not a constant, but it doesn't change that much, does it? And if the expansion is accelerating then $H_0$ is going up, implying the age of the universe $1 / H_0 $ was higher in the past than it is today.


Calculate age of the Universe at 7 billion and 28 billion years old

If $1 / H_0 $ is about 14 billion years, then what happened when the universe was half its current age? Is the empirically determined $H_0$ supposed to have been twice its current value? And when the universe is twice its current age is Hubble's parameter half its current value? That would predict expansion is slowing. But expansion is actually speeding up. That implies that $ H_0 $ is getting larger, and because $1 / H_0 $ gets smaller in that case, the universe is growing younger. I asked this question already here, but it was marked a duplicate. I know that Hubble's Parameter is not a constant. That doesn't actually clarify the answers to these questions.


Is Hubble's constant really constant?

How does Hubble's constant resembles age of universe? Isn't universe getting old each day? How can a constant be a reciprocal of age of universe? Hubble's value must be variable, isn't it?


Electromagnetic energy with P and M

In many books, for a linear dielectric medium in which we have Maxwell's equations and the relationships $${\mathbf D}=\varepsilon({\mathbf x}){\mathbf E}$$ and $${\mathbf B}=\mu({\mathbf x}){\mathbf H}$$ we find formulae for the electromagnetic energy as $$U=\int d{\mathbf x}\frac{1}{2}({\mathbf E}\cdot{\mathbf D}+{\mathbf B}\cdot{\mathbf H})$$ in e.g. J.D. Jackson's book. Suppose, instead, we have Maxwell's equations and the ${\it general}$ linear relationship $${\mathbf D}=\varepsilon({\mathbf x}){\mathbf E}+{\mathbf P}$$ and $${\mathbf B}=\mu({\mathbf x})({\mathbf H}+{\mathbf M}).$$ We are willing to suppose that we know what the energy for making the polarization $\mathbf P$ and magnetization $\mathbf M$ appear are - but want to know what the electromagnetic energy is. Think for example that we are only moving a bunch of permanent magnets around and want to know how the energy changes when we do. We don't care about the condensed matter cost of making the magnets and the medium has no hysteresis and is linear as above. What then is the formula for the electromagnetic energy? Do you have a reference for this (likely) more than century old result?


Can classical systems exhibit "strong coupling"?

Does the concept of strong coupling mean anything in a classical setting? If strong coupling means just an inability to apply perturbative methods to the Hamiltonian, then obviously yes, we can provide examples of "classical" systems that cannot be handled perturbatively, such as granular materials, dusty plasmas, etc. I have only rudimentary QFT knowledge (and strong coupling is originally a QFT concept), but my intuition from my background in spectroscopy tells me that there is more to the term strong coupling than just the need for nonperturbative methods. The example I am most deeply familiar with is Forster energy transfer or FRET in biophysics. You're in weak coupling if the non- radiative rate is small compared to the radiative rate. But you're in strong coupling if the non-radiative rate is so large that you can get energy migration, where multiple energy transfer processes occur in series. In that case, the excitation is delocalized and you have to treat the network of interacting atoms as a single entity. I was under the understanding that delocalization was an essential feature of the term "strong coupling". Now, back to the original question, to what degree is it appropriate to use the term strong-coupling in many-body classical systems, such as granular materials, dusty plasmas, and dense colloidal glasses? These systems are classical in the sense that there is complete decoherence, and hard sphere potentials are the rule. In one sense, I want to say yes, the kinetic energy terms carry little information and most of the energy is stored in potential energy terms, but I am still hung up on the issue of delocalization. Can anyone help set me straight?


In famous Einsteins Photoelectric effect, Why does intensity of light doesn't raise the kinetic energy of the emitting electrons?

The work function of any metal is no doubt constant for it is related to electromagnetic attraction between electrons and protons. However on increasing the intensity of any light source the kinetic energy of the emitting electrons must increase, mustn't it? Let us assume there is only 1 electron in a metal surface. Let hf be the energy required to expel it out with a velocity 'v'. Again let us increase the intensity of the source keeping frequency constant. Now ' hf ' will change to 'nhf' where n = no. of photons striking on electron at a same time. Since work function is constant the only variable must be 'v' letting it increase its K.E. This clearly shows Kinetic energy of emitting electrons is directly proportional to the intensity of light source. If kinetic energy depends upon the intensity, stopping potential for a particular frequency of light for a particular metal is a variable quantity. It is true that on increasing the intensity the no.of photo electrons will increase. But what if the no.of electrons in a metallic plate is constant. Suppose I have only two electrons in a metal, on increasing intensity i.e on increasing the no.of photons , the no.of photons colliding from different sides simultaneously may increase . So, K.E of emitted electron must increase on increasing intensity, mustn't it? But experimental data doesn't show this. What is wrong? Help me out


Has Bose-Einstein theory been considered for dark matter? The theory would explain why no measurable radiation is emitted due to zero temperature--its lack of interaction with other matter and its gravitational lensing. Yet I haven't seen it considered. Am I missing something or have I totally misunderstood dark matter and/or Bose-Einstein theory?


How was it possible to beat a soldier with a full plate armor?

From my point of view, it seems that a soldier armed with a **full metal plate armor** was almost invulnerable at the time their opponents yielded swords, spears or bows. I understand that it couldn't be the case, but I'm not sure about the physics behind it. More specifically:   1. How could an archer beat this soldier? Would it matter whether the archer aimed at more vulnerable spots at the soldier, or any arrow shooted with full power would do?   2. How could a swordsman or a spearman beat this soldier? Would the impact of these weapons suffice?   3. Just out of curiosity, a modern gun would penetrate into the plates that easily? One headshot would do? I'm thinking on a soldier wearing an armor like this one below, or even more bulkier. ![Italian suit of Armor, c.1450](http://upload.wikimedia.org/wikipedia/commons/thumb/3/34/Italian_- _Sallet_-_Walters_51580.jpg/378px-Italian_-_Sallet_-_Walters_51580.jpg)[1] [1] Courtesy of Wikipedia


Static Friction on a Circular Road

I was recently brushing up on my dynamics, when I saw the interpretation of how a car moves in a circular road. Apparently, the car can stay in a circle due to static friction acting inwards. But I don't seem to understand exactly why this force exists and where it originates. What kind of motion causes it to exist?


Why isn't jumping against a wall an elastic collision?

According to this calculator http://www.abecedarical.com/javascript/script_collision1d.html when low mass object hits high mass object it is reflected gaining opposite velocity almost the same as initial velocity. If I jump onto the wall why my body is not reflected? I know that collision is not fully elastic but it should be at least similar.


Inertia and momentum

Question #1: Why does speed have nothing to do with inertia? Question #2: If a car hits a steel wall and stops, where did the momentum go?


If wave packets spread, why don't objects disappear?

If you have an electron moving in empty space, it will be represented by a wave packet. But packets can spread over time, that is, their width increases, with it's uncertainty in position increasing. Now, if I throw a basketball, why doesn't the basketball's packet spread as well? Wouldn't that cause its uncertainty in position to increase so much to the point it disappears? EDIT: I realize I wasn't clear what I meant by disappear. Basically, suppose the wave packet is spread over the entire Solar System. Your field of vision covers only an extremely tiny part of the Solar System. Therefore, the probability that you will find the basketball that you threw in your field of vision is very small.


What does temperature look like at the subatomic level?

I am trying to get a better understanding of the definition of temperature at the subatomic level. I have a background in molecular biology with some college physics, but no deep quantum mechanics background. Everything I've found on the web (Wikipedia, Google Scholar) seems to use 'temperature' very loosely as just " _agitation of particles_ ": more movement/agitation of particles equals higher temperature. But what exactly does this mean? The reason I'm asking is because the use of "particles" in relation to temperature seems to just mean **atoms**. The increase in agitation of _atoms_ is equal to an increase in temperature. But I am asking because I don't know if this is true. So atoms are made out of protons/neutrons/electrons. Protons and neutrons are _composite particles_ , each made up of 3 _elementary particles_ : quarks. Also, each of these examples I've mentioned are _matter_ particles, but other particles like photons are massless. So how do they fit into temperature? Basically, how do the different subatomic particles (both composite and elementary) relate to temperature?


Is Moon really rotating about its axis?

Wikipedia says yes but on Newtonian arguments. From general relativistic point of view Moon is not rotating but moving along geodesic trajectory. And like parallel transport of a vector (pointing to the direction of motion) on a surface of a sphere along equator, from outside it seems like rotation. So, would Moon still be rotating if we remove the Earth? Would Foucault pendulum on Moon detect its rotation?


Find true anomaly given period, eccentricity and time

Is it possible to find the true anomaly of an object in a Kepler orbit given the orbital period of the object, the orbital eccentricity and the time? Assuming a two body system and the mass of the orbiting body is negligible. I'm doing this computationally so I'd like to be able to place an object on an orbital ellipse with as few evaluations as possible. I ask about these orbital elements specifically because at the time in the program that I need to calculate the true anomaly, I have these values on hand already. ## Edit I've been searching extensively for an answer to this, and found these four steps to solving position as a function of time. The thing I'm having most trouble with is solving Kepler's equation -- it seems like there should be a simpler way to do this -- I can find the velocity at each apsis, and the orbital period relatively easily. As the eccentricity approaches 1 the orbiting object would spend more time near apoapsis and less time near periapsis -- but I can't seem to put that into numbers.


What makes General Relativity conformal variant?

I have a question regarding the well known fact that General Relativity is not a conformal invariant theory or to put it in other words about the fact that it is conformal variant: What are the physical assumptions in General Relativity about the gravitation that make the resulting formulation of gravity to be conformal variant? I am not asking about a mathematical reasoning because that is obvious: One performs a conformal transformation and he can check that the action and field equations are not invariant under this transformation. This mathematical part is obvious for me. I am asking what is the physical assumption that make this happen? What physical property of gravity (which is absent in Electromagnetic interaction) makes it conformal variant (at least in GR)? is it weak equivalence principle? is it the requirement of the Corresponding principle? or what? there should be some physics behind this, it can't be a mere mathematical coincidence ...


A puzzle from 'The origin of the hidden supersymmetry'

In the paper arXiv:1004.5489 The origin of the hidden supersymmetry, the author use {Qa,Qa}={Qb,Qb}=2H, {Qa,Qb}=0 for N=2 hidden SUSY, which is different from what I was taught: {Qa,Qa}={Qb,Qb}=0, {Qa,Qb}=2H. I think they are controversially different. Is any of them wrong? If not, could anyone help by explaining some more?


in an organic semiconductor, what is the average distance travelled by an exciton?

In an organic semiconductor, what is the average distance travelled by an exciton up to recombination? How is this value related to the morfology / structure of the organic semiconductor?


Where does energy go when performing a useless effort?

I went to school one day, so I thought I was able to get this simple one.. but it looks like I'm not anymore. :( One lonely little spaceship is resting into space. It has a small fuel capacity that it suddently burns to run away and finally reaches a speed $v$. Its tank is now empty, its kinetic energy $E = \frac{m v^2}{2}$ is exactly what it had in reserve, and it is equal to the work performed by the constant force $F$ it has just applied to itself over the run distance $d$ with $W = Fd = E$.. Now, instead of being free, it is tied with a rope to a wall. It does try to run away, burns its fuel.. But there is nothing to do, the rope is too strong. And it ends up with an empty tank and a null speed, a null kinetic energy, and the work $F$ has performed has been null all the time, for the rope's $-F$ has prevented any acceleration.. Where is its energy gone? The entire universe had energy $E$ all the time, the spaceship had $E$ initially (chemical potential) now $0$.. what is the system that had energy $0$ initially and now $E$, and why? what happened? If it is only friction and heat within the rope and the wall, does it mean there is no way modeling a "perfect rope" without assuming that some energy is lost?


Landau's derivation of a free particle's kinetic energy- expansion of a function?

I was reading a bit of Landau and Lifshitz's Mechanics the other day and ran into the following part, where the authors are about to derive the kinetic energy of a free particle. They use the fact that the Lagrangian of this particle must be the same (or at most, differ by the total time derivative of a function of co-ordinates and time) in different inertial frames. > _We have > $L'=L(v'^2)=L(v^2+2\mathbf{v}\cdot\boldsymbol{\epsilon}+\boldsymbol{\epsilon}^2)$. > Expanding this expression in powers of $\boldsymbol{\epsilon}$ and > neglecting terms above the first order, we obtain_ > $$L(v'^2)=L(v^2)+\frac{\partial L(v^2)}{\partial > (v^2)}2\mathbf{v}\cdot\boldsymbol{\epsilon}.$$ I think I'm ok with all the physics in this section. What I don't get is just the part I quoted above (so maybe this post is better suited for the math site, but since this is book is so physics-y I thought I'd post it here). My math is pretty rusty, so I'm not really sure- how do the authors expand the function to arrive at the above expression? It reminds me a bit of a Taylor expansion, but not very much. What's the process used to arrive at it?


What is the physical interpretation of the fermion field Hamilton?

I am still unclear what the hamilitonian of a quantized field is, but what I do know is the hamilitonian of the boson field is defined as \begin{align} H_{\text{boson}} &=& \frac{1}{2}\sum_{\mathbf{k},\mu=-1,1} \hbar \omega_{\mathbf{k},\mu} \Big({a^\dagger}^{(\mu)}(\mathbf{k})\,a^{(\mu)}(\mathbf{k}) + a^{(\mu)}(\mathbf{k})\,{a^\dagger}^{(\mu)}(\mathbf{k})\Big) \\\ &=& \sum_{\mathbf{k},\mu} \hbar \omega_{\mathbf{k},\mu} \Big({a^\dagger}^{(\mu)}(\mathbf{k})a^{(\mu)}(\mathbf{k}) + \frac{1}{2}\Big) \\\ &=& \sum_{\mathbf{k},\mu} \hbar \omega_{\mathbf{k},\mu} \Big( N_{\mathbf{k},\mu} + \frac{1}{2}\Big) , \end{align} however something that appears scarcely mentioned in the literature is the Hamiltonian of the fermion field. Interpolating from the boson field Hamilton, my guess is the Hamiltonian of the fermion field would be \begin{align} H_{\text{fermion}} &=& \frac{1}{2}\sum_{\mathbf{k},\mu=-1,1} \hbar \omega_{\mathbf{k},\mu} \Big({b^\dagger}^{(\mu)}(\mathbf{k})\,b^{(\mu)}(\mathbf{k}) + b^{(\mu)}(\mathbf{k})\,{b^\dagger}^{(\mu)}(\mathbf{k})\Big) \\\ &=& \frac{1}{2}\sum_{\mathbf{k},\mu=-1,1} \hbar \omega_{\mathbf{k},\mu} \Big({b^\dagger}^{(\mu)}(\mathbf{k})\,b^{(\mu)}(\mathbf{k}) - {b^\dagger}^{(\mu)}(\mathbf{k})\,b^{(\mu)}(\mathbf{k})+1\Big) \\\ &=& \sum_{\mathbf{k},\mu} \hbar \omega_{\mathbf{k},\mu} \Big( 0_{\mathbf{k},\mu} + \frac{1}{2}\Big) . \end{align} I found $H_{\text{fermion}}$ to be very different from $H_{\text{boson}}$, because $H_{\text{boson}}$ depends on the number of bosons $N_{\mu,\boldsymbol{k}}$, however $H_{\text{fermion}}$ is independent of the number of fermions in each of its states, but rather seams to equal the vacuum energy only. Is $H_{\text{fermion}}$ real and what is its physical interpretation?


Can induced-emf be divided?

A conductor's area is reduced by half or possibly more. I calculated the induced-emf based on the original area unreduced, does it make sense to divide the induced-emf value to the reduction factor? Say the area was reduced by half,therefore, diving the induced-emf value by /2? I assume by reducing the area the induced-emf should be less?


what happens when I roll a gyroscope along its axis of spin

Say:   1. I have a gyroscope that is spinning in the xy plane along the z axis.   2. I then roll its spinning axis by some angle theta Now I know the gyroscope will resist my attempting to change its axis of rotation. The question is:   1. When I perform work to change the axis of rotation.   2. Where will that work be transferred ? ... will the rotation speed of the disk slow down? To clarify: see the diagram below, the axis of rotation is around the Z axis - the disk is in the x-y plan. Now say the gyroscope is in a frame that allows it the disk's rotation axis to rotate in x-z plane -- which means the pivot is in the yaxis -- so assume there is a frame around the disk and there is a rod along the y axis on wither side of the disk attached to the frame. So, if I apply a fore in the -x direction at point 'a' the gyroscope will rotate and I will have to move the force a distance delta-x so clearly work is being done (lets assume no friction). The question is where does this work go?                                     /|\ z-axis                    | 'a'   |                    |       |                   / | \     |______\ x-axis                  \ _ /      \     /                    |         \                    |         _\| y-axis      Thanks.


Why do floating objects attract?

Objects floating on the surface of calm water seem to be attracted to one another and tend to come together in a drift. Is this attraction like gravity?


Calculating the Potential from the E-Field

I find that often times I'll be tripped up by questioning whether or not I can do something mathematically, and be unable to come up with a satisfying answer. This is, unfortunately, one of those times. I'm told: > A uniform electric field, $\vec{E} = E_0\hat{x}$. What is the potential, > expressed using cylindrical coordinates, $V(s,\phi,z)$? My first course of action is: We know... $$|r| = \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2} = x = E_0$$ $$\tan^{-1}{\frac{y}{x}} = \theta = 0$$ $$z = z = 0$$ So the electric field only has a component in the $\hat{r}$ direction. Now, we know that $\vec{E} = -\nabla V(r, \phi, z) = - \frac{\partial V}{r} - \frac{1}{r}\frac{\partial V}{\theta} - \frac{\partial V}{z}$ So, I think "Oh. I just have to integrate." ... but over what? Do I integrate three times, once w.r.t. radius, then phi, then z? I'm pretty sure that won't give me the right answer. If I decide to express $\nabla V$ in terms of cartesian coordinates, I get $- \nabla V(x,y,z) = E_0 \hat{x}$ ... but the question still remains. I feel like this is definitely the _easy_ part of the problem, and I can often do the more complicated parts—it's just small things like this often throw me off. How would I go about extracting the potential from either of those equations? I know I have to integrate, but... where?


Who was the R. Dolen behind Dolen-Horn-Schmid duality?

Can you provide biographical information for R. Dolen, known for Dolen-Horn- Schmid duality?