I'm not sure whether to place this in the maths or physics forums, but it sortof follows on from my thanksgiving question.

Consider a system where the total energy equals the sum of potential and kinetic energies. The KE is proportional to the square of the speed as normal, but the PE reduces as speed increases.

Such a system might be a ball rolling down a hill. As it rolls down it picks up speed and KE, but its height decreases so its PE decreases with speed.

The attached is a display plot of such a system.


Now the interesting thing is that our present day system of mathematics depends upon functions being single valued. Modern maths cannot cope with the question "what speed corresponds to a given energy", for which there are two answers.

However electrical and other engineers cheerfuly use the fact that nature displays both answers simultaneously during the generation of hydro electricity.

This comment has bearing on wave mechanics (classical and quantum) and relativity as the real world situation lead to a discontinuity in the governing physical equations.

 

Now the interesting thing is that our present day system of mathematics depends upon functions being single valued. Modern maths cannot cope with the question "what speed corresponds to a given energy", for which there are two answers.

Are you kidding me. Math is the language of all sciences without math you wouldn't have the ability to understand , classify , calculate... physical quantities. Or even have a reasonable way to logical display a certainity.

Their are multivalued function 's f(x,y,z.... ) even multivalued vector valued functions ,....etc

Their is no inadequacies in math it is just the inability to know all the factors in nature.

 

Arithmetic by itself can't exactly cope with QM, but it becomes insufficient in applications much simpler than modern/theoretical physics

 

Their is no inadequacies in math

At least know what you are talking about.

Godel's theorem?

f(x,y,z.... )

This is not a multi valued function.

 

Arithmetic by itself can't exactly cope with QM, but it becomes insufficient in applications much simpler than modern/theoretical physics

Perhaps you could explain further as I didn't understand this.

 

I'm not sure whether to place this in the maths or physics forums, but it sortof follows on from my thanksgiving question.

Consider a system where the total energy equals the sum of potential and kinetic energies. The KE is proportional to the square of the speed as normal, but the PE reduces as speed increases.

Such a system might be a ball rolling down a hill. As it rolls down it picks up speed and KE, but its height decreases so its PE decreases with speed.

The attached is a display plot of such a system.


Now the interesting thing is that our present day system of mathematics depends upon functions being single valued. Modern maths cannot cope with the question "what speed corresponds to a given energy", for which there are two answers.

However electrical and other engineers cheerfuly use the fact that nature displays both answers simultaneously during the generation of hydro electricity.

This comment has bearing on wave mechanics (classical and quantum) and relativity as the real world situation lead to a discontinuity in the governing physical equations.

I completely don't understand the point you're trying to make. I can kind of see it has something to do with knowing things we aren't supposed to know... and maybe something to do with wave mechanics. But otherwise could you please clarify or restate your point?

-blazed

 

In math

The function y = f(x) is not allowed to have more than one value of y for any given value of x.

Equations for which this happens are officially not functions.


Yet I can see everyday occurences of physical processes (and some esoteric quantum ones) which break this rule.

 

In math

The function y = f(x) is not allowed to have more than one value of y for any given value of x.

Equations for which this happens are officially not functions.


Yet I can see everyday occurences of physical processes (and some esoteric quantum ones) which break this rule.

PE(v) is a function, KE(v) is a function, and Total E(v) is a function. I just woke up so I'm a bit slow. Do you have a better example of a physical process we fail to represent with a function? A function, may I remind you, is only a mathematical concept we use to represent, well, anything we want, but mostly to represent reality.

This is exactly like your other example with the Fourier series. The Fourier series is a tool WE came up with to represent signals in such a way to help us understand them, or use them in machinery. It doesn't mean the real signal is actually a sum of sines...

I think this is coming down to perspective. You are choosing to try and defy math when it's not necessary. You can either work with what we've got, or you can come up with a better system to represent the world...

-blazed

 

I think he's refering to phenomena like diffraction patterns, and is confused about the correlation between the wave function, and the actual photon..
A photon has a destiny (in a metaphorical sense, atleast from our POV).. It is a packet of energy that cannot replicate itself, if you fire it at a two-slit apparatus you would only measure one impact position [most probably in regions expected to be intensified].. If you fire a series of identical photons at the same slits though, with a delay between them, so they could not interfere with each other, over time you would delelop the expected diffraction pattern..
There is a definate outcome, we just don't have the data necessary to pinpoint a definate expected value, so we use the wave function to graph our certainty..
Do you watch Numb3rs? All that behaviour calculation crap could also be represented by a similar function.. Masses of humans are predictable when you average it all out, individuals on the otherhand are often predictable, but othertimes not at all..

 

First, math can deal with relations of all kinds, functional and non-functional, quite well.

Second, whether a particular problem can be viewed as a functional or non-functional relationship often depends on choices, such as the co-ordinate system, the frame for reference and it’s construct and in physics, your particular interpretation of the system.

Finally, I don’t agree with your example. The function of interest would be the total energy function E = KE + PE which would, given the complete description of the system, reveals everything you need to know and would certainly allow you to relate a given energy to a given speed.

 

Well hello Bill, perhaps you would like to expand upon your statements?

I don't recall mentioning relations or functionals, and what are non-functionals, other than that they are (presumably) not functionals?

I displayed a graph (physics definition) showing E =KE+PE, exactly as you describe. So what's to not agree with?
Only difference is that if I ask my graph the question

At what velocity does a particular total energy pertain I get two answers for every value of total energy except one. Which answer do I choose and why?

It may interest you to know that engineers at the Hoover Dam use the exact same equations to predict the depth of water flowing in the outfall channel of the dam as a function of the distance along the channel.
Both nature and these engineers have no trouble selecting the correct depth for the water at any section of the channel. This includes the section at the critical point where the water surface shows a step and physically posesses two depths. This is known as a hydraulic jump.

If you came to visit me you could see a really spectacular famous natural version of this called the Severn Bore where a wall of water, up to 30 feet high rushes up the Bristol Channel at around 60 miles per hour.

Another way of modelling this is called a soliton or singleton wave.

 

Your graph shows the magnitude of the velocity, which is often the result of setting appropriate 'boundary conditions'..
Like Billo said, its up to the person analyzing the problem to determine appropriate cordinate systems, and boundary conditions etc..
Consider for instance that if your graph was extended to the left it would still be a valid function that yields only one value for each value on its independant axis, but if you were graph v with respect to E then you would technically be facing a relation, and may need to set bounds or find a toolset that can cope with the difficulty..
Problems can't always be looked at arbitrarily..
A good example about how failing math can get rescued by toolsets is the squeeze theorem in basic calculus..
Physicists are always looking for mathematical toolsets that allow them to express their ideas..

 

Lest us forget that we can graph more than one function on a single graph? Or we don't need to use the Cartesian plane. We can graph in polar or parametric form (among others).

Math, again, is simply our way of representing the world. Right now we have the tools to do that. Math can change, as it has in the past. Were we to discover something we could not represent with math, we would expand math to be able to do so. That's the beauty of a self-defined system like math or logic. Furthermore, if math fails to represent the world, it's simply that math is either inadequate or being used incorrectly.

I still feel your examples fail to demonstrate a situation where math and physics don't agree...

I'm sure there exists one, but it certainly isn't one engineers are using at all times. I could be wrong, but at this point I don't think you've proven it or explained it correctly, if that's the case.

-blazed

 

Well hello Bill, perhaps you would like to expand upon your statements?

I don't recall mentioning relations or functionals, and what are non-functionals, other than that they are (presumably) not functionals?

I'll try, but I’m not sure if you’re serious.

A function – is functional relation. A relation is a mathematical construct that relates two or more expressions. Those that relate a domain to a range with a one to one mapping are considered mathematically functional and are called functions.

{functions} Î {relations}
​

So you were talking about relations, but I think you knew this. Please feel free to look up the precise definitions of relations and functions.

As an example of choosing your coordinate system correctly, think of a particle in a circular orbit with radius ‘R’ around the origin. I we only observe it in the XY plane its mathematical model is:

R=x^2 + y^2
​

Which is a relation but not a function. However, if we observe that the position is time dependent then we can look at it separately in X and Y we will get:

y=Rcos(wpt)
x=Rsin(wpt)
​

Which are functions and together form a single vector function of time.
We could also look at it as a vector in polar coordinates:

r=R
j=wpt

Again, functions.

I displayed a graph (physics definition) showing E =KE+PE, exactly as you describe. So what's to not agree with?

Only difference is that if I ask my graph the question

At what velocity does a particular total energy pertain I get two answers for every value of total energy except one. Which answer do I choose and why?

Your plot shows 3 functions, KE, PE and KE+PE (dashed line). If you remove the plots for KE and PE you will get one value for the total energy for any given speed. Hence my comment “The function of interest would be the total energy function E = KE + PE which would, given the complete description of the system, reveal everything you need to know and would certainly allow you to relate a given energy to a given speed.”

It may interest you to know that engineers at the Hoover Dam use the exact same equations to predict the depth of water flowing in the outfall channel of the dam as a function of the distance along the channel.

Both nature and these engineers have no trouble selecting the correct depth for the water at any section of the channel. This includes the section at the critical point where the water surface shows a step and physically posesses two depths. This is known as a hydraulic jump.

In hydraulics water is treated classically and is considered continuous. It therefore has infinite depths. It’s just doing different things at different depths. Also, a hydraulic jump is not a true step function, it just cannot be. Engineers may indeed treat it that way for the sake of simplicity but Vanderwaal’s (sp) forces between the water molecules will ensure that there will be layers of turbulence, no matter how thin, between the streams of different velocities.

Do you have a link to a site that has more information on this?

If you came to visit me you could see a really spectacular famous natural version of this called the Severn Bore where a wall of water, up to 30 feet high rushes up the Bristol Channel at around 60 miles per hour.

We have a similar event that occurs (used to anyway) in the Petitcodiac river in Moncton.

 

Maybe I was not clear when I said this

Their are multivalued function 's f(x,y,z.... ) even multivalued vector valued functions ,....etc

multivalued is usual used to indicate that a given equation has 2 or more output's for one input ...etc.

But some old book's use this term as multivariate function's. Either way you are correct it was a bad choice of words.

I was just trying to get at the fact that math is just a language to represent are logic thoughts , proof's ,...etc in science and other subjects.

As for the Godel's completness theorem stuff. This was proven in MATH.
And basically say's that in any system of logic their will always be something that cann't be proven true or false.

But this was total not what I was trying to get at.
Plus we can always change or create a new math subject based on a given set of axioms that suit are scientific phenomena. Usual this is what applied mathematic's does. It models the math after nature ...etc etc

Your question is not the correct one to be asking because so what if it isn't a function. Hell the circle isn't a function provided that we don't restrict the range or domain.

And if you really feel uncomfortable their are books that just call an equation a function weather it is multivalued or not. But Normally mathematicians want to split these categories up and classify them by the properties. Numbers --> natural , fraction , real , complex , ...etc

Math is just a language to represent are scientific thoughts.

PE we usually take the inital starting point to be at infinity or a fix place completely arbitrary.

And you can always restrict the domain / range of a non-functions to
a function.

So I am just rambling on but I don't really get the post anyway.
Hopeful you got it now.

 

I am not trying to be disingenuous, Bill; and I am sorry that I have not explained my point very well, Blazed.

However I feel that, although you guys are correctly interpreting my graph you are missing the wood for the trees.

Funnily enough Triggernum seems to have the opposite problem of missing the trees for the wood.

Consider for instance that if your graph was extended to the left

I suggest you take a closer look at my graph and description of then physical variables.

The vertical (energy ) axis corresponds to zero velocity. If the graph were extended to this would imply negative velocity with the ball going backwards up the slope!

This sort of suggestion is precisely what this thread is meant to bring out.

We do not have any information as to what rules might apply to the left of the energy axis. So it is wrong to attempt to draw conclusions by extrapolating from the right hand side.

At the energy axis the ball's velocity is zero, as is the KE. The Total Energy therefore matches the PE exactly.

As the ball begins to roll the ball gains velocity and therefore KE in direct proportion to the square of the velocity. Meanwhile the PE diminishes linearly with velocity as the height above datum reduces.

The Total Energy follows the PE line down, gradually diverging as the KE begins to add significantly to the total.

Consequently at some point the KE overtakes the PE as the main energy component and the Total Energy curve now turns back upwards, coming increasingly close to the KE parabola and eventully matching it at infinity.

I'm afraid that's all there is to the Total Energy curve, folks.

This system can support an indefinite energy increase for TE and KE and decrease for PE,
to the right, not left, with no change of the mechanics (governing equations) of motion.

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.....?

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 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.

 

But, studiot, all you're saying is that this is not a one-to-one function... and changing the position on the horizontal axis only changes the input. We're allowed to change the input instantaneously (or close to instantaneously), we do it all the time.

I also disagree with your evaluation of the graph/situation. You said you're describing a ball rolling down a hill. A ball's potential energy is dependent on its height, not on its velocity (or speed). Then again, if you're talking about speed, then you're not talking about velocity. If that's the case then it's impossible for v to be less than 0. If you are talking about velocity, then velocity is only a vector, and negative velocity would would have absolutely no bearing on the potential energy. And since kinetic energy is directly related to the square of the velocity, it shouldn't cause a problem there either.

I still don't completely understand what you're getting at. I'm missing both the forest and the trees... I just see a plain... bad analogy, but you get my point.

-blazed

 

I am aware of the mathematical heirarchy of generality and the definitions of

Distribution, generalised function, function, functional, operator, relation etc.

I am also aware that these definitions produce a wealth of sound maths.

However they are not the only possible definitions and I am simply questioning if they are all appropriate in Physics.

I don't want to play with words and this is the Physics forum. So I am using the Physics definitions, where they are different from the maths ones . For example the term 'graph theory' means something very, very different in maths and physics.

Physics and Mathematics are different or there would be no point distinguishing them.
I can assure 'Mathematics' that Maths is much much more than just the handmaiden of Physics. Much maths was indeed developed to explain and handle the physical world. But much, particularly recent maths was not.

There are many constructs in the mathematical world that have no physical manifestation (that we know of). Equally there are physical phenomena for which we do not have adequate mathematical description. The recent painful introduction of Fractal Geometry was a good reminder that we do not yet know everything.

The attached sketch shows the difficulty of trying to extrapolate physical laws to a region where we have no information as to their validity.

The first graph shows a common explanation as to why it is 'impossible for a particle to travel faster than c'

The graph of the apparent relativistic energy becomes asymptotic to infinity at c.

But what assumption can we make about the mechanics beyond this asymptote?

The second graph shows a similar mathematical function which does something unexpected, and different from the assumption in the 'Light Cone' theory.

You may (not) recognise it from my poor sketch - it is the tangent function, but there are many other candidates.

Transiting through a hydraulic jump or the Mach barrier produces similar discontinuities in our mathematical description.

 

I also disagree with your evaluation of the graph/situation. You said you're describing a ball rolling down a hill. A ball's potential energy is dependent on its height, not on its velocity (or speed). Then again, if you're talking about speed, then you're not talking about velocity. If that's the case then it's impossible for v to be less than 0. If you are talking about velocity, then velocity is only a vector, and negative velocity would would have absolutely no bearing on the potential energy. And since kinetic energy is directly related to the square of the velocity, it shouldn't cause a problem there either.

It's important that we agree about the mechanics of a simple system, before moving on to a complicated system.

The further the ball moves from left to right, the further it will have moved along the x axis (a distance variable), the further it will have perforce moved down the slope (another distance variable) and the more speed or velocity it will have acquired since it is subject to constant acceleration. All of these are proportional so I could equally have labelled the horizontal axis distance or speed.

Direction is irrelevant here so there is no difference between velocity and speed. Sorry if that caused confusion.

 

This system can support an indefinite energy increase for TE and KE and decrease for PE,
to the right, not left, with no change of the mechanics (governing equations) of motion.

KEinitial + PEinitial + Wexternal = KEfinal + PEfinal
As a function this works for 'any' interval on your 'independant axis' (v).. This isn't a probability distribution any more than trig functions are..