Jerk is defined as \(\frac{da}{dt}\) the rate of change of acceleration. Do you think that in the REAL WORLD, it is possible to have jerk with an infinite value? For e.g., an instantaneous change in the magnitude of a force on a body.
I'm sure someone read the thread title and thought "Have you met AlphaNumeric?"..... Anyway.... Yes, at least in theory. When two objects collide in a simplistic model (like 2 balls rolling along a surface) the force is instantaneous, which means it has infinite strength for zero time. These are represented by Dirac delta functions. If some function f'(t) experiences a instantaneous 'jerk' of \(k \delta(t-t_{0})\) then it means at \(t=t_{0}\) f(t) will jump by a value of k. For example, if \(\ddot{x} = k \delta(t-t_{0})\) then at \(t=t_{0}\) \(\dot{x}\), the velocity, will jump by k.
I don't think so. I'd be curious to see an equation for an infinite jerk which produces a reasonable physical displacement equation.
Object suspended by monomolecular cable in vacuum. A laser cuts the cable in less than 10 ns, leading to a jerk of at least a billion meters per second^3. Does someone have a better way to estimate the time or jerk in this setup? Taut monomolecular chain is severed by same laser, causing chain to snap back like a stretched rubber band. In a very short time the acceleration goes from zero to a high number (in human scale units). But I don't have an estimate of magnitude. X-ray scattering off free-electron. Currently this is modeled as infinite acceleration and thus infinite jerk. If string theory is correct, the answer is not so simple. I think unless there is a modeled physical upper limit to jerk, one has to admit the possibility of infinite jerk even if one cannot demonstrate it.
Funny thing is, it's not possible to have infinite velocities. You can't have infinite KE. It's conceivable that we coulld have infinite acceleration for an instant. But any longer and there'll be infinite work done as well. Infinite jerk in theory will exist only for an instant as well.
Hmm, I was thinking in terms of infinite jerk => infinite acceleration => infinite velocity => infinite momentum => infinite energy => broken physics.
actually, if you have accleration of constant magnitude that switches direction instantly, you will get saw-tooth velocity, and quadratic oscillations. Not possible to have a quadratic oscillator without infinite jerk.
Conventionally, all "infinite jerk" means is that the velocity or acceleration changed a finite amount in zero time. "Unbounded jerk" means that there is no lower limit for the time acceleration (or velocity) needs to change by a finite amount OR that in a given finite time there is no upper limit to how much (proper) acceleration can change. It's my position that unless you can demonstrate the laws of physics require jerk to be bounded, then we have unbounded jerk and if we have unbounded jerk then there is no conceptual reason to model velocity as piecewise-smooth with certain events self-consistently modeled as infinite jerk.
Idealized (2-D) circular roller on idealized piecewise-straight surface. If the height of the surface is \(h(x) = \left{ { \begin{eqnarray} h_0 & \quad \quad \quad & x < 0 \\ \frac{(L - x) h_0 + x h_1}{L} & \quad \quad \quad & 0 \le x \le L \\ h_1 & \quad \quad \quad & L < x \end{eqnarray}} \right.\) On the surface, the component of the force in the direction of the surface is \(f(x) = \left{ { \begin{eqnarray} 0 & \quad \quad \quad & x < 0 \\ Mg \frac{h_1 - h_0}{\sqrt{(h_1 - h_0)^2 + L^2}} & \quad \quad \quad & 0 \le x \le L \\ 0 & \quad \quad \quad & L < x \end{eqnarray}} \right.\) Which has infinite jerk at x=0 and x=L
Yeah we could do the same thing with most piecewise displacement functions, but eram was asking about "reality" and you said yet piecewise displacement functions cannot be physical for 2 reasons... 1) A known limitation of the theoretical approximation: Force and movement are not discrete. Reality is not two billiard balls "connecting" to transfer energy. It's a buildup of the distant repulsive forces of the respective EM fields emitted from the charged particles making up the billiard balls. "Contact" does not occur instantaneously, and therefore discontinuous acceleration functions are approximations only. 2) A theoretical problem exists anyway: If we allowed for discontinuous displacement, velocity, and/or acceleration functions we would break relativity. This is because we could then switch frames theoretically without experiencing acceleration due to your suggestion that the infinite acceleration is avoided by lasting zero length of time.
Oops, left that important word out. Not an issue for point-like particles, like electrons in the Standard Model. That's an argument against discontinuous position, not velocity or acceleration; and you misuse "frames" as the term is use in relativity. Velocity may indeed be discontinuous without any harm being done to relativity, as with semi-classical analysis of Compton scattering (i.e. billiards with electrons and photons).
No, that isn't the case. Frames are not something an object carries with itself, they are conceptual things, a way of describing the system. A ball moving at constant speed has a natural choice of reference frame, the one in which the ball is stationary. When that ball hits something, be it an instantaneous blow or not, that frame is unchanged, it's just now the ball isn't at rest in that frame and there's another frame where the ball is at rest (assuming all the interactions have been completed). Two particles hitting one another and instantly changing their velocities is fine. There's no action at a distance, no faster than light instant communication across a distance, everything happened at a point. There would be faster than light issues if you were considering large objects, not point particles. Then you have to consider that when the front of 2 balls collide it takes some time for the signal to propogate through their volumes, to the ball is squeezed slightly. This is why there's no absolutely rigid objects in special relativity, they would violate light speed restrictions.
I'm sorry but you guys are confusing the map of mathematics with the territory of reality (in fact, you're pulling out different maps to defend your POV). Common sense says point-like particles cannot collide (without infinitely graduated field interaction, making your point of instantaneous velocity change moot anyway). And when I say relativity is broken by instantaneous velocity change I'm not talking about a c-violation, I'm talking about a change of frame for an observer which is not experienced by that observer. The non-inertial experience of an observer is what differentiates the younger twin from his elder in the Twin Paradox, for example. What you're suggesting is that physics could theoretically change for an observer who would claim that he is inertial when he is not, which breaks relativity.
I disagree -- I say it is the geometric structure of space time. In curved space-time there may be two inertial paths that cross at two points in space-time allowing for a Twin Paradox without acceleration involved. In Minkowski space (or many other suitably flat space-times) acceleration is trivially needed to ensure the separating world-lines meet at some future point, and that requirement too is geometric in nature; but it is not the crucial element of the Twin Paradox.
But you're ignoring the consequence, which is that an observer subjected to acceleration without experiencing it would claim that he is inertial and that the laws of physics are changing. This essentially makes the first postulate of relativity meaningless, because we could claim that the laws of physics are changing all of the time but that we as observers are experiencing instantaneous frame changes to offset them. You also didn't address the fact that point particles cannot collide, or that the SM recognizes that it is actually tapered EM field interaction (as opposed to any sort of discrete collision) which underlie billiard balls, etc. I agree that the approximating math models suggest infinite jerk and infinite acceleration are permissible but I'm surprised that you continue to contend that they are physical.
How does any of this follow? Instantaneous acceleration is still acceleration. The first postulate of relativity states that the laws of physics all take the same form in inertial reference frames. This way of stating the postulate could be considered partially circular, as inertial coordinate systems are specifically those in which the laws of physics take their canonical form (at least, that's how I like to define them), though there's an understanding that inertial coordinate systems should all move at constant velocity with respect to one another. Either way, a reference frame attached to an instantaneously accelerating observer is not inertial. Newton's first law does not apply in such a frame, for instance: you would observe an instant velocity change in any free particle, assuming you even survived the acceleration. Your argument is a bit like saying an instantaneous rotation would be somehow detrimental to the principle of rotational symmetry in physics, of which relativity is essentially a generalisation.
