Principle of least action

Factual Questions
1

I’m reading The Theoretical Minimum, and am stuck on the derivation of the Euler-Lagrange equations.

To derive it, Susskind discretizes the functional.

Action=sum of L( (x[sub]n+1[/sub] -x[sub]n[/sub])/delta t, (x[sub]n[/sub]+x[sub]n+1[/sub])/2)*delta t

He then minimizes the action with respect to one x[sub]n[/sub], specifically x8.

Only two of the terms have x8, so

Action=L((x9-x8)/(delta t), (x8+x9)/2)(delta t)+ L((x8-x7)/(delta t), (x7+x8)/2)(delta t).

Here’s where I lose him. Take the derivative of each side wrt x8.

dA/dx8= (1/delta t)( -dL/dx dot|n=9 + dL/dx dot|n=8) + (1/2)(dL/dx|n=8 + dL/dx|n=9)

He doesn’t show any intermediary steps and I haven’t been able to work it out. Any help?

Sorry about the notation. dL/dx dot is supposed to be the partial of the Lagrangian wrt the velocity and |n=x is the preceding expression evaluated at x. Other than that hopefully you can figure out what I meant.

2

You should try one of the many math sites. The notation is preventing me from even trying to figure this out

3

Can you write this out on paper and just post a picture of it? That way the notation will be something people are familiar with, and they might be able to help. It still won’t make any sense to me, but other people could probably help you.

4

Ok, so here’s what I was asking with readable equations http://i44.tinypic.com/2vlmkpk.png.

Basically I don’t know how he got from the sum of the Lagrangian terms to the last equation. Hopefully that’s more clear than my chicken scratch.

5

Expand the sum over n:

Action = ( L((X8 - X7)/delt), (x7+x8)/2) + L((X9 - X8)/delt), (x8+x9)/2) + … ) * dt
where the … are all the terms that do not depend on x8.

For clarity, rewrite as

Action = (L(a,b) + L(c,d) + …) * dt
where
a = (x8-x7)/dt
b = (x7+x8)/2
c = (x9-x8)/dt
d = (x8+x9)/2

(a and c are xdot at different points, and b and d are x at those points.)

So
dAction/dx8 = ( dL(a,b)/da * da/dx8 + dL(a,b)/db * db/dx8 + dL(c,d)/dc * dc/dx8 + dL(c,d)/dd * dd/dx8 ) * dt

Now there are no … terms since they do not depend on x8.

da/dx8 = 1/dt
db/dx8 = 1/2
dc/dx8 = -1/dt
dd/dx8 = 1/2

so

dA/dx8 = (dL/da - dL/dc) + (dL/db + dL/dd) / (2 dt)

Which is pretty much what you have, apart from a factor of dt (typo?). Hopefully close enough.