Sup

I see it in the Legendre transformation, which states:

f*(p) = sup(p*x-f(x))

What is the "sup" function? What does it do? How is it used? What is the point of using it? How does the Legendre transformation work? Can you please provide examples (preferably with functions that variables can be plugged into)?

What does p represent? I know that in transformations, f(x) is usually mapped onto a different variable, but in this particular transformation, doesn't really seem to be doing anything to f(x), because it's external to f(x).

Also, if the sup function is the supremum function, does that imply that the curve is full of maxima? This is really confusing to me, and I feel like I need to understand this before I can start to grasp thermodynamics.

 Unrelated question

Out of curiosity: If I differentiate x(t) = cos(ω*t), ω>1, an infinite number of times, does it eventually diverge? In other words, if x'(t) is the velocity, x''(t) is the acceleration, x'''(t) is the jerk, ... and the infinite derivative is the infinite change in some quantity of motion, does that imply that a simple oscillator can have an infinitely large change in some quantity describing the motion?

Unrelated Question: I think after acceleration it stops. That's like comparing going to the 2nd to 3rd to 4th dimension infinity times. I don't think we know that.

If a force is variable, then the acceleration is also variable. Why should it "stop" after acceleration? The only reason that acceleration is considered the stopping point is that Newton's second law states F = m*a, but that doesn't mean that x'''(t), or any further derivatives don't exist.

Also, acceleration and any of its derivatives exist in R^3, so I don't see what you're trying to say here with your "higher dimensions". And by the way, increasing the number of dimensions isn't difficult a difficult task. Simply assign a scalar value (temperature, energy, luminance, time, etc.) to every point in space. The more variables assigned, the more dimensions. That also means that each derivative can be assigned to a dimension, but what's the point of doing this? It's not a true Cartesian coordinate.

I've found the Wikipedia description quite clear on the way that the Legendre's transform works.

On the unrelated question:
The effect of large order derivatives on the original function decreases on 1/n! (coming from the taylor expansion), so even if your derivatives are increasing on w^n that "loses" to the factorial and thus it converges. Deriving infinite times has no meaning as well, it has about the same meaning as saying "if we take an infinitely large number for x, does x^2 diverge?"

I didn't understand the Wikipedia article. I checked there before making this thread. I also checked Wolfram Mathworld, and it looked surprisingly different from the Wikipedia article. I understood the Wolfram one a bit more, but I still don't really understand why it's used.

Unrelated question:
f(x) = cos(ω*x)
f'(x) = -ω*sin(ω*x)
f''(x) = -ω^2*cos(ω*x)
f'''(x) = ω^3*sin(ω*x)
f^4(x) = ω^4*cos(ω*x)
Etc.

x=0 will cause:
f(0) = 1
f'(0) = 0
f''(0) = -ω^2
f'''(0) = 0
f^4(0) = ω^4

Etc. In fact, the pattern will continue: 0, -ω^6, 0, ω^8, 0, -ω^10...

If f is differentiated infinite times, at its peak, it will have ω^∞ = ∞? I'm not saying, "plug in this" to see if it diverges. Even the sine sequences will have a maximum at ω^n.

What is the "sup" function?

As stated in the wikipedia article:
"sup(S) is defined to be the smallest real number that is greater than or equal to every number in S."

What does it do?

Read above.

How is it used?

Sup is not a specific algorithm - it is just a definition/function.

What is the point of using it?

If you need the smallest real number that is greater than or equal to every number in set or subset, you use sup.

How does the Legendre transformation work?

http://en.wikipedia.org/wiki/Legendre_transformation

Can you please provide examples (preferably with functions that variables can be plugged into)?

Google shows the second link to be this:
http://www.student.fizika.org/~nnctc/legendre.pdf
which answers things nicely.

What does p represent?

A function or a number - anything that really makes sense within the context. If you wonder what it usually means, look up the answer above.

I know that in transformations, f(x) is usually mapped onto a different variable, but in this particular transformation, doesn't really seem to be doing anything to f(x), because it's external to f(x).

Above. Besides, f(x) might be a function, so not really. But in usual application, as mentioned before, look above.

Also, if the sup function is the supremum function, does that imply that the curve is full of maxima?

No. A function doesn't imply anything.

Out of curiosity: If I differentiate x(t) = cos(ω*t), ω>1, an infinite number of times, does it eventually diverge?

It doesn't diverge, because the infinite derivative is not defined for that function.

In other words, if x'(t) is the velocity, x''(t) is the acceleration, x'''(t) is the jerk, ... and the infinite derivative is the infinite change in some quantity of motion, does that imply that a simple oscillator can have an infinitely large change in some quantity describing the motion?

This is a whole other question, because you're now taking a mathematical aspect and assuming it makes sense to apply it in a physical situation as a "description" of the system. As for your question, the infinite derivative of the position x(t) does not make sense and is not used in modern physics - and no, as the infinite derivative is not defined, we cannot do so.

This is a whole other question, because you're now taking a mathematical aspect and assuming it makes sense to apply it in a physical situation as a "description" of the system. As for your question, the infinite derivative of the position x(t) does not make sense and is not used in modern physics - and no, as the infinite derivative is not defined, we cannot do so.

do not like.

You can keep taking derivatives forever, and when you finally do reach the infinith derivative, it can be whatever you want. Anything continuous on C^(inf) has such a derivative, but since it's continuous the derivative is not infinite. And anything past C^(3) isn't really useful, but special functions that can be described by infinite power series (trig, trig h, bessel.... most others) will have a infinith derivative.

I've actually exhausted (downloaded and tried to comprehend) most of the internet sources regarding the Legendre transformation. Most of them say that it's used within a thermodynamic or mechanical context, and then proceed to apply it without explanation of how it works. Some of them, such as the one Devil Sunrise found, show the geometric interpretation in such an abstract way that I don't really understand. However, none of them show them within basic mathematical contexts.

 Old

I'm looking for a very simple mathematical definition, which just isn't available anywhere. (Or if it is, I've yet to find.) I feel like an actual, simple example will do much more for me than a complicated definition. Try to avoid the sup function if possible, because I know that it can be avoided. As a simple example, I attempted to transform:

df = 2x*e^(5y)*dx + 5x^2*e^(5y)*dy

Here is my work.
f = x^2*e^(5y)
df = w*dx + z*dy
w = 2x*e^(5y)
z = 5x^2*e^(5y)

Let g = f - w*x
g = x^2*e^(5y) - 2x^2*e^(5y)
g = -x^2*e^(5y)
dg = -2x*e^(5y)*dx - 5x^2*e^(5y)*dy

If this works properly:
dg = -x*dw + z*dy

dw = 2*e^(5y)*dx + 10*x*e^(5y)*dy
z*dy = 5x^2*e^(5y)*dy

dg = -2*e^(5y)*dx - 10*x*e^(5y)*dy + 5x^2*e^(5y)*dy
dg = -2x*e^(5y)*dx - 5x^2*e^(5y)*dy

Okay. They match, so my transform worked as it should. Now... what is the point of this procedure? You still need both variables. Also, I'm not sure how to transform it back. Supposedly, the Legendre transform is its own transform, but I haven't tried it yet. I'm not sure what's going on.

Also, I don't really understand the notation that most of these sources use:

(dy/dx)_y

I'm not sure what this means.

Edit: So I finally figured out how to use the Legendre Transform:

 Image

My new questions are:

1. How does this fit into the sup function? What's the point of writing the f*(p) = sup(px-f(x)), when the Legendre Transform takes f'(x,y) = wdx+zdy and turns it into: g'(x,y) = -xdw+zdy?

2. I read in one paper that the Legendre Transform of a parabola is a parabola, but a parabola exists in two-dimensional space and is a function of one variable. How do I apply the Legendre Transform to single variable functions?

3. What is the geometric interpretation of switching the differentials? What is convexity? Why is convexity a requirement for the Legendre Transform?

4. In my f(x,y) = x^2*e^(5y) example, how is g'(x,y) any more useful than f'(x,y)? Both g'(x,y) and f'(x,y) depend on x and y.

5. How is this process done in three variables?