However if you draw a line parallel to the horisontal axis it represents a specific value of energy. This line intersects the total energy curve in two points. So the system has two states, call them state A and state B, for any given total energy.

In the macro world of billiard balls and slopes the ball has never been known to switch spontaneously between state A and state B.
However in the quantum world of micro particles.....?

Hmmm, not sure about this. If you are looking at speed as the indepedant variable, then total energy is a function. What you talking about here seems to indicate that you are looking at total energy being the independant variable. If that is the case then, certainly, speed is not a function of energy.

Also in the macro world of fluid mechanics the Hydraulic jump is just such a switch. The mechanics are different before and after the switch so applying the mechanics of before will get you the 'wrong' answer.

I think I see what you're getting at, but the hydraulic jump (HJ) is not really a switch. There is a definite transition in which an awful lot of energy is lost to turbulance. The jump occurs when conditions change such that the laminar flow rate of the water exceeds the the new wave velocity, or if an obstruction causes turbulance that destoys the laminar flow. The maths I have seen on the HJ (and there is not much, fluid dynamics was never my area) seems to center around such things as the change in critical velocity, the position of the change and the flow rate. I suppose you could model it around a discontinuity in the critical velocity, but it would not give 100% accurate predictions.

Anyway, that's not really your point, but I think it might do to demonstrate that mathematical models are never really complex enough to fully describe real systems. Maybe that's where your delemma comes from.

I will expand, in a separate post, on the mechanics of the jump for Bill, in such a way that it does not matter whether the fluid particles are macro or micro or whether they are stuck together with araldite or Vanderwalls-ite, or held apart by rods or Maxwell's Demons or other apparatus. Their centres of mass, which determine the mechanics, do not change.

'sokay, I just did not get what you meant originally. I'm okay with hydraulic jumps (but no expert).

 

mathematical models are never really complex enough to fully describe real systems.

That's exactly the point, or part of it.

Part, because sometimes the maths is misleading when applied to real things. We should get the Physics right and then develop the maths to suit, not the other way around.

Part, because sometimes there are several mathematical possibilities, but only one will describe reality.

Part because maths itself can be based on some very dodgy ground.

As a for instance let us go back the Fourier series I mentioned once and has been thrown back at me several times.

All of the theoretical mathematical justification for FS is based upon the set theory premise that

If A ,B are members of the working set, then A+B are also members. We extend this indefinitely to allow the fourier components to sum to the function of interest and this works well until we try a humble square wave.

Nothing wrong with the summation it still works.
However we have stepped outside our set and therfore outside our rulebook.

The square wave is not a member of the class of continuous functions mathematicians use to prove the technique, in fact it is not even a true function.

 

That's exactly the point, or part of it.

Part, because sometimes the maths is misleading when applied to real things. We should get the Physics right and then develop the maths to suit, not the other way around.

Part, because sometimes there are several mathematical possibilities, but only one will describe reality.

Part because maths itself can be based on some very dodgy ground.

As a for instance let us go back the Fourier series I mentioned once and has been thrown back at me several times.

All of the theoretical mathematical justification for FS is based upon the set theory premise that

If A ,B are members of the working set, then A+B are also members. We extend this indefinitely to allow the fourier components to sum to the function of interest and this works well until we try a humble square wave.

Nothing wrong with the summation it still works.
However we have stepped outside our set and therfore outside our rulebook.

The square wave is not a member of the class of continuous functions mathematicians use to prove the technique, in fact it is not even a true function.

I think I finally am ... somewhat... getting at what you're saying. And I for the most part agree.

Let us even take the example of the square wave:

When we solve for the fourier series of the square wave we solve it for the function:

\(f(t) = 0 for 0 < t < \frac{T}{2},1 for \frac{T}{2} < t < T\)
Where T is the period.

We ignore the line connecting the up and down portions of the square wave because we KNOW that those portions aren't possible as part of a function!

-blazed

 

But mathematics are the tools needed to examine the physical world.. Thats kind of like asking somebody to build a house in order to earn the privledge of using tools..
We model the physical world mathematically, we apply the model to physical situations, we take note of situations where the mathematical predictions of our model don't match observation, and return to the theoretical drawing board in hopes of coming up with a more encompassing/accurate model.. During that time though we take full advantage of the predictions we can make with the flawed model in scenarios where it is sufficient for our purposes..
Gravity is the perfect example.. Cavemen knew how it worked on a rudimentary scale, and used that knowledge to improve their lives without math per se.. Newton vastly improved the model.. He expanded the scope of how we percieve it, and gave us equations that yield very accurate results in any common scenario.. Newton's gravity starts to fail in extreme scenarios like the orbit of Mercury, or black holes though.. Luckily we have a better model of gravity to work with in GR.. (Its also worth noting that while GR could make Newton's approximations obsolete, it hasn't..)
Now GR is problematic at the quantum scale, and physicists are picking sides about how to resolve it..

 

It's interesting to note he same discussions are still going on that we discussed 30 years ago. We need not go to the likes of FS and hydraulic jumps to show the difference between the physics and the math we use to try to describe it.

Heck, I can't think of a real situation where f=ma will give an absolutely accurate answer.

But it's really that the two, math and physics, are completely different disciplines. Sure, physics has driven a lot of development in math, most notably calculus, but calculus and the physics that gave rise to its development stand apart.

I wouldn’t say that the maths itself is dodgy, but certainly our use of it can be. Physicists are famous for giving mathematicians fits! We are always pulling the guts out of their beautiful theories because some ‘bit’ of it gets us close enough to where we want to be. QM is a perfect example of that.

 

discussed 30 years ago.

30 years ago we didn't have string theory.

String theory limits the minimum sizes of various physical quantities, (time, length etc)

Much of string theory proofs depend upon calculus, particularly integral calculus.

But the very proofs of calculus depend upon taking limits -the epsilon / delta argument. This cannot be truncated as required by string theory.

Blazed, nevertheless a square wave, extended to infinity in both directions contains an infinite number of discontinuities and therefore fails to meet the criterion for even a piecewise continuous function.
Also you need some equals signs in your expression.

 

Yes, I know. I was in a rush when I wrote that...

-blazed

 

What Billo is trying to say is that the concept of inadequate definition in the application of math to physics is old news..
To make a bad analogy in response to your String theory conundrum, there are always current events in the news, but at the same time its all old news..

 

Truth, like wine, improves with age.

I didn't claim any novelty to the facts, nor do I see the age of the questions as relevant.

Nor has anyone offered an answer to my original question which may be repharased

Provide a mathematical expression for the behaviour at a discontinuity in a physical system.

 

The pdf file I posted in your FS thread goes into quite a bit of detail regarding that particular question..

 

30 years ago we didn't have string theory.

Dinosaurs, no, but string theory was alive and well. Maybe not as entrenched as today. It was young a brazen, but its roots were very well developed and by 1980 the name was being used with familiarity.

Hey, I know you youngsters are all really brilliant, but whose shoulders did you think you were standing on?

 

.Do you mean:

Provide a mathematical expression for the behaviour at a

hypothesised

discontinuity in a physical system.

Because, quite honestly, I am not familliar with a real physical system where there is an empiricle discontinuity. There sure are mathematical models that approximate reality that have those nasty habits, but ... this seems to be soething we cannot observe with certainty.

 

I gotta stop thinking that 30 years ago was closer to 1970..