/sci/ - Science & Math

>Chapter 1: Equations of Motion
Okay, so in the beginning Landau must set up his the basic concepts in terms of which much of the book will be expressed.
As we’re doing classical mechanics, many quantities are familiar with everyone, or at least the words are recognized. Now many concepts are translated to mathematical symbols and thereby made somewhat quantifiable.

>§Generalized co-ordinates
Position, velocity, acceleration, coordinates and the equations of motion are defined (or let’s say they are mentioned).

• This section is short, we’re expected to know what co-ordinates are. Landau doesn’t really explicitly draw a distinction between points/numbers determining a system and trajectories in time, though the co-ordinate space.

>§The principle of least action
So interestingly, Landau doesn’t want to start out with Newton, but directly formulates mechanics in the Lagrangian formulation. Even more, we start with the action and the The principle of least action.
The postulate is that „it turns out“ that we can model physical systems by assigning it a function L, the Lagrangian, and the physical path is one which satisfies an extremal condition for an integral over L, bounded by two points in time.
He goes on to show that this implies the Lagrangian equations of motion, which can be seen as a formulation of Newtons laws.
The arguments are arguably heuristic - variation of a function with respect to functions/whole trajectories are performed like variations with respect to real parameters.
He goes on to discuss the Lagrangian of uncoupled or loosely coupled systems: Addition of sub-Lagrangians.
He comments on multiplicative factors of the Lagrangian and addition of total derivatives.

• I think I like starting mechanics with the Lagrangian perspective, but I’m not perfectly sure if I like the action principle -that- much. Would it be bad to start with the Lagrangian equations of motion? Would it be arbitrary looking, that equation?
Maybe a point is that there are more theories, field theories in particular, that can be written down via a the principle of least action, for some action. There will also be differential equation, but they’ll generally be different -looking- than the one for q in classical mechanics.

• With the subsystems and additivity, I find it confusing what’s causation and correlation here. He doesn’t go on to say that we define uncoupled sub-Lagrangians to be added, he seems to argue that this arises. I don’t really see that - it’s eventually only justified by the equations of motions being independent. Maybe that’s what he means.

• Interestingly, we haven’t encountered numbers now, although it’s clear that L (generally?) takes values in the reals. I’d be interested to see Lagrangian mechanics (or rather the analog) fully introduced on a grid.

• He seems to be stating (using?) that for an action at a point of minimum, any variation <span class="math">\delta q[/spoiler] will lead to a grows in the functional. What about continuous variations that leave the Lagrangian invariant, why not comment on it?

>§Galileos relativity principle
Said principle is introduced. Inertial systems too. As well as the law of inertia.

• This is again very short - maybe for the better.
I expect that conceptually, this is the very hardest part of mechanics.
There are some very strong opinions on what a theory of physics is and what it should or can do, and in my experience those different perspectives lead to heated discussions on how to set up notions such as coordinate dependence and the possibility for making good sense of inertial systems and so on. When is a particle free? Can we figure out when there are no forces acting on a body. Is this mere part of setting up the math to describe a system? I’m purposely not going into that at this point now.
People who think the issue is simple should stick to that line of though for the time being, I guess — I expect the chapter including Noethers theorem will have more substance and be related.

• I see Landau likes to introduce heavy principles on the fly, to some extent. At least we’re several pages in and only now ran into that principle. That’s somewhat interesting to me.

>§The Lagrangian of a free particle
It’s shown that the principles imply we can write the Lagrangian (energy) of a free particle as <span class="math">(m/2) v^2[/spoiler].
Then he states that for several particles, we just have a Lagrangian that is the sum.

<span class="math">T = \sum_a m_a v_a^2 / 2 [/spoiler].
(Kinetic energy)

The equations of motion for the free particle give a straight line.
Then some ad hoc comments on other coordinate frames.
And notably, the opinion that it’s our duty to make presentation simple.
(At least in the german translation I’m reading the way is formulated is more than a call for practicality, it’s written like moral rule.)

>§The Lagrangian for a system of particles
„It turns out that“ the general form of L is

<span class="math">L = T - U ( r_1, r_2, … ), [/spoiler]

where U is the potential energy, capturing the interaction between particles in an otherwise closed system (free of other external forces).
There is again some discussion on the invariance of the Lagrangian.
We get Newtons equation with

<span class="math">F_a = - ∂ U / ∂ r_a . [/spoiler]

Then the form of the general Lagrangian in general coordinates is written down.
We got a non-closed equation and the notion of a homogenous system.
He notes boundary conditions and that friction losses, if not negligible, leave the realm of classical mechanics.

Exercises.

(solutions are on the pages too, I'm not gonna post these)

Other problems in the same vain welcomed.

Thank you, please do not stop

In case that's not enough, I'll go on the weekend - as I said, I don't know about the pace.
Also tell me if that's too much explicit talk (it's work for me, of course, but this way there is reference to the topic and not everybody is just in their own mind)

>>7156949
> Then he states that for several particles, we just have a Lagrangian that is the sum.

...


>§The Lagrangian for a system of particles

Aren't "several particles" a system of particles? Or does he mean that they are not interacting with each other?

Landau's style is quite percuilar. There's that common joke about Lifshitz who comes to Landau and says "Hey Lev, I'm afraid that I forgot those ten pages of calculations in the tram", and Landau replies "Don't worry, we'll write as usual: 'It's obvious that...' "

>>7156977
Both a here a "systems of particles", i.e. several, collecitons of those.

He mentions the kinetic energy being a sum already in that chapter because it's the one where he introduces the mass m as multiplicative constant of L...
...and he want to make the point that while the multiplicative factor of the overall L doesn't matter* (you can view it as giving just the unit in which terms you express the numerical value of L), the relative mass values of a sum such as

<span class="math">m_1 v_1^2 / 2 + m_2 v_2^2 / 2[/spoiler]

>>7157033
More reason for good discussion then :D
It's clear that starting with 30 pages Lie-group theory would make it more clear what's we're dealing with when we use the big words I have raised caution about - but in fact I think one doesn't need to even know about them at first. On the opposite, I actually think that making lots of notes on ones ideas about what to try, what approaches to develope, where things could lead, how things could be perceived etc...if you write them down before you know more, you might create ideas that would get lost or never be discovered once you learn the canonical theory.
I read a lot, but I try to be cautious that I don't get trapped in thinking in terms of relations others have discovered decades before me - that way I'll never have an original idea.
I said it in another thread 2 days ago: I think learning even wrong physics is a good thing, just learn the right thing later. To do so, discuss what you think.

Btw. I'll leave the exercises open, so that there is actually something to do. I'll comment on them at the very least before I make another thread, which I don't want to do before 7 days.
Depending on what people say, I can go faster though this.

Also, to the guy with Kampen: I've looked into the book and it's good in the sense that it's 50% exercises. I could literally just drop them and make it a Putnam kind of thread series for statistics.
Depends on the audience of course, I have a thesis to write too.

>he want to make the point that while the multiplicative factor of the overall L doesn't matter*
Oh, I forgot to come back to the Asterisk:

The overall multiplicative constant to L doesn't matter in classical mechanics, as

<span class="math">L/2=m v^2 / 4[/spoiler]

is just the old Lagrangian with the mass measured with units divided by two.
But note that once you pass to path integrals

<span class="math">\sim \int e^{\int L[\phi] dt} D \phi [/spoiler]

the issue is more subtle.

>>7157112
It's not exactly subtle, as the multiplicative constant besides L is the inverse planck constant. Of course when we take the classic solution the value of \hbar doesn't matter

>>7157033
Sometimes I think that's basically every theoretical physicist ever (when teaching, at least).
I heard about a lecturer at my uni skipping steps in calculations with "=...=", where "..." is three pages worth of calculation.

I might be interested in participating here, as I still have the first three volumes sitting here and never really gotten around to it. But I am a bit too busy with other things right now (beyond shitposting on /sci/, I mean), I might join in maybe in a week or so. Will look out for threads.

>>7157118
>It's not exactly subtle
I'd say it's more subtle when you say you absorb multiplicative changes into a constant which governs validity of expansion - v.s. the classical case where it doesn't matter.
Haven't seen the hbar being tracked through all expressions like canonical momenta etc., in the standard toy models and theories. Would be good to know which observables even depend on it, and how.

Related: I like the idea of
>hey, let's take the classical limit!

<span class="math">P=i\hbar ∂/∂x \to P=0[/spoiler]

Oh..

>>7157125
Just fap less and save the collected minutes every day.

>>7157137
Classic observables don't depend on it, obviously. What it tells is that momenta, which in classical mechanics is just another degree of freedom (and thus EOMs are second order in time), actually has an oscillatory "internal structure", which finesse is governed by \hbar. So true EOM (schroedinger's equation) is first order in time.

>>7157150
I'd put the full emphasis on the degrees of freedom/required initial condition. The order of the differential equation is essentially arbitrary, you can rewrite the second order one as first order equation.

Noob here. In section 2 he shows that if L, L' differs by a total time derivative of a function of coordinates and time. I assume the converse is true too?

>>7157263
I think you didn't finish the first sentence.

>>7157286
never mind i got it

Just did the exercises with all the Lagrangians with all the gross sin's an cos's.

Somebody motivated to look how Wildberger defines his Quadrance and Spread and write the Lagrangians down in terms of those :D

http://en.wikipedia.org/wiki/Rational_trigonometry#Spread

where do i need to be mathematically to be able to understand these books? currently in lin alg and calc II.

>>7157337
I think the current first book needs only calculus, it's for physics students in the second year. (?)

On the other side, as I've noted above he'll just go ahead and use variation with respect to a whole curve (as opposed to a value of a real parameter) without mention.
Read into it, ask about the concepts that you can't associate to anything you know.

Ok guys I'm probably retarded but I don't get pic related at all

1) how does he do that expansion?
2) could anyone explain the whole "The second term...write it as 1/2mv^2"?

>>7157439
The argument is that you have, for a function <span class="math">f[/spoiler],

<span class="math">f(x+c_1(x)·d+c_2(x)·d^2+c_3(x)·d^3+...) [/spoiler]

<span class="math">\sim f(x) + \frac{∂f(x)}{∂x}·c_1(x)·d+...[/spoiler]

when expanded in small d.

Use f=L, x=v^2, d=epilon and c_1(x)=2v.

If the deviation should be a total derivative (hence v=dr/dt itself), then ∂L/∂v^2 must be constant, hence L is at most linear in v^2.
He introduces m as proportionality factor.

>>7157326
Okay, I’ve tried Wildbergerian coordinates and it doesn’t make things all too nice.

The single pendulum with hanged up at the origin
<span class="math">(0,0)[/spoiler]
has coordinates
<span class="math"> r = (x, y) [/spoiler]
In polar coordinates,
<span class="math"> x = d\, \sin ( \phi ) [/spoiler]
<span class="math"> y = d\, \cos ( \phi ) [/spoiler]
so
<span class="math"> x’(t) = d\, \cos ( \phi ) \phi’(t) [/spoiler]
<span class="math"> y’(t) = -d\, \sin ( \phi ) \phi’(t) [/spoiler]
with sin^2+cos^2=1 the kinetic energy is m/2 times
<span class="math"> r’ (t)^2 = d^2 \, \phi’(t)^2 [/spoiler]

To be fair to the Burger, we’d have to write
<span class="math"> y = d \sqrt{1-\sin ( \phi )^2} [/spoiler]
etc.

Looking at the Wikipedia page, Wildberger works with Quadrance, which is defined as the I’ve-not-taken-the-root-of-pytagoreas distance
<span class="math">Q(a,b) := (a_1-b_1)^2+(a_2-b_2)^2[/spoiler]
and the Spread, which is the opposite divided by the hypotenuse, but with quadrants instead of distances. Hence it’s the square of the sine of the angle.
Hence
<span class="math">x = \sqrt{Q \, s} [/spoiler]
<span class="math">y = \sqrt{Q \, (1-s)} [/spoiler]
That’S cute enough. But now
<span class="math">y’(t) = \sqrt{Q} s(t)^{-1/2} s’(t) [/spoiler]
<span class="math">y’(t) = \sqrt{Q} (1-s(t))^{-1/2} s’(t) [/spoiler]
and the kinetic energy is m/2 times
<span class="math"> r’ (t)^2 = Q/4 \, s’(t)^2 \, / \, (s(t)-s(t)^2) [/spoiler]

>That fraction
Yeah that’s something rational, I see…

With s=sin^2, we’d get
<span class="math"> s’(t)^2 = 4 \sin(\phi(t))^2 \cos(\phi(t))^2 \phi’(t)^2 [/spoiler]
and the nasty terms would cancel again.

>>7157487
Okay, I see that denominator
<span class="math"> \frac{1}{4} \frac{1}{s-s^2} [/spoiler]
is one over his fourth "spread polynomial", which is also on the Wikipedia page (pic related)

Those are cute creatures, related to the Chebyshev polynomials <span class="math">T_n[/spoiler]
http://en.wikipedia.org/wiki/Chebyshev_polynomials
via
<span class="math">1 - 2S_n(s) = T_n(1 - 2s).[/spoiler]

fun fact: I like Chebyshev polynomials as the Russians use them to capture data for heat capacities of gases (I work in chemical kinetic theory atm.)
The Americans, maybe in the cold war, made up their own polynomials and those are called NASA polynomials.
http://combustion.berkeley.edu/gri_mech/data/nasa_plnm.html

>>7157439

I've gone through Landau's mechanics before, and if you're ever wondering "How did he get from here to there?" the first thing to try is a Taylor series.

>>7156943
>Maybe a point is that there are more theories, field theories in particular, that can be written down via a the principle of least action, for some action.
This. the action is more fundamental than the lagrangian, equations of motion, etc.

>>7157538
If you say "fundamental", do you mean "versatile", or something else?

>>7157538
>the action is more fundamental than the lagrangian, equations of motion, etc.

huh?

>>7157538
Here's a thought I had on page two, and it relates to why I said what I said in the second post:
The line of thought in the book (if I recall it right) is the following. Consider an L and two configurations <span class="math">q_1==q(t_1)[/spoiler] and <span class="math">q_2==q(t_2)[/spoiler] at times <span class="math">t_1[/spoiler] resp. <span class="math">t_2[/spoiler] and hold them fixed. The trajectory we're after is one connecting the two and such that
<span class="math">\int_{t_1}^{t_2}L(q(t),q'(t),t)[/spoiler]
is extremal.
Agreed?

From this he derives the differential equation. That equation is expressed in terms of the function L alone.

Now take the differential equation as starting point. For initial condition <span class="math">q_1==q(t_1)[/spoiler] and any <span class="math">q_1'(t_1)[/spoiler], the differential equation, step by step, tells you how the path of the trajectory <span class="math">q(t)[/spoiler] evolves.
But from this perspective it's not apparent that the solution <span class="math">q(t)[/spoiler] defined by the differential equation from the initial condition will pass though <span class="math">q_2==q(t_2)[/spoiler].

I find this confusing, the differential equation perspective has less baggage, in a way.

>>7157538
Here's a thought I had on page two, and it relates to why I said what I said in the second post:
The line of thought in the book (if I recall it right) is the following. Consider an L and two configurations <span class="math">q_1==q(t_1)[/spoiler] and <span class="math">q_2==q(t_2)[/spoiler] at times <span class="math">t_1[/spoiler] resp. <span class="math">t_2[/spoiler] and hold them fixed. The trajectory we're after is one connecting the two and such that
<span class="math">\int_{t_1}^{t_2}L(q(t),q'(t),t)[/spoiler]
is extremal.
Agreed?

From this he derives the differential equation. That equation is expressed in terms of the function L alone.

Now take the differential equation as starting point. For initial condition <span class="math">q_1==q(t_1)[/spoiler] and any <span class="math">q'(t_1)[/spoiler], the differential equation, step by step, tells you how the path of the trajectory <span class="math">q(t)[/spoiler] evolves.
But from this perspective it's not apparent that the solution <span class="math">q(t)[/spoiler] defined by the differential equation from the initial condition will pass through <span class="math">q_2==q(t_2)[/spoiler].
What if it's totally unphysical that <span class="math">q_1[/spoiler] and <span class="math">q_2[/spoiler] be connected, even if L can be minimized? What if the local spray doesn't point from <span class="math">q_1[/spoiler] anywhere in the direction of <span class="math">q_2[/spoiler]?

I find this confusing, the differential equation perspective has less baggage, in a way.

>>7157568
both. its the more 'physical' quantity, and more easy to generalize.

there are field theories with symmetries that are only manifest on the action, not the lagrangian. take for example the U(1) theory that defines EnM...

>>7157684
Yup. the DE tells you that the solution is locally minimizing, which doesn't mean that it satisfies the boundary conditions. you must turn it into a boundary value problem in order for that part to work.

>>7157684
>posting 3dpd

>>7156961
Need some help with problem 4. How do he get <span class="math">dl_1^2 = l_1^2 d \omega^2 + a^2 sin^2 \omega d \phi^2[/spoiler]?

>>7158330
not omega, theta

>>7158316
>posting Chinese girl cartoons.

>>7158330
I'll go for a run now and come back to you later.

.
.
.

Generally, since I can't gauge if we're 3 or 8 people in this thread, could people plx leave a vote here on participation

http://strawpoll.me/3979300

and here on pace

http://strawpoll.me/3979316

>>7158330
I'll go for a run now and come back to you later.

.
.
.

Generally, since I can't gauge if we're 3 or 8 people in this thread, could people plx leave a vote here on participation

http://strawpoll.me/3979300

and here on pace

http://strawpoll.me/3979316

>>7158389
supa smat

at least twice for 2

>>7158389
Since I couldn't vote for two things, just wanting to let you know that more emma stone is appreciated.

thanks based mm

>>7156952
Solution for the lazy/curious.

>>7158438
Don't you need some Langrange multipliers to constrain the position of the second particle?

>>7158448
Constrained to what?

>>7158455
The other particle
Now you've described two separate particles who dont interact with eachother

>>7158461
but they're constrained by the condition that the lengths of the two bars are constant.

>>7158330
The second l1 you wrote should also be an a.

The length element arises like this, because the geometry is such that m never leaves a sphere of radius a, and
>a·d angle
and
>a·sin(angle) d otherangle"
both just come from the formula that the arc-length is the radius times the angle. Once the radius is a and once it's (by geometric considerations) a·sin(angle).

Without thinking, the result is also what you get if you take the formula under (4.5), which expresses dl^2 in spherical coordinates, and plug in r=const=a

>>7158438
I was thinking about the Euler-Lagrange equations in this pic at the bottom and was thinking another argument for starting with the action S
(as landau does, and as opposed to starting with the differential equation direct)
is that if we know that however we rewrite one Lagrangian in other coordinates
(like we do her in the exercises, where we think in Cartesian coordinates first, and then switch to angles, or Wildberger snizzle, ect.)
then we just get a new expression for the Lagrangian - which we can think of as a perfectly new characterization of the problem stated in terms of angles, and the maximum principle indeed say that optimizing S is using one and the same differential equation.

>>7158399
supa smat??

In any case, looking at the polls
http://strawpoll.me/3979300
http://strawpoll.me/3979316
up to now, only 2 to 4 people seem to be up to it, that might not be worth it

>>7158883
>up to now, only 2 to 4 people seem to be up to it, that might not be worth it
stop being so defeatist

>>7158883
The subject matter is a bit dense for 4chan tbh. It might be better just to keep it to random discussion/demonstration threads. You seem to know your stuff so just let us know if you find anything particularly interesting (I can't promise I'll be able to contribute anything useful but I'll try :3).
>What if it's totally unphysical that q1 and q2 be connected
I assume <span class="math">q_{1}[/spoiler] and <span class="math">q_{2}[/spoiler] have to be defined on a set of possible solutions. Something like an Ergodic principle would probably help too (i.e. given enough time, all possible coordinates will be "explored").

>>7160398
>I assume q1 and q2 have to be defined on a set of possible solutions.
I don't know what you mean by "defined on a set of possible solutions".

In any case, I've since clarified my question.
I don't like Landaus text on page 2 in this regard, not in the translations at least.
I found this example which makes the issues clear:
If your boundaries are <span class="math">q_1=(3,7)[/spoiler] and <span class="math">q_2=(3,-3)[/spoiler] in the real plane,
if moreover your Lagrangian is one making straight lines into the solutions of the equations (the standard geodesics),
and if in your setting the plane is punctured on the connecting line, e.g. your q-space is
<span class="math">M={\mathbb R}^2 \setminus \{(3,2)\}[/spoiler]
then there can't even be an extremal curve.
(interestingly, if you take M to be the sphere like pic related, then removing one point will not kill the existence of the extremal curve, just switch from shortest to longest circle)

If on the other hand the space is geodesically complete
http://en.wikipedia.org/wiki/Geodesic_manifold
then you'll find a solution for any q1 and q2.
An error I made yesterday was in mentally fixing the velocity initial condition <span class="math">q'(t_1)[/spoiler] together with q1 and then it naturally seemed that a direct Euler-Lagrange differential equation approach would provide far less physical paths than the action S-approach, where for each starting point, you get a line for each endpoint.

I don't think it's too dense, actually. There are enough people here capable for it, but they probably just come to 4chan to have a laugh and masturbate. But it's a better place for discussing than StackExchange, for example. Or fast paced irc channels.
I started out with the idea because I want to complete a set of notes on physics that I'm writing, but if only two people are tuning in and if in turn I have to write all the replies myself, I'm not going to continue writing summaries like the one in the beginning of the thread.

>>7160398
>I assume q1 and q2 have to be defined on a set of possible solutions.
I don't know what you mean by "defined on a set of possible solutions".

In any case, I've since clarified my question.
I don't like Landaus text on page 2 in this regard, not in the translations at least.
I found this example which makes the issues clear:
If your boundaries are <span class="math">q_1=(3,7)[/spoiler] and <span class="math">q_2=(3,-3)[/spoiler] in the real plane,
if moreover your Lagrangian is one making straight lines into the solutions of the equations (the standard geodesics),
and if in your setting the plane is punctured on the connecting line, e.g. your q-space is
<span class="math">M={\mathbb R}^2 \setminus \{(3,2)\}[/spoiler]
then there can't even be an extremal curve.
(interestingly, if you take M to be the sphere like pic related, then removing one point will not kill the existence of the extremal curve, just switch from shortest to longest circle)

If on the other hand the space is geodesically complete
http://en.wikipedia.org/wiki/Geodesic_manifold
then you'll find a solution for any q1 and q2.
An error I made yesterday was in mentally fixing the velocity initial condition <span class="math">q'(t_1)[/spoiler] together with q1 and then it naturally seemed that a direct Euler-Lagrange differential equation approach would provide far less physical paths than the action S-approach, where for each starting point, you get a line for each endpoint.

I don't think it's too dense, actually. There are enough people here capable for it, but they probably just come to 4chan to have a laugh and masturbate. But it's a better place for discussing than StackExchange, for example. Or fast paced irc channels.
I started out with the idea because I want to complete a set of notes on physics that I'm writing, but if only two people are tuning in and if in turn I have to write all the replies myself, I'm not going to continue writing summaries like the one in the beginning of the thread.

>>7160235
doubt.jpg

>>7160445
Well, I'm going to be doing the excersizes in any case. I'll post my solutions to them next time.

I have two questions:
- You mention >>7156949 that 'boundary conditions' leave the realm of CM. Why is that? Can't you handle any holomorphic BC with CM by introducing a lagrange multiplier?

- I get your point about the action principle now. I guess LL should have mentioned that the ODE is not equivalent to it; rather, being a solution to the euler lagrange equations is implied by being an extrema of the action.

So are there cases where the global minimizer is not a local minimizer? For example when you want to minimize -x^2 in the interval [-1,1], the global minimum is not a critical point. does this issue show up in mechanics?

>>7160474
You parsed the sentence different than intended:
>(He notes boundary conditions) (and that friction losses, if not negligible, leave the realm of classical mechanics.)
The leaving the real part was only with respect to friction, or heat effects in particular.

The issue with global/local minimum/extrema is, as far as I can see, only in the direction that piecing together local minima need not need to a global minimum - like the long arc around the sphere is not the shortest path between points, although if you look locally, for any small interval you go along a minimizing path.

Went for a run, again :), and came to a conclusion regarding what I missed when connecting the action and the Euler-Lagrange equation (ELE) approach.
First I noticed that the initial value data to the ELE, q1 and q1', modulate the solutions of the ELE: Give my a pair of values and amongst all solution, one curve is specified.
On the the other hand, the action takes endpoints, i.e. q1 and q2, and doesn't care about q1'.
Now q1 and q1' are both data at t1, and q2 alone, the point in space, doesn't tell you q2. Hence I want to point out that really only (t1,q1,q1') resp. (t1,q1,t2,q2) modulate the solutions.
When comparing the solutions they specify, one must keep in mind that a solution to the ELE don't aks you for how long you want to rid along the curve. Meanwhile, the action concerns a fixed endpoint - a choice of end point and time!
I thought about R^3 and how q' is the limit of (q2-q1)/(t2-t1) for small t2.
My conclusion is that to match the two perspectives, and the data in particular, one should really take pic related to heard and maybe express the connection in those terms. Btw. I have my notes available online here: http://axiomsofchoice.org/euler-lagrange_equations
(Btw. I thought about the solution parametrization space because it (moduli spaces) pops up heavily in the topos approach to field theories I've been reading about. Related, a buzzword: Teichmüller space)

>>7157361
>>7157337
Landau's textbooks were written for graduate students. You can still give them a go, but don't expect to gleam much.

>>7158438
why the fuck would you not use the angles as your gen coordinates?

>>7157361
>>7157337
Isn't there a derivation of the EL equations?
That requires at least a weak understanding of calculus of variations.

>>7157101
>Also, to the guy with Kampen: I've looked into the book and it's good in the sense that it's 50% exercises. I could literally just drop them and make it a Putnam kind of thread series for statistics.

I started reading too. Pretty good so far, though rather advanced. He treats the topic very differently from how a pure mathematician would usually do it. The exercises are quite interesting and some of them definitely not trivial. I already found one I couldn't solve.

>>7160956
If your motivated you can do such kinds of thread about it. I'd definitely join in. Something which would be a challenge like the statistics book would be better for me too.
I think Landau is (in principle) a good idea because it's such a general interest topic, and while I had courses on the subjects already years ago, I never read into this particular series. I'd continue if I had a way of knowing we'd actually go through with it.

>>7160906
As I said above (I think here >>7156943), his use of variation of paths is like he was dealing with variation of real numbers. It's done completely heuristically.

>>7160956
Guy who suggested Kampen here.
I'll start a thread on it next weekend; I'm planning on covering about 10-15 pages per week.

The treatment is very 'physics-y'; I'm not sure if I like it, but I think learning it this way is good for intuition.

>>7160968
>Something which would be a challenge like the statistics book would be better for me too.
What statistics book?

>>7161045
I meant the Kampen book. Statistics as in Statistical physics, where I learned about the van Kampen approximation.
Okay, so you'll actually do that? Good. I'll tune in and happily discuss the exercises with you two or more guys.
So do you want to go with that book or the mathier one you suggested?
(Since the other guy is reading Kampen already, I'd suggest to stick with Kampen, actually)

I'll also make a second Landau thread at least next week (defeating my defeatist attitude, I try.)

>>7161257
>So do you want to go with that book or the mathier one you suggested?
We'll start with Kampen, but after chapter 1 or 2, I think I'll take a look at the mathier one, and suggest readings from it to complement Kampen.

It seems that Kampen has more material, but just done in a less rigorous way. At the very least, we should cover section 1 of the mathier one, because it goes through a rigorous but seemingly accessible treatment of stochastic processes, so we can use the language later.