Functional Derivative done right!

Epitaph

See this is why I'm going to Boulder, CO. 
'Cause I keep coming across stuff coming out of there ...
that does things right and makes sense!

Source

(sorry I stole'd your copyrighted stuff man)

Definition of functional derivative

This is exactly the explanation I was looking for. <3. Thank you for telling me what I need to know...

L(h) is the differential

then showing me exactly how I wanted to use it!

What he derives is the same problem as the QFT for noobs exercise problem. (Euler-lagrange looking thing ex1.3 p17). It matches the result I got for the functional derivative, not what the QFT book says!

See? There is a derivative term multiplied by the (delta) function h, that the physics people ignored!! I wasn't completely wrong!

Maybe it goes to zero if we extend the limits of integration to infinity? Or maybe it is very small compared to the other terms? Who knows... *smh*... physics people... They need to go learn some math or something.

Delta functions

These delta functions are starting to scare me. They are so creepy.

p18

source: QFT for the gifted noob

Attempt:

For the first part (1.44)

Write the first phi function as a trivial functional.
Use the functional derivative definition.
Evaluate one of the delta functions, dropping the integral and replacing the dummy variable with either x or y.

For the second part (1.45)

Do the same thing, but then get stuck near the end because it's more complicated.
Here the phi dot is a derivative, that we can rewrite in terms of phi using the definition of a derivative.
Then we can introduce variation and simplify terms, but I can't get a d/dt out of the mess.
So I'm thinking somehow, when I do the h-> 0 limit of the definition of phi derivative, somehow I can get a d/dt out of it?

If instead, I treat the variation as epsilon * d '(t' -t), I can't integrate by parts to get rid of the delta function derivative, since the other function is the delta function and I would have to take the derivative of that... then I'd get another delta function derivative...

Functional Derivatives

Euler-Langrange looking thing

This calculation looks wurry wurry emportunt, because we have a Lagrangian looking function g(y, f(x), f'(x)) that we are integrating in the functional, and we want to find its functional derivative.

Why is that important? I think it has to do with a way to describe the motion of an object, given its equation of state.

p17

 Using their definition of functional derivative gets tricky because it involves delta functions.

p12

 source: QFT for the gifted noob.

Here's my attempt:

I vary f(x) by epsilon times a delta (y-x) term. By analogy I vary the f '(x) term by epsilon times delta '(y - x).
Next, I Taylor expand the function g to get first order in epsilon terms.
In order to evaluate the delta '(y-x) term, I integrate by parts.
 
This gets me close to the answer. I'm skeptical of the math involved, but physics explanations usually are dubious about doing things that make math sense. Problem is I get an extra term.

I'm getting a partial derivative term with respect to f'(x) multiplied by a delta function at y=x, which if that equals zero somehow, then my result equals the desired Euler-Lagrange term for the partial derivative.

Same thing for the g(y, f, f', f'') case; there's an extra term dg/df'' *delta ' (x' - x) in my calculation, that were it equal to zero for some reason, I would get the prescribed dosage of partial derivative terms in the result.

Delta functions are confusing

deltafunc(x_0 - x_1)

 
 
source : http://www.physicspages.com/2014/11/08/functionals-and-functional-derivatives/

What does this mean? First, it's like the derivative rule for polynomials except the second derivative creates this delta function between two variables. What does that even mean? When two variables have the same value, the integral of the delta function equals 1?

Q2: Second, the result of the first differentiation is not another functional, it is a function, so how can we differentiate it as a functional?

A2: A function can be thought of as a trivial functional:

So the two variabled delta function is saying, pick out the function 6 f(x0) when integrated and x1=x0.... ok what monstrosity of mathematics is that pseudo-function?

delta function is ... what is that term? f(a) = f(-a), you can switch the x and the x0. d(x - x0) = d(x0 - x)

Functional derivatives

Showing the functional derivative of the integral of a two variable function is the same as the partial derivative of that function.

Seems kinda wrong, but I can't find a correct solution online so... going to go with what I got until someone tells me the right answer.

Quantum Field Theory for the Gifted Amateur. Ebook. Tom Lancaster, Stephen Blundell. New York, NY : Oxford University Press, 2014. 9780191779435 0191779431

Decline in class attendance

This lecture is getting to be too shouty for my ears. It was also a pretty bad lecture, imo. The enthusiasm is starting to turn into glorying over undergraduate students' admiration. Just the facts, Jack.

Most people don't stick with an entire lecture series to the end. They just watch a couple of videos.



 

There's also a few fawning students who I want to start throwing out of windows.
 

 

Now I need to visit the VD clinic. I'm going to keep going with the lectures, but I need some kind of brain contraceptive to keep out intellectual AIDS. 

Are all functionals integrals?

Q: Are all functionals integrals?

A: Yes.

As long as we're working within a complete inner product space (that is, a Hilbert Space) with an inner product defined by (f,g)=∫Kf(x)g(x)dx(f,g)=∫Kf(x)g(x)dx, the Riesz representation theorem allows us to conclude that "all functionals are integrals". – Omnomnomnom Sep 9 '14 at 4:26

Source:

http://math.stackexchange.com/questions/924427/are-all-definite-integrals-considered-functionals

From Fermat's Principle to Snell's Law

Fermat's principle of least time postulates that light takes the fastest route between two points.

When light is traveling at an interface between two mediums, the speed light travels changes based on the material.

(Light moves at a constant speed within the same medium no matter how fast the source of light is moving relative to an observer. A beam of light from a flashlight moving towards you has the same speed of a beam of light from a flashlight moving away. That's Einstein's special relativity. What I mean by the 'speed of light changing' is that the constant speed of light is slower in a dense material than it is in a vacuum.)

A beam of light that enters a dense material will bend to travel a shorter distance through the slower region of dense material. When it goes from a dense region to a lighter region, why does it bend in the opposite direction? I don't think Snell's law really explains that with a physical rationale... it's just like the symmetry of the mathematics setup by the light to dense case implies that the opposite would happen, say for example like if you could travel that beam of light backwards.

But the principle of least time, kind of... kind of... has a reasoning to explain why light bends outward from dense to light. It says that given any two points A and B, light travels the path that takes the least time. So if we pre-select points A and B on opposite sides of a interface between two materials, then we can figure out the fastest route to navigate between them using Google Maps.

This is really a cheat, because, well... how do we know when we start at A we will end up at B ahead of time? We don't, we are just asking: "if there is a beam of light that goes from A to B, what would be the path it takes?" We discover at what angle to the interface does that light travel. We say, that if this light that traveled from A to B obeyed this law, then all light beams also obey this law. Then we can make a forward prediction to figure out where we would end up, if we all we knew was we started from A at an angle to the interface.

The derivation from Wikipedia is straightforward and common place. So of course I could figure it out on my own, right? Nah, I looked that shit up after I spent an hour diagramming the interface and attempting to brute force labeling everything and joining labels with basic kinematics.

The idea is straightforward. Write an expression for the time it takes the light to travel from A to B, in terms of the speed of light in each region and the distance traveled. Then take the derivative of that expression in terms of a parameter that controls at which angles theta1, theta2 does the light cross the interface. Set that derivative equal to zero to find the set of angles that minimizes the time.

I got lost in all the lengths, couldn't figure out which one to unify the expression. I thought I had to write time in terms of the angles theta1, theta2 before and after light crosses the interface, then take a partial derivative in terms of each theta.

My unsuccessful kinematics attempt. Do not copy this!

There's a more on-target variable to vary. It's the position y1 at which the light crosses the interface. This is a single variable, that can easily differentiate the expression with, and it fully describes the angles theta1, and theta2, since the distances x1, x2 along the component parallel to the light's motion are fixed by the geometry of the setup. Likewise, the total vertical distance y (= y1 + y2) is fixed, so that the crossing point y1 also describes the length y2 the light travels after the interface.

Once you figure out to use the crossing point as the independent variable, the rest follows just using Pythagorean Pythagoras and SOCATOAn SOCATOA: basic trigonometry. Yeah you need to remember the chain rule in order to take the derivative, but that's a first year calculus level of being careful.

So here's my write-up of the Wikipedia derivation, from Fermat's Principle to Snell's Law.
You should refer to Wikipedia instead of me, but this blog is my refrigerator so I will put my homework on it!