(reading your maths now because I have been disturbed by the case of how to deal with the lebniz integral rule for at least 4 years if we cannot interchange limtis and integration)

(and 4 years later in the present, I want to find the answer to my disturblance)

I have a reasonably detailed proof of the EL equations in personal notes somewhere.

@0celo7 Pffffft

Beware I wrote this like 4 years ago

it has weird notation

@ACuriousMind the sound of you farting in 0celo7's general direction? :-)

@ACuriousMind How does that work without assuming $L$ is analytic?

@0celo7 Why would the analyticity of $L$ matter? You're not claiming that $L$ has a power series, you're just using Taylor's theorem to first order - the remainder is always of order $h^2$ no matter the properties of the function (well, it has to be once continuously differentiable).

@JohnRennie That's a possible interpretation

@ACuriousMind I always forget about Taylor.

@ACuriousMind We had a string theorist from cal tech give a guest lecture in physics, he derived the EL equations for us

@0celo7 nice

@ACuriousMind Of course I hijacked the lecture and discovered he didn't know shit about the nitty gritty

of course.

@ACuriousMind former student at my hs

@ACuriousMind this was still before my disillusionment with physics

After this QM course I'm even more jaded

I really should have taken modern algebra...

I wish I could do college over again

I should have taken two of the classes this semester last year

Taken modern algebra

And advanced linear algebra

@0celo7 For line 1 $\begin{equation} \delta E[\phi]=\delta\int dxV[\phi]=\int dx(V[\phi+\eta]-V[\phi])\sim\int dxV'[\phi]\eta \end{equation}$ is it wrong to view it as analogous to some kind of differentiation under the integral sign, and that the correct way to view it is a definition which is justified by that the functional is differentiable as shwon in p.4?

Otherwise the derivation flows nicely

This is drivel isn't it?

Or am I missing some subtlety here?

@Secret No, that's literally the definition of the functional differential + Taylor's theorem.

ok

Btw, without reference to any physics context, is there an argument to motivate why we want to extremise that functional derivative?

@Secret get a legit calc of var book if you're so worried about this

@Secret yes

the Poisson equation is solved rigorously using a variational technique

@Secret Just look at pages 55-57 of Arnold

@Secret Riemannian geometry uses a lot of techniques from calc of var

@ACuriousMind Interesting lemma: a minimal submanifold of $\Bbb R^n$ is noncompact unless it's a point.

@0celo7 I don't think that's interesting :P

@Secret Landau's classical mechanics chapter 1 gives a nice explanation for how you get $T - V$ based on symmetry and the PoLA, only a few pages, Galilean symmetry motivates $L = T$ for a free particle, then just adding some function $U$ for a closed system of interacting particles to account for interactions, so you have $L = T + U$, however working out the EL equations, in order to match it up with Newton's law you want $U = - V$, but you don't need Newton's laws, it's just formalism

That Noether paper is a bit mental btw, iirc it does Noether's theorem for n'th derivatives of functionals of several variables

Gel'fand's book is the cleanest nicest run through of what you need for mechanics I've seen

@ACuriousMind Well, what's confusing me is the minimal surface equation

there's no reason for the volume functional to be finite

so everything is purely a formal manipulation I suppose

I don't even understand those functional derivatives, seems like lazy physics tricks to do honest math badly

Greiner does them, as does Hatfield, just seems like a lazy trick to do something you can do far easier just doing it normally

@bolbteppa Ok, and the calculus of variations were explained well by Arnold. I will check Gel'fand's later because I cannot seemed to find his book without the name

Are Lagrangians and lagrange multiplers related, or they just happened to have the word "lagrange" in it?

[some random things in topology]

so there are parameters in the lagrangian that played the same role as $\lambda$ as the $f-\lambda g$?

Think of Gelfand as the mathematical basis for Landau's mechanics, and then Arnold as a rewrite of that using linear algebra and updating it

Yeah, Gelfand mentions them, so does Oprea's Diff Geom book iirc

@0celo7 Line 19, why when the partial derivatives of H bounded will result in the integral of them to vanish?

Something like, extremizing the function $S(y) = \int_a^b F(x,y,y')dx$ subject to the constraints $y(a) = y_a, y(b) = y_b, \tilde{S}(y) = \int_a^b G(x,y,y')dx = L$ is the same as extremizing $\int_a^b (F + \lambda G)dx$, need to revise it

Note Noether comes about when you let the endpoints vary too (just for $S$ not related to Lag Mult)

@Secret Ask me tomorrow, I'm really busy today

@MAFIA36790 Both funny and sad to see, how the greens suddenly switch topic hearing that the nuclear energy doesn't produce CO2.

@JohnRennie It doesn't even rise to the level of drivel. More over, both the topic and the strange habit of posting a new comment under the answer ever few minutes even if no one is interacting echo a couple of earlier users.

@dmckee are you there?

@yuggib uhhh

@0celo7 wanna answer a couple questions about thermodynamics

Definitely not.

@Obliv And you were asking for me? Thermo is not my strong suit, and I have to teach in just a few minutes.

It's okay @dmckee if you're busy

This is good: Convert Latex to MS Word equation.

@IceLord : oh yes I do.

@IceLord : I don't use them much in my answers. But you do see a few expressions here and there.

Now I'm writing a word document. It's good to be able to convert a latex expression into a word equation fairly easily.

@IceLord I don't know that either.

@IceLord ok, I dunno who you are tho

Who?

@IceLord isn't that just positive reals including $0$? o.o pretty basic quantifier notation imo

why is your name icelord now lol

first year students taking analysis..

i can't take it until i'm a junior 8)

so next year probably.

This is your first semester in college, right?

are you in the U.S/

?

Well I'm sure you'll do great :) make sure to make friends and not get stuffed too hard in the books.

wait you said you were taking a QFT course? wtf..

not sure how you were allowed to take it :D

oh okay

That's the best option :p

@icelord do you by any chance know how energy is transferred between particles (through conduction/radiation)? Or is this a poor question to ask because particles cannot 'conduct' energy?

Like macroscopically, conduction comes from 'touching' things. I do not know what this really means in a smaller context

Oh you're the Canadian one.

GL on QFT, I'm not helping you this time.

Heya

doing Brazilian exercises

@0celo7 Push ups?

Riemannian geometry

@ACuriousMind I have discovered a truly marvelous property of geodesic balls.

Too small for this chat box, and you're not worth of the proof.

@BernardMeurer Your countryman Mr. do Carmo has written the most magnificent book

@0celo7 Didn't expect any less

@BernardMeurer you always say Brazilians are scum

@0celo7 Yeah, but we make nice stuff