newtonian mechanics - Two objects exert forces on each other, will the reactions affect them?

Two objects go in against each other, and then they collide, will object 1, exerting force 1, necessarily get on it a reaction equal in magnitude and opposite in direction?

EDIT:- In my book it only says "for every action there is always opposed an equal reaction". But after reading on Newton's third law on Wikipedia, I read this "The action and the reaction are simultaneous"; it's not an intuitive statement (it sounds more like an object gets a reaction after an action), but anyway that solves my problem. If you care to explain how the two forces are always simultaneous please do.

Answer

It sounds like you may have been confused because intuitively, you would think a reaction occurs after the action, in response to it. As you've found, that is not what Newton's third law is saying. The reaction force is not a response to the action. For this reason, some people don't like the statement "For every action, there is an equal and opposite reaction."

A better statement of Newton's third law is to say that forces always occur in pairs. It is impossible for object A to exert a force on object B without object B also exerting a force on object A. The two forces are simultaneous, of equal magnitude, and in opposite directions.

Mathematically, this is stated as

$$\mathbf{F}_{AB} = -\mathbf{F}_{BA}$$

schroedinger equation - Nonlinear dynamics beneath quantum mechanics?

Yesterday I asked whether the Schroedinger Equation could possibly be nonlinear, after reviewing the answers and material given to me in that thread I feel like my question were adequately answered.

However could there still be nonlinear dynamics underneath quantum mechanics that could potentially explain the "weirdness" of quantum mechanics? I know of atleast one serious physicist who is developing an approach in this direction, Tim Palmer with his Invariant Set Postulate.

Just like the Liouville equation is linear, but there are "deeper" nonlinear dynamics beneath it, couldn't the Schroedinger Equation be the same? This is atleast one of Tim Palmers motivations for developing his approach and is also what Dirac said way back in the days.

I suspect this is what David Zaslavsky meant yesterday when he responded to my other thread (Could the schroedinger equation be nonlinear) when he said:

That being said, there's no reason that the "true" theory underlying QM would have to be linear. In fact, for exactly the reason you pointed out (i.e. that general relativity is nonlinear), it's commonly believed that we will need some kind of nonlinear theory to properly explain the universe at its most basic level.

Any views / opinions / thoughts / theories ?

fluid dynamics - Why does the gas cloud of an underwater gunshot pulse?

I have been watching some slow motion video and I was intrigued by the slow motion underwater gunshot. The first moments of the video go as expected. The gun fires and a cloud forms in front of the gun. Then the bullet rips through the water, leaving an open space (which I assume is a cavitation vacuum.) Soon after, the rip closes, and the gas bubble shrinks (I'm assuming this is due to the cooling and/or compression of the hot high pressure gasses.) Then something funny happens. The bubble pulses a few times and almost appears to emit light. Each pulse is accompanied by a noise.

The video can be seen here from start, or just showing the bubbles .

What is this pulse effect and/or what causes it?

EDIT: Here is another great video for a possible explanation:

energy - Non-resonant but efficient frequencies

I understand that if the frequency of a driving force coincides with the natural frequency of an oscillator (say a pendulum), the rate at which energy is transferred to the same is maximized. However, there may be other frequencies that are not so efficient but do transfer energy to the system, i.e. the latter absorbs it.

The usual graphs plot intensity or amplitude versus frequency and they have the look of a steep triangle. So they point at the frequencies surrounding the natural one as the most effective. But what about frequencies that are sub-multiples of the natural frequency ($f_o$), like $f_o/2$, $f_o/3$ and so on. For example, when swinging I take impulse not every time I reach a peak but every other time. It seems to me that a force operating at this frequency should attain as much amplitude increase as the natural one, in every shot, though of course it shoots at lower rate.

Thus I understand that this actor does not shine up in a graph plotting against intensity (which is power/surface, so it has a time-dependence, ok?) but I don't see why these frequencies are not merited in a graph plotting purely against "amplitude", like this one:

A reason may be damping (the effect is wiped out before it can consolidate), but what if the oscillator were ideally free of damping?

Anyhow, leaving aside the graphs, can it be said that those are, after the natural frequency itself, the most effective frequencies in terms of increasing amplitude?

Answer

The details they don't put in physics textbooks, but that you likely learn from engineering experience is that resonance depends not only on internal structure of the system but how energy flows in and out. A resonant system tends to trap energy and that energy may or may not necessarily be admissible at the resonant frequency of the system. It depends on the internal structure.

Although the more common means of energy transfer for a swing is a cycle by cycle push in one direction, it's also possible to have a person in front also deliver a push such that the rate of energy input to the system is doubled. Each input, the same amplitude but 180 degrees out of phase with one another. You show only the amplitude frequency response of your swing (pendulum) system but there is also a phase component and this illustrates how the 180 degree out of phase push works. The phase plot below shows that approaching from lower frequency the near zero degree phase signal is admissible, and approaching from high frequency, 180 degrees. The change in phase across resonance is very sharp when the system is high Q (very little damping, very low energy loss).

For practical purposes the swing system depends on the swing's bearings and drag forces at some point to reach an energy flow that is equal and opposite to the rate of energy input from the push. Otherwise the swinger will eventually loop over the top! In principle for linear systems zero damping means all the energy entering the system stays there, and the resonant peak approaches infinite amplitude. But for practical, real systems there are nonlinearities that limit trapping of energy. Energy has a tendency to find a way out sometimes breaking the system (like the Tacoma Narrows Bridge collapse).

For the swing system the structure is such that energy rate and phase (in the case of two people pushing) must be specific but that's not necessarily true for all resonant systems. Consider the singing rod that's often used in physics demos on resonance. The energy in this case is supplied by slip-stick friction between rosin coated fingers and the surface of the rod; essentially broad band colored noise vibrations entering the rod's surface. In this case the internal structure of the rod filters out and concentrates energy from the noise input at the natural frequency of the rod. The rod admits and traps only a narrow band of the input excitation. The remaining band of frequencies are mostly dissipated as heat at the rod's surface.

thermodynamics - How do greenhouse gases trap heat?

I am looking for a molecular-level understanding of the greenhouse effect.

What is it about the carbon-dioxide molecule (and methane, and water, etc) that is different from other gasses (particularly, N₂ and O₂) such that it works in the atmosphere to trap heat?

Is it, say, the distance between nuclei in the molecules relative to the wavelengths of infrared light? Dipolarity of the molecule? A combination of various factors?

Answer

To absorb infrared light, a stretching or bending vibration of the molecule must change the molecule's dipole moment. In $N_2$ and $O_2$ there is no dipole moment regardless of how you stretch the bond. On the other hand, O=C=O can change dipole moment by the C moving toward one O and away from the other O, or by bending with the C becoming a vertex of an obtuse angle. Water and methane molecules can also change dipole moment.

hilbert space - State-operator map, and scalar fields

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned https://physics.stackexchange.com/q/215060/ which was wrong question. By studing Ginsparg's applied conformal field theory now I become familiar with the concept of operator state map. Which indicates that between the state in $R\times S^1$, cylinder and operator in $R^2$, plane, there is a one-to-one map. $i.e$, following conformal map we can make one to one map between them. \begin{align} \xi = t+ix, \quad z = \exp[\xi]=\exp[t+ix] \end{align} here $\xi$ is a cylinder's complex coordinate, and $z$ is a plane's complex coordinate.

Now i am curious about the field between them. For example, for scalar field $\phi(t,x)$ in cylinder after conformal map how this changes in plane? $i.e$ From the conformal map, combination of $t,x$ maps to specific value of $z$, and scalar field is dependent of $t,x$ thus it should be function of $z$ in the other side. I want to know how this works in detail.

Answer

Under conformal mapping z=>w(z) and $\bar{z}$=>$\bar{w}(\bar{z})$ a field of conformal dimension(h,$\bar{h}$) transforms as $\tilde{\phi}(w,\bar{w})=(\frac{\partial w}{\partial{z}})^{-h}(\frac{\partial \bar{w}}{\partial\bar{z}})^{-\bar{h}}\phi(z,\bar{z})$..

research level - The bijective correspondence between a symmetric polynomial and edge excitation of the fractional quantum hall droplet

I am recently reading Xiao-Gang Wen's paper (http://dao.mit.edu/~wen/pub/edgere.pdf) on edge excitation for fractional quantum hall effect. On page 25, he claimed that it is easy to show that there exist a bijective correspondence between a symmetric polynomial and edge excitation of the fractional quantum hall droplet. As we all known that Laughlin state is a zero-energy eigenstate for Haldane pseudopotential. And it is easy to see that if a symmetric polynomial times the Laughlin wave function, then that increases the relative angular momentum for particles, thus that wave function is still a zero-energy eigenstate for Haldane pseudopotential. However, Wen claimed that the reverse also holds, but I am not quite convinced by his argument in his paper. Does anybody know how to rigorously show that the reverse is also true, that is every zero-energy eigenstate is of the form of a symmetric polynomial times the Laughlin wave function?

Answer

Looks like I have to answer this question :-)

Let me first answer the math question: Every zero-energy eigenstate is of the form of a symmetric polynomial times the Laughlin wave function.

To be concrete, let us consider an $N$ boson system, with delta-potential interaction $V=g\sum \delta(z_i-z_j)$ where $z_i$ is a complex number describing the position of the $i^{th}$ boson. The zero energy state $\Psi(z_1,...,z_N)$ satisfies $\Psi(z_1,...,z_N)=P(z_1,...,z_N)exp(-\sum_i |z_i|^2/4)$ where $P$ is a symmetric polynomial that satisfy $\int \prod_i d^2 z_i \ \Psi(z_1,...,z_N)^\dagger V \Psi(z_1,...,z_N) =0$.

Now it is clear that all the zero energy state are given by symmetric polynomial that satisfy $P(z_1,...,z_N)=0$ if any pair of bosons coincide $z_i=z_j$. For symmetric polynomial this implies that $P(z_1,...,z_N) \sim (z_i-z_j)^2$ when $z_i$ is near $z_j$. The Laughline wave function $P_0=\prod_{iarXiv:1203.3268.

However, a physically more relevant math question is: Every energy eigenstate below a certain finite energy gap $\Delta$ is of the form of a symmetric polynomial times the Laughlin wave function for any number $N$ of particles. (Here $\Delta$ does not depend on $N$.)

We only have numerical evidences that the above statement is true, but no proof.