# B You normally go through B and C when passing from A to D

1. Jul 21, 2017

### Vir27

I am wondering if there is a term for a certain phenomenon. I tried to search (here, google, asking folks whose coursework had more physics courses than mine [1 :) ]). But I have trouble coming up with specific terms to search, which is the original question anyway?

I found several topics (about teleportation ;) ) here that discussed it directly...! ...but did not name it :) I have understood from searching, though, that this phenomenon, while common, is not guaranteed by known physical laws. That's just fine; that's not what I'm asking about. So I don't want to entice people unnecessarily to tell me about the circumstances of wormholes, teleportation, or passing through walls if you lean long enough. These things seem like exceptions to the "rule" I am asking about, but I am asking if there is term for the rule, not how to bend it. Thanks :)

The phenomenon is that, generally, you can't get from any point A to any point D without also passing through all points B and C in between.

Isn't there a handy term for referring to this phenomenon? For what it matters, it seems intuitive to me that this phenomenon is fundamental to observed experience, like inertia, the zeroth law of thermodynamics, or causality (though, again to admit, perhaps this phenomenon is not as reliable for physicists after all). Therefore, I wonder didn't somebody ever name it?

2. Jul 21, 2017

### olgerm

Do you mean that trajectory is continuous?
In one dimensional space it means that you cannot go from any point $x_A$ to any point $x_D$ without also passing through point $x_B$ if $x_A>x_B>x_D \vee x_A<x_B<x_D$

3. Jul 21, 2017

### Vir27

Yes, those look like other ways to put it to me.

4. Jul 21, 2017

### jbriggs444

The "intermediate value theorem" provides that guarantee -- providing that the function is continuous.

5. Jul 21, 2017

### Vir27

Possibly you're thinking I'm asking about points mathematical "space" instead of points in physical space, jbriggs444? (lol, I hesitated to use a math analogy in the first place because it might mislead people in this exact way ;) )

Is there a physics version of the intermediate value theorem?

6. Jul 21, 2017

### jbriggs444

The mathematics version applies if position is a continuous real-valued function of real-valued time. But that is a very big if.

7. Jul 21, 2017

### olgerm

If you assume that Newtons 2.law is correct, mass is constant (real number) and force is always finite (real number), then position must be continuous function and intermediate value theorem applies to it.

8. Jul 21, 2017

### Vir27

Finishing(?) the original question
So this seems a decisive answer. The "trajectory is continuous" phenomenon doesn't have a name (in current usage, at least: see below).

Further, I am hearing that this isn't as interesting a find to the informed physicist as to me, the amateur, because the phenomenon itself is 1) contingent on an above "very big if," and 2) not pertinent or reliable from a conceptual view. One does find it a pertinent and reliable phenomenon for doing, you know, motion in practice in the same way that inertia is pertinent and reliable On the way up to the third floor, the elevator always does pass the second, but never the fifth. I never do open a door at work and--surprise--I'm in Mexico City. I personally am not aware of a practical situation in which one would do better not to take it for granted, though I'm open to hear. Still, the relevant information for my question seem to be that if it were a phenomenon worth modern physics' having a term for, then this term would have come to your notices as handy already. That's good enough for me.

Point of Order: Respect Poe's Law :) That paragraph may sound sarcastic, but no sarcasm is intended. It's kind of hard to express judgment on such a counter-intuitive matter without sounding facetious.

I wonder if it might have an archaic term, though. I say this because the factors that make it pertinent and reliable in practice have been there since before folks were making physics terms, obviously. Unless I'm mistaken, the notions which problematize seeing position and time as continuous, real value functions are relatively recent in the physics world? So there was quite a lot of time in which folks might have found this phenomenon handy enough to name.

Supposing that I am concluding correctly from the above about there is no term for this and why, then an answer I got on GameFAQs.com seems to come closest to naming it: "In a way the whole of classical mechanics is this property."

So I'm hearing from olgerm that, if you were to set about explaining how come "trajectory is continuous" in physics terms, you'd say it follows from Newton's Second Law?

"If... mass is constant (real number) and force is always finite (real number), then position must be continuous function and intermediate value theorem applies to it."

Let me see if I understand that by trying to unpack it.

I reckon why we might accept constant, finite mass and force as entailed by the situation I propose.

"mass is constant (real number)"

An object would not have to go through point B on its way to point C if at point B it stopped being an object. Thus, we need that mass be constant.

"force is always finite (real number)"

An object would not have to go through point B on its way to point C if it "were not anywhere" on account of it increasing the rate at which it changed position infinitely? And that can't happen anyway because:

Force = mass * acceleration

The product of two finite quantities is finite, I suppose. (I don't know the name for that rule either ;) .)

Infinite acceleration is precluded by the finite speed of light being proved insuperable. While... infinite mass is impossible because it would also imply infinite energy because E=mc^2 and there is no infinite energy because then clearly energy was not conserved?

"then position must be continuous function"

This one was harder. I am trying to understand this by thinking about a position-time graph of a function. If there were a break in the graph, that would mean there was a point for which "you can't get there from here." But there is nowhere an object can't get from here because for any arbitrary y value position on the graph, I can draw a line from any other position to get there at some slope of velocity. So this is why you tell me the function must be continuous, if mass doesn't give out at some position (and in spacetime mass is conserved) to make a discontinuity in the graph and if the impossibility of infinite velocity (in spacetime energy is conserved) precludes asymptotes. Is that right?

Well, if that is right that it a corollary of Newton's Second Law and such, and if it is right that it isn't a pertinent phenomenon for modern physics, it would still be easier to refer to if it had a name! :) "Olgerm's continuity of trajectory" may serve well :)

Caveat
In physics, despite the notions which I put together to explain to myself the relevance of finite mass and acceleration, the "very big if" line indicates that my argument is invalid.

Last edited: Jul 21, 2017
9. Jul 22, 2017

### Tom.G

"Reality is contiguous; perception of Reality, not so much."

10. Jul 22, 2017

### Vir27

Is perception of reality not part of reality? If it is neither unreal nor contiguous, is reality contiguous? :)

11. Jul 23, 2017

### Vir27

Could anyone say whether I have understood olgerm, or else speak to #8? Much appreciated.

Being new here, I am not sure if there just isn't much to say about this "continuity of trajectory" thing or what. It seems like it's a question for physics to me.

12. Jul 23, 2017

### jbriggs444

Classical physics assumes that objects have three dimensional positions that vary as continuous and differentiable functions of time. The laws of classical physics are expressible as differential equations. If you do not make those assumptions, differential equations do not work. Experiment at the normal human scale shows that objects closely follow the laws of classical physics. So we accept the assumptions of continuity and go about our business.

At the scale of the very quick and the very small, the world may not fit these assumptions. It is not possible to measure an object's position continuously and exactly. This is the world of quantum mechanics. Our best models at this scale still use a continuum, but they do not require that particles always have an exact position.

It is not known whether the universe is really discrete or continuous. Experiment does not (and, perhaps can not) say one way or the other.

13. Jul 23, 2017

### olgerm

I did not mean that, but it is also true, that if mass of an object suddenly become 0, then it basically disappeared. In classical physics masses of objects do not change at all that way.
You are wording it wrongly, but basic idea is correct.
No.

$\begin{cases} \vec x(T_1)=x(T_0)+\int_{T_0}^{T_1}(dt_1 \cdot(v(T_0)+\int_{T_0}^{t_1}(dt_2 \cdot \vec a(t_2)))) \\ \vec a(T_1)=\frac{\vec F(T_1)}{m} \end{cases} \Rightarrow \vec x(T_1)=x(T_0)+\int_{T_0}^{T_1}(dt_1 \cdot(v(T_0)+\int_{T_0}^{t_1}(dt_2 \cdot \frac{\vec F(t_2)}{m})))$
From these equation follows, that if F and m are finite real numbers, then x must be continuous function.

If $\frac{F}{m}$ is second derivative of that function (curvature of graph). If second derivative is everywhere finite, then function itself must be also contnuous.

14. Jul 23, 2017

### jbriggs444

Stronger: If the second derivative is everywhere defined, the first derivative must be everywhere defined and continuous. If the first derivative is everywhere defined, the function must be everywhere defined and continuous.

Last edited: Jul 23, 2017
15. Jul 23, 2017

### Vir27

Ahhhh, thanks for the detailed replies, fellows! I appreciate your time.

From jbriggs, I see why there is this "very big if" and why the question is slightly askew of proper physics discussion. I remotely of knew those things about there being a tension between classical and quantum mechancis, but I did not properly apply the information to my question. I had been reading here in other threads about the Born probability of passing through walls, about which I know nothing, and I just assumed that the very big if was something I don't know about like that. Thanks for humoring me, then.

From olgerm, I am not in context for the more involved expressions there, but that's alright. I am very glad you responded because I worked at it to try to follow what you were saying. Unless you tell me I'm mistaken, I feel it's fine I don't puzzle out the context for the larger expressions you cited. I am glad you included the last bit: now you mention it, I remember the "second derivative test." That was not going to occur to my mind, but acceleration as the second derivative of motion and rearranging for F/m=a makes sense!

Thanks, Physics Forums.

16. Jul 25, 2017

### GreyNoise

I couldn't resist the parting comment that your original question brings to my mind, what are called in physics, the equations of continuity; they arose originally in fluid mechanics but the concept shows itself in electrodynamics and energy as well. I lifted a paragraph from a Wikipedia article:

"Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy is fixed. This statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement ..."

Just remember that Wikipedia is not authoritative, but that doesn't mean it cannot be helpful; it just means that you must corroborate what you find there with an authoritative or reliable content source.

Anyway the equations of continuity are a physics concept that buttress the conservation laws with the restriction that, "... you can't get from any point A to any point D without also passing through all points B and C in between." So in answer to your question, "Isn't there a handy term for referring to this phenomenon?", I will pitch the that the answer is yes, the equations of continuity (albeit that is not a term, singular).

17. Jul 26, 2017

### Vir27

That's a good page. If I had found that page, I'd have been satisfied without asking. Thanks!

Thanks for posting!

18. Jul 27, 2017

### Buzz Bloom

How about tunneling. Doesn't this phenomenon imply that a particle moves from one location to another without ever occupying any points in between?

19. Jul 27, 2017

### Vir27

I really don't know. Is it skipping points or taking an alternate, presumably shorter set?

20. Jul 27, 2017

### jbriggs444

It is detected here. It was detected there. Anything else is philosophy, not physics.