/sci/ - Science & Math

And no I just found it off of a random site

Yep it's just a simple little problem I just like to give a brain teaser once in a while.

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Good Luck,
MM
THIS IS YOUR ONLY CLUE:
EZENIMITH


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There was this highly intelligent but socially dazed persona Ron Maimon on StackExchange, who years ago did (try to) explain to me how renormalization theory is inherently tried to stochastics and fractals.
There seems to be a direct string on thought that leads from Feynmans PhD thesis
en.wikipedia.org/wiki/Wheeler%E2%80%93Feynman_absorber_theory
to his formulation of the pah-integral formulation of quantum mechanics to
en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula
Ron also answered on how the commutation relations [x,p]=ih, clearly an algebraic relation of operators, manifests in the path integral approach, where there are no operators.
It ought to be a special case of the Ito lemma:
http://en.wikipedia.org/wiki/It%C5%8D%27s_lemma

>Seeing your pic, Klenke is a pretty good book btw. Why don't you read him in German?

I do on an off.
Before I left for Easter holidays I jumped by the library, saw that there’s now an english translation and thought I take it with me. I love that particular format of books, I couldn’t say no.

>The guy who suggested it wanted to start a thread. If he's too busy, I'll make one myself on Monday.
Don’t know what he’s up to, you can do it.
I read the first 30 pages of the book and made two-three exercises. I like the lines-on-a-triangle rant/warning :)

>>7173298
One key word might be
>Where do stochastic integrals play a role in the action principle?

In the last posts in the thread last week I reported on „extremal action“ vs. „Euler-Lagrange“, I was reciting my thoughts on the different boundary data for the two.
<span class="math">t_0, q_0, t_1, q_t[/spoiler] for the variation
vs.
<span class="math">t_0, q_0, \frac{∂q(t_0)}{∂t}[/spoiler] for the differential equation
and how the one translated to the other.
I was cooking up examples where you could write
<span class="math">\frac{∂q(t_0)}{∂t} \approx \frac{q_1-q_0}{t_1-t_0}[/spoiler]
I was on the train on thursday and tried to work out the following:
If instead of
<span class="math">L=(m/2)(∂q/∂t)^2[/spoiler]
you use m/2 times a finite approximation of the derivative, what is a corresponding kind of differential equation and is the straight line still the extremal solution? What is the theories of such finite mechanics?
With an expansion of the finite different I’m lead to a Lagrangian with higher order than q’ and at this point I dropped that line of thought.
But the finite differences of Lagrangians appear in the computation schemes to path integrals where the classical solutions are extremal w.r.t. the integrators.
For stochastic process, a direct connection is

en.wikipedia.org/wiki/Onsager%E2%80%93Machlup_function

>>7172455
But I mean you can use that too for the general theorem (the Wikipedia page on it states it like that). In fact they have the general invariant term there, but I didn't post it because I don't really "feel" it either, and you can look it up yourself.

Btw. just to have mentioned it, there is "a version of the Noether theorem" for path integrals, i.e. quantum mechanics where you leave remove your paths from least action value a bit.
http://en.wikipedia.org/wiki/Ward%E2%80%93Takahashi_identity
I'm interested in it because the whole action principle talk form the first week thread lead me to read up on stochastic integrals. What's your background anyway?

>>7172421
>but I still don't "see"
If <span class="math">q_a=(3,7,0)[/spoiler] and <span class="math">q_a\mapsto q_b=q_a+\delta q[/spoiler], then <span class="math">q_b[/spoiler] is close to <span class="math">(3,7,0)[/spoiler] if <span class="math">\delta q[/spoiler] is an "infinitesimal generator" (would need Lie-group theory to make this formal). The mapping from <span class="math">q_a[/spoiler] to <span class="math">-q_a[/spoiler] may be a symmetry of the system, but then <span class="math">\delta q=(-6,-17,0)[/spoiler]. Noethers theorems gives you functions C(q,q',t) (conserved along trajectories that are solution of the equation of motion) for small <span class="math">\delta q[/spoiler].

I may post a fleshed out version of the theorem ITT,
but I'm currently on vacation and all the elaborate and clear physics books I like are German and not available to me right now.

>>7172339
and I should add that p in the classical newtonian approach is also mostly used a a function <span class="math">{\mathbb R}\to {\mathbb R}^3[/spoiler].

>>7172292
The input to the theorems are generators for smooth coordinate transformations and flipping the system is discrete and doesn't result in a conserved quantity.

Here's one intuition for the theorem (but take it will a grain of salt, it's just my mental picture):
-By the minimal action principle, the Lagrangian L determines the price you have to pay for various paths and the physical ones are the effective ones.
-The value L(q,q',t) changes when you change q, q' and/or t in most directions, but there are some in which it's invariant (pic related).
-The equations of motions are about finding a trajectory (which is a function dependent on time).
Now if the Lagrangian has an invariance in one direction (the lines of same height in the picture), then a movement in time along such a line doesn't bother the system and hence this movement is a legal mechanical trajectory. It's movement about which nature doesn't care, price-wise.
For example, if your system has translation invariance, then whatever initial momentum value you choose (initial conditions), the system is okay with your physical quantities moving along those translational direction while time passes on. Hence invariance of translation leads to any initial momentum being left alone.

>>7172306
If your mechanical system is the manifold <span class="math">{\mathbb R}^3[/spoiler], say, then a trajectory <span class="math">tr[/spoiler] is a function <span class="math">tr : {\mathbb R}\to {\mathbb R}^3[/spoiler].
For any time <span class="math">t_0 : {\mathbb R}[/spoiler], we have <span class="math">tr(t_0) : {\mathbb R}^3[/spoiler].
The Lagrangian <span class="math">L[/spoiler] is a function of position, velocity and time and takes real values, i.e.
<span class="math">L : {\mathbb R}^3 \times {\mathbb R}^3 \times {\mathbb R} \to {\mathbb R} [/spoiler].
People write <span class="math">L(q,q',t)[/spoiler] and mean the above. Here <span class="math">q,q',t[/spoiler] are 7 numbers. But people also write <span class="math">q[/spoiler] for the trajectory, i.e. a function from the reals into the manifold.
E.g. in the action S, the integrand is <span class="math">L(q(t),q'(t),t)[/spoiler], which is a function in
<span class="math">L : {\mathbb R} \to {\mathbb R} [/spoiler].

The canonical momentum (being part of the Euler-Lagrange equations) is <span class="math">\pi(q,q',t):=\frac{∂L}{∂q'}[/spoiler] and this is a function of type
<span class="math">{\mathbb R}^3 \times {\mathbb R}^3 \times {\mathbb R} \to {\mathbb R}^3 [/spoiler].
The Newtonian momentum
>p=mq'
(and here comes part of the main sloppyness people commit) is sometimes used as a function of the above type, and sometimes as a function of type
<span class="math">{\mathbb R}^3 \to {\mathbb R}^3 [/spoiler]

If I write q(t), people generally mean the trajectory, i.e. a function in <span class="math">{\mathbb R}\to {\mathbb R}^3[/spoiler], but formally it should be the value of <span class="math">{\mathbb R}^3[/spoiler].

i.e. whenever I write
<span class="math">\frac{d}{dt}C=0[/spoiler]
above, I really mean
<span class="math">(\frac{d}{dt}C)(q)=0[/spoiler],
where
<span class="math">q:{\mathbb R}\to M[/spoiler]
is not just any trajectory (with M the manifold in which your curves inside the mechanical system lie), but one that is a solution of the equations of motion.

>>7172279
ad: for the easiest case, consider <span class="math">L=(m/2)\,q'^2[/spoiler], where the canonical momentum <span class="math">\pi(q,q',t)[/spoiler] is actually the Newtonian momentum
<span class="math">p(q,q',t):=mq'[/spoiler],
(as function of q', not as (co-)tangent vector)
and the equations of motion <span class="math">\frac{d}{dt}q'(t)=0[/spoiler] literally say the conserved quantity (along the path) you look for is just this.
<span class="math">C(q,q',t)=\pi(q,q',t)=p(q,q',t)=mq'.[/spoiler]
(it's a little tiresome to keep functions of q and q' apart form functions of time. In the equations of motion I used the tangent vector q'(t) and below I use the same symbol q' for the velocity coordinate of functions like L. Everybody does this, but it's actually a little shitty)

>>7172255
Maybe I should have written down my thoughts after all.

Yeah, he doesn't present the theorem explicity (oddly, maybe), but only computes special cases.
In essence, if <span class="math">\pi[/spoiler] is the canonical momentum of the system (derivative of L w.r.t. q') the theorem uses the equations of motion
<span class="math">\frac{d}{dt}\pi=∂L/∂q[/spoiler]
to get the result
L has an invariance property => there is a function <span class="math">C(q,q',t)[/spoiler], so that
<span class="math">\frac{d}{dt}C(q,q',t)=0.[/spoiler]
He does the computation for the mentioned special cases
>(Noethers theorem, explicit calculations for energy, momentum, angular momentum
explicitly in Chapter II.

Okay, so I’m opening up a thread for round 2 of Landau.

Thread from last week:
https://archive.moe/sci/thread/7156940/

With about 20 pages for the week, we are though Chapter II (Noethers theorem, explicit calculations for energy, momentum, angular momentum, mechanical similarity and some consequences) and in the kids of Chapter III (approaches to solving the equations of motion).
I’m not gonna do a summary this wee, as people rather responded to what I asked that what I stated last week.
However, if one dares me, I can do it if ya like.

Despite medicare discussion last week (the thread died on day 3 or so), reading up on Lagrangians etc. got me thinking on several issues and I’ve closed some of my gaps.
At the moment I’m interested in stochastic calculus and Ito integrals, which over the path integral are topically not far to all those issues. Looking forward for discussions and questions. I’ll come on with questions/thoughts on the exercises as soon as someone else does.

>>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.)

>>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.

>>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)

>>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

>>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

>>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

>>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.

>>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.