Question

Time to derive the Euler-Lagrange equation! @Astrophysics

Answer 1

woot woot

Answer 2

Alright so first off, I'll throw this up here:

\[\int_a^b \sqrt{1+(y')^2}dx\]

What's his integral? What function y minimizes this integral, any ideas?

Answer 3

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Answer 4

c'mon gogogooogogoogoogoogoogogoggo none of my questions should take longer than 5 minutes, if you don't know you don't know, just say something and let's move forward I don't want this to drag on cause this is quite long.

Answer 5

Ahhh no I have to have participation for this one every step of the way, otherwise once you're lost it's hopeless since this is quite difficult. If you gotta do your homework though we can do this some other time if you like.

Answer 6

Naw it's cool, lets do it!

Answer 7

straight line

Answer 8

Ok! Also anyone else who wants to chime in and try to answer my questions feel free! Ok so back to what's up:

That integral is arc length, what function y(x) minimizes the value of that integral given an interval [a,b]? A straight line, y=mx+b as they say! (At least in Euclidean space, let's not worry about nonEuclidean space for now hehe ;P )

So we can rewrite this integral as:

\[J[y] = \int_a^b \sqrt{1+(y')^2}dx\]

The [ ] square brackets mean that the input is a function and the output is a number. This is a functional! So in our case we know that when y=mx+b this will make \(J[y]\) to be a minimum, as in some sense the derivative will be equal to zero here in some sense! We'll get into this soon!

Answer 9

Now we can look at a more general case,

\[J[y] = \int_a^b F(x,y,y')dx\]

Where now F(x,y,y') could be a bunch of other stuff you'd like to integrate such as some stuff about a curve that will let you travel under the force of gravity in the shortest amount of time or it might represent a hanging chain. We'll get more into the details of this later, the key thing is that our goal here is to find in general:

What function y minimizes the integral J[y]?

Any ideas or questions before I get into it?

Answer 10

So far so good!

Answer 11

The idea is to get a function that minimizes/ maximizes the integral, so in a nutshell we're looking for the equation of a line correct?

Answer 12

you guys don't have a life.

Answer 13

Only when the integral has \[F(x,y,y')=\sqrt{1+(y')^2}\]
For instance F=T-V could be minimizing the action along a path.

Answer 14

Right right, so this is the principle of least action, so if we have a path, quick sketch, it could be any path