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mimno committed Dec 6, 2017
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I generally tell the story that the action principle is another way of getting at the same differential equations -- so at the level of mechanics, the two are equivalent. However, when it comes to quantum field theory, the description in terms of path integrals over the exponentiated action is essential when considering instanton effects. So eventually one finds that the formulation in terms of actions is more fundamental, and more physically sound. But still, people don't have a "feel" for action the way they have a feel for energy.
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I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.). I have this book from Saletan and Jose , but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics. Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing.
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Yes, in the appropriate limit. Roughly, the study of geodesic motion in the Schwarzschild solution (which is radially symmetric) reduces to Newtonian gravity at sufficiently large distances and slow speeds. To see how this works exactly, one must look more specifically at the equations.
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Maybe no symplectic geometry or forms here, but this book has a LOT to offer:
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What happens if you lean on bike while trying to ride straight? It happens that you turn on same side you lean. Why, because you have to generate centrifugal force and lean back up if you do not want to fall down. So if you lean left you have to make left turn in order to generate force which will turn you straight position. If you turn too much left you will eventually fall on right because of too much centrifugal force kicking you other side.
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I have an Hamiltonian problem whose 2D phase space exhibit islands of stability (elliptic fixed points). I can calculate the area of these islands in some cases, but for other cases I would like to use Mathematica (or anything else) to compute it numerically. The phase space looks like that : This is a contour plot make with Mathematica. Could anyone with some knowledge of Mathematica provide a way to achieve this ?
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Structure and Interpretation of Classical Mechanics ( table of contents ) certainly deserves mention. It might not have as much differential geometry as you'd like, though they have a followup article titled Functional Differential Geometry .
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I think it is correct your appreciation if Gauss law holds in a two dimensional world, then the electrostatic force should be inversely proportional to the distance between charges. However, I'm not at all convinced that Gauss law could be true in a two-dimensional world because $F_{q}=k \displaystyle \frac{qq'(r-r')}{|r-r'|^{3}}$ is a consequence of a 3-dimensional space and since we derive Gauss law from such a force law (to be precise, from its electric field), we can not assume the validity of Gauss law independently from a 3-dimensional space.
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Pure beaten gold, any edition, paperback. It never leaves you. Lagrangian approach. See the reviews on Amazon L D Landau (Author), E.M. Lifshitz (Author)
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There is also Feynman's approach, i.e. least action is true classically just because it is true quantum mechanically, and classical physics is best considered as an approximation to the underlying quantum approach. See or l . Basically, the whole thing is summarized in a nutshell in Richard P. Feynman, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. II, Chap. 19. (I think, please correct me if I'm wrong here). The fundamental idea is that the action integral defines the quantum mechanical amplitude for the position of the particle, and the amplitude is stable to interference effects (-->has nonzero probability of occurrence) only at extrema or saddle points of the action integral. The particle really does explore all alternative paths probabilistically. You likely want to read Feynman's Lectures on Physics anyway, so you might as well start now. :-)
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Yes, absolutely. In fact, Gauss's law is generally considered to be the fundamental law, and Coulomb's law is simply a consequence of it (and of the Lorentz force law). You can actually simulate a 2D world by using a line charge instead of a point charge, and taking a cross section perpendicular to the line. In this case, you find that the force (or electric field) is proportional to 1/r, not 1/r^2, so Gauss's law is still perfectly valid. I believe the same conclusion can be made from experiments performed in graphene sheets and the like, which are even better simulations of a true 2D universe, but I don't know of a specific reference to cite for that.
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Squeezed light can be generated from light in a coherent state or vacuum state by using certain optical nonlinear interactions. For example, an optical parametric amplifier with a vacuum input can generate a squeezed vacuum with a reduction in the noise of one quadrature components by the order of 10 dB. A lower degree of squeezing in bright amplitude-squeezed light can under some circumstances be obtained with frequency doubling. Squeezing can also arise from atom-light interactions. References:
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I learned today in class that photons and light are quantized. I also remember that electric charge is quantized as well. I was thinking about these implications, and I was wondering if (rest) mass was similarly quantized. That is, if we describe certain finite irreducible masses $x$, $y$, $z$, etc., then all masses are integer multiples of these irreducible masses. Or do masses exist along a continuum, as charge and light were thought to exist on before the discovery of photons and electrons? (I'm only referring to invariant/rest mass.)
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Trained as a pure mathematician, I see claims about the mass of a galaxy and other such huge measurements that are arrived at experimentally, and I just have to scratch my head. I know this is a bit of a vague question--but does anyone have a good resource for something like "measurement in astrophysics" or an introductory history of how astrophysicists figured out how to make measurements of this kind?
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Rest mass of elementary particles is not quantized: in the standard model, the masses are free parameters of the theory; they must be measured and introduced in the model experimentally. However, the mass of, say the Hydrogen atom is given by the mass of its constituent (proton and electron whose mass are given) minus the binding energy which is quantized.
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There are a couple different meanings of the word that you should be aware of: In popular usage, "quantized" means that something only ever occurs in integer multiples of a certain unit, or a sum of integer multiples of a few units, usually because you have an integer number of objects each of which carries that unit. This is the sense in which charge is quantized. In technical usage, "quantized" means being limited to certain discrete values, namely the eigenvalues of an operator, although those discrete values will not necessarily be multiples of a certain unit. As far as we know, mass is not quantized in either of these ways... mostly. But let's leave that aside for a moment. For fundamental particles (those which are not known to be composite), we have tabulated the masses, and they are clearly not multiples of a single unit. So that rules out the first meaning of quantization. As for the second, there is no known operator whose eigenvalues correspond to (or even are proportional to) the masses of the fundamental particles. Many physicists suspect that such an operator exists and that we will find it someday, but so far there is no evidence for it, and in fact there is basically no concrete evidence that the masses of the fundamental particles have any particular significance. This is why I would not say that mass is quantized. When you consider composite particles, though, things get a little trickier. Much of their mass comes from the kinetic energy and binding energy of the constituents, not from the masses of the constituents themselves. For instance, only a small part of the mass of the proton comes from the masses of its quarks. Most of the proton's mass is actually the kinetic energy of the quarks and gluons. These particles are moving around inside the proton even when the proton itself is at rest, so their energy of motion contributes to the rest mass of the proton. There is also a contribution from the potential energy that all the constituents of the proton have by virtue of being subject to the strong force. This contribution, the binding energy, is actually negative. When you put together the mass energy of the quarks, the kinetic energy, and the binding energy, you get the total energy of what we call a "bound system of $\text{uud}$ quarks." Why not just call it a proton? Well, there is actually a particle exactly like the proton but with a higher mass, the delta baryon $\Delta^+$. Technically, a $\text{uud}$ bound system could be either a proton or a delta baryon. But we've observed that when you put these three quarks together, you only ever get $\mathrm{p}^+$ (with a mass of $938\ \mathrm{MeV/c^2}$) or $\Delta^+$ (with a mass of $1232\ \mathrm{MeV/c^2}$). You can't get any old mass you want. This is a very strong indication that the mass of a $\text{uud}$ bound state is quantized in the second sense. Now, the calculations involved are very complicated, so I'm not sure if the operator which produces these two masses as eigenvalues can be derived in detail, but there's basically no doubt that it does exist. You can take other combinations of quarks, or even include leptons and other particles, and do the same thing with them - that is, given any particular combination of fundamental particles, you can make some number of composite particles a.k.a. bound states, and the masses of those particles will be quantized given what you're starting from . But in general, if you start without assuming the masses of the fundamental particles, we don't know that mass is quantized at all.
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One common way of making these measurements is gravitational lensing. Basically, astronomers look at some distant object which is located directly behind the galaxy in question. Since the galaxy is so massive, it bends the light from the more distant object around it, so we see an image of the object displaced by some angle from where it actually is in the sky. For a distant object in the right position, we can see multiple images, one from light deflected to the left and one from light deflected to the right. Measuring the angular separation between them allows you to compute the angle by which the light was bent, and in turn to determine the mass of the galaxy required to produce that deflection. For galaxies that are less massive but closer, close enough to resolve the spectra of individual stars, we can measure the Doppler shifts of the spectra of stars on the advancing side of the galaxy and on the receding side, and taking the difference gives (twice) the tangential speed of the stars at that radius, which is related to the amount of mass contained within that radius. So measuring the spectra of stars at the edge of the galaxy, or in e.g. a globular cluster that orbits the galaxy, can give you the total mass contained in the galaxy.
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First, I want to say upfront that this question need not dissolve into arguments and discussion. This question can and should have a correct answer, please don't respond with your opinions. GNUplot is very pervasive in Physics research. Many of the plots appearing in things like PRL and JPB are made in GNUplot. Why is this the case. There are much more modern tools for doing these types of graphs, and from my uninformed position, it appears that this would be easier. One obvious and relevant first point to make is that GNUplot is free and opensource. I respect this, but do not expect that this is the primary reason. I am hoping for answers that specify what GNUplot can do that can be achieved in other programs efficiently, say Mathematica. If there are none of these, and the reason is simply tradition/resistance to change, that is also fine, but I expect there must be some specific tasks one wishes to perform. Thanks!
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As many others said, the Sun feels the same force towards Earth as the Earth feels towards the sun. That is your equal and opposite force. In practice though the "visible" effects of a force can be deduced through Newton's first law, i.e. ${\bf F} = m{\bf a}$. In other words, you need to divide the force by the mass of the body to determine the net effect on the body itself. So: ${\bf F_s} = {\bf F_e}$
 ${\bf F_s} = m_s {\bf a_s}$
 ${\bf F_e} = m_e {\bf a_e}$
 therefore, $m_s {\bf a_s} = m_e {\bf a_e}$
 and ${\bf a_s} = {\bf a_s} \frac{m_e}{m_s}$
 Now, the last term is $3 \cdot 10^{-6}$! This means that the force that the Earth enacts on the sun is basically doing nothing to the sun. Another way of seeing this: $F = \frac{G m_s m_e}{r^2}$
 $a_s = \frac{F}{m_s} = \frac{G m_e}{r^2}$
 $a_e = \frac{F}{m_e} = \frac{G m_s}{r^2}$
 $\frac{a_s}{a_e} = \frac{m_e}{m_s} = 3 \cdot 10^{-6}$
 Again, the same big difference in effect. Regarding the centripetal force, it is still the same force. Gravity provides a centripetal force which is what keeps Earth in orbit. Note It's worth pointing out that the mass that acts as the charge for gravity, known as gravitational mass is not, a priori, the same mass that appears in Newtons's law, known as inertial mass . On the other hand it is a fact of nature that they have the same value, and as such we may use a single symbol $m$, instead of two, $m_i$ and $m_g$. This is an underlying, unspoken assumption in the derivation above. This is known as the weak equivalence principle .
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What would be the implications to the Standard Model if the Higgs Boson hadn't been found with the LHC? Also, if the Higgs Boson had not been found with the LHC, would it have been successfully proven as non-existent? Or would we just wait for an experiment with higher energy ranges?
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Penrose's Conformal Cyclic Cosmology‎ is such an example. There is late time acceleration due to Lambda, but no inflationary period in the beginning. The late universe is equated to the beginning (of the next big bang cycle) through a conformal factor which maintains the physics.
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Wikipedia actually has a very nice graphic with this information (which roughly agrees with what I remember hearing from people "in the know"): The point is that there are both lower and upper bounds on the mass of the Higgs boson. The LHC should be able to cover pretty much the entire range that has not yet been searched, so if it doesn't find the Higgs, we can be fairly confident that something is wrong with the Standard Model. Now, the question is, what could be wrong? Well, there are various possibilities. At the simple end, it's possible that there is more than one Higgs boson. The simplest possible model has only one Higgs boson, and for obvious reasons that's the model that many people are hoping is correct, but it's perfectly possible that there could be a multiplet of several Higgs particles instead. If there is more than one, I'm not sure how exactly that would change the lower and upper bounds on the mass range, but I believe that there is some possibility that if there is a Higgs multiplet, all the particles could have higher masses than we would be able to detect. (I used to know more about this but it's been a little while) At the other extreme, it could be that the whole theoretical framework of the Standard Model is incorrect. That seems pretty unlikely, since pretty much every prediction the SM has made has turned out to be spot on (except for the presence of the Higgs, of course, but that's still an open question). There are definitely alternate theories waiting in the wings that will be receiving quite a bit more attention if the Higgs is not found.
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In order to calculate the average speed you have to weight the time of the different parts of the trip, and not with the distance covered in the same parts! So the basic formula you hate to use is : $v_{avg}=S_{tot}/T_{tot}$ If your trip is divided into two parts - $S_1$ covered at speed $V_1$ and $S_2$ covered at speed $V_2$ - what you can't do is : $V_{avg}=\frac{V1\times S1+V2\times S2}{S_1+S_2}$ (i.e) actually what you did with yours: $\frac{1}2(40\ mph+60\ mph) = 50\ mph$, since in your example $S_1=S_2$. Whereas what you can do is : $V_{avg}=\frac{V1\times T1+V2\times T2}{T1+T2}$ That, given your input, can be written as $\frac{S_1+S_2}{S_1/V_1+S_2/V_2}$, which is indeed equal to $\frac{S_1+S_2}{T_1+T_2}$
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I agree with David and his disagreement. My experience and personal opinion is that GNUPlot is simpler and quicker for the real time analysis and then u already have the graph in gnuplot.. why would u bother to change it ;)
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Deuteron ( 2 H) is composed of a neutron (spin-1/2) and a proton (spin-1/2), with a total spin of 1, which is a boson. Therefore, it is possible for two deuterons to occupy the same quantum state. However, the protons and neutrons inside are fermions, so they proton/neutron shouldn't be able the share states due to the exclusion principle. Why it does not happen?
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The main problem here is this: Newton gives us formulas for a force, or a field, if you like. Einstein gives us more generic equations from which to derive gravitational formulas. In this context, one must first find a solution to Einstein's equations. This is represented by a formula. This formula is what might, or may not, be approximately equal to Newton's laws. This said, as answered elsewhere, there is one solution which is very similar to Newton's. It's a very important solution which describes the field in free space. You can find more about this formula -- in lingo it's a metric, here: The fact that they are approximations fundamentally arises from different factros: the fact that they are invariant laws under a number of transformations, but mostly special relativity concerns - in other words, no action at a distance - is a big one.
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Pulsars are a label we apply to neutron stars that have been observed to "pulse" radio and x-ray emissions. Although all pulsars are neutron stars, not all pulsars are the same. There are three distinct classes of pulsars are currently known: rotation-powered, where the loss of rotational energy of the star provides the power; accretion-powered pulsars, where the gravitational potential energy of accreted matter is the power source; and magnetars, where the decay of an extremely strong magnetic field provides the electromagnetic power. Recent observations with the Fermi Space Telescope has discovered a subclass of rotationally-powered pulsars that emit only gamma rays rather than in X-rays. Only 18 examples of this new class of pulsar are known. While each of these classes of pulsar and the physics underlying them are quite different, the behaviour as seen from Earth is quite similar. Since pulsars appear to pulse because they rotate, and it's impossible for the the initial stellar collapse which forms a neutron star not to add angular momentum on a core element during its gravitational collapse phase, it's a given that all neutron stars rotate. However, neutron star rotation does slow down over time. So non-rotating neutron stars are at least possible. Hence not all neutron stars will necessarily be pulsars, but most will. However practically, the definition of a pulsar is a "neutron star where we observe pulsations" rather than a distinct type of behaviour. So the answer is of necessity somewhat ambiguous.
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