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Lagrangian formalism

Anonymous Sun 20 Oct 14:24:42 2019 No.11076697 View Reply Original Report

Quoted By: >>11076828 >>11077016 >>11077022 >>11077090 >>11077799 >>11078189 >>11078696

Looking for some good resource on this. Preferably something that is pretty careful and formal.

When I've looked around I've seen a lot of shitty material like: Look it works for a harmonic oscillator, hence it must apply for arbitrary systems.

Some key points I want to see addressed:
* Why is action extremized?
* How can we take partial derivatives wrt q and q dot separately, can we really keep one constant and vary the other (I mean I guess we can but I'd like to see why)
* What are the smoothness requirements on L and q - and why? How do we know that our system has sufficiently smooth solution?

Anonymous

Anonymous Sun 20 Oct 2019 14:29:45 No.11076710 Report

Quoted By: >>11077734

Take a look into VI Arnold's classical mechanics book.

Anonymous

Anonymous Sun 20 Oct 2019 15:35:23 No.11076828 Report

Quoted By:

>>11076697
For your second point, think of the lagrangian lying in an abstract space called the phase space. This phase space is generally 6N-dimensional, with N being degrees of freedom of the system, typically particle positions, and the factor of 6 coming from 3 spacial dimensions and the fact that position and velocity (or sometimes more commonly momentum) are being specified.
If we simplify things to a one-particle system, you can imagine the phase space as giving just enough information to fully specify future evolution of the system. Imagine a particle in some sort of sloped potential at some point in time lying at a point but having no velocity, then imagine an identical particle at the same position but with a velocity up the potential well. Clearly the two systems will evolve differently. Basically specifying just the position or just the velocty isnt enough. When you then imagine adding an extra dimension for velocity/momentum you can see how evolutions become curves in this space, and then it makes a lot of sense to differentiate with respect to each seperately.

Anonymous

Anonymous Sun 20 Oct 2019 16:50:17 No.11077016 Report

Quoted By:

>>11076697
read something on calculus of variations for fucks sake.

Anonymous

Anonymous Sun 20 Oct 2019 16:52:04 No.11077022 Report

Quoted By:

>>11076697
look into calculus of variations

Anonymous

Anonymous Sun 20 Oct 2019 17:14:58 No.11077090 Report

Quoted By: >>11077734

>>11076697
>Why is action extremized
That is what the euler lagrange equations do. They find the local max/min of a functional.
The real question is how do you beat the laws of physics into that form.
The answer: start with the E-L equations, figure out how to make the general solution express newtons laws, then work backwards to derive the meaningful concept of action.
> How can we take partial derivatives wrt q and q dot separately
It is an abuse of notation.
If it were L(a,b,c), there would be no ambiguity in what is meant by dL/da, dL/db, and dL/dc.
You have a recipe for the action, L(x(t),x'(t),t), that depends on x, x', and t.
Instead of dL/dx and dL/dx' it might be less ambiguous to write L_1 and L_2 to indicate that it is how x and x' appear in the recipe for L that matters and not that x and x' are related.
This is also ambiguous since somebody could have written the action as L(t,x(t),x'(t)) in which case dL/dx = L_2 and dL/dx' = L_3.
You want the specificity that referring to a specific argument of L gives but you also want the generality to not have to write the arguments of L in a specific order.
Hence the notation.

Anonymous

Anonymous Sun 20 Oct 2019 21:05:17 No.11077734 Report

Quoted By: >>11078031

>>11076710 Thanks a lot! I do love the yellow books, I'll read Ch3 for sure
>>11077090
> Part 1
But then why L=T-V?
> Part 2
I feel almost, but not 100% convinced. Thanks for the explainer!

Anonymous

Anonymous Sun 20 Oct 2019 21:26:13 No.11077799 Report

Quoted By: >>11077846

>>11076697
Look into differential geometry. p and q are basis vectors of the tangent space of the underlying manifold. In that way, they are different directions which you can derive wrt independent of each other, since they build an orthogonal basis.

Also, the Lagrangian is the geodesic equation. Solutions are geodesics, or in other words curves that are stationary paths in phase space regarding action.

Anonymous

Anonymous Sun 20 Oct 2019 21:43:37 No.11077846 Report

Quoted By: >>11079011

>>11077799
I hope I will understand what you are saying later, but I don't currently.

Anonymous

Anonymous Sun 20 Oct 2019 23:06:41 No.11078031 Report

Quoted By: >>11078106

>>11077734
>But then why L=T-V
From newton, F(x) = mx''
F is the derivative of -V(x)

Using the convention L = L(x,x',t), Euler-Lagrange gives:
(L1) - (L12)x' - (L22)x'' - (L32) = 0, or equivalently, (L1) - (L12)x' - (L32) = (L22)x''
"Beating" newton into this form gives
L22 = m
L2 = mx' + f(x,t)
L = (m/2)(x')^2 + f(x,t)x' + k(x,t)
where f and k are arbitrary functions of x and t.

F(x) = (L1) - (L12)x' - (L32) = (f1)x' + (k1) - (f1)x' - (f2) = (k1)-(f2)
Letting k(x,t) = -V(x) and f(x,t) = 0 gives equality.
So L(x,x',t) = (m/2)(x')^2 - V(x) allows Euler-Lagrange to solve Newton.

If you let F depend on time, V(x) becomes V(x,t).

Anonymous

Anonymous Sun 20 Oct 2019 23:26:06 No.11078106 Report

Quoted By:

>>11078031
Interestingly, you can let f(x,t) = any h(x) and Euler-Lagrange will still solve newton.
The new lagrangian would be L = T - V + h(x)*x'

Anonymous

Anonymous Mon 21 Oct 2019 00:02:20 No.11078189 Report

Quoted By:

>>11076697
I have very little background on either the calculus of variations or theoretical. Most of the math I know is on the discrete side; physics is just a light hobby for me, so take all of this with a grain of salt.

Susskind's excellent lectures for non-practicing physics-interested folks (The Theoretical Minimum) are on Youtube for free. Discussion of the Lagrangian form begins in the third lecture, here:
https://www.youtube.com/watch?v=3apIZCpmdls
Very little of the calculus of variations is required to produce the Euler-Lagrange equations; he derives it with only a bit of fiddling with infinitescimals; nothing too fancy.

If Susskind is correct, and not speaking loosely in his lectures, that the Lagrangian formulation works is taken as a very fundamental law of physics: Hamilton's principle (https://en.wikipedia.org/wiki/Hamilton%27s_principle).

Essentially, it seems to be true that for any known laws of motion/time-evolution that describe a measurable physical system, there exists a Lagrangian whose stationary action corresponds with those laws of motion. Nothing mathematically requires the universe to be this way to be consistent; it's just been observed to behave this way. According to him, this is so pervasive that physicists have more or less stopped looking for descriptions of physical systems that don't have a corresponding Lagrangian. That is, the set of physical laws and the corresponding Lagrangian are more or less taken as interchangable, in practice.

El Arcón

El Arcón Mon 21 Oct 2019 03:07:42 No.11078696 Report

Quoted By:

>>11076697
Maybe instead of googling "Lagrangian formalism" try googling "the Euler-Lagrange equation." This is about the most common thing is advanced class mech and every derivation of I have ever seen answers all those questions you ask.
>derivation of Euler-Lagrange

Anonymous

Anonymous Mon 21 Oct 2019 05:36:20 No.11079011 Report

Quoted By: >>11079033

>>11077846
I was kind of tired and now that I read it after a night's sleep, it's definitely my fault.

Anonymous

Anonymous Mon 21 Oct 2019 05:43:01 No.11079033 Report

Quoted By: >>11079200

>>11079011
That’s not what they meant sperg.

Anonymous

Anonymous Mon 21 Oct 2019 07:14:01 No.11079200 Report

Quoted By:

>>11079033
Just saying my formulation sucks. I know what he meant.

>they
Is it appropriate for me to say "dilate" now?

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