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15:00
It took a while, but I've finally surpassed Qmechanic for Suggested Edit queue: physics.stackexchange.com/review/suggested-edits/stats!!!
Congrats!
If a shaft has torques A and B on left side and right respectively, and a distance as x going from left to right, what would the internal torque in the shaft be? Would it be A or B? I'm trying to imagine cutting the shaft and then balancing torque with an internal torque but I'm not sure which side you would consider
@Slereah how does one pronounce Henneaux?
I'd pronounce it eno
/eno/ in IPA or something
not hen-ee-ax?
or hen-ee-ox?
15:05
Nah, -eaux is o in french
and french doesn't really have an /h/ sound anymore
it's just a leftover from the frankish days
well if it's supposed ot be an o sound, why don't they use an o?
Why is it knight and not nite
or nit
because it's pronounced kuh-nig-it
All hail the Monty.
@0celo7 Your definition of $w$ is not coordinate-invariant if $u$ is not scalar.
15:14
basically it's the usual thing where a word looks normal in the era it's used
but then the pronounciation changes, but not the spelling
I mean, sure all that works if you treat $g^{\mu\nu}$ just as an array of functions, but at the end of the day it's still a tensor and picking out particular components from it is still non-sensical unless the resulting expression is properly covariant.
that's what you get for not using the bundle
Like a god fearing christian
@ACuriousMind it's an array of functions in a chart.
But if you change the coordinates, you're not in the same chart
and I'm doing all of this in a chart
@Slereah the conservation law you get at the end, $dJ_X=0$, is again covariant
working in coordinates is not a crime
15:21
the conservation law is just gonna be the conservation law of $n$ scalar fields tho
the conservation law is a PDE, stop trying to insert epistemology
Well when you try to flow the metric tensor it will just act on it as something like $g_{\mu\nu}(x) \to g_{\mu\nu}(f(x))$ or something
Which from a PDE perspective is fine I suppose
But that's not the proper transformation law for a tensor
it leads to a covariant conservation law, regardless
but somehow it's wrong
unless you can point to a mathematical error in what I have done, then, well, physics is wrong
@Slereah The point here is not whether it's the "proper law" or not. The problem is that if 0celo7's derivations were valid, then regardless of what his "non rotation rotation" actually "is", it would still prove a conversation law that's false.
Well no
It's just a conservation law on scalar fields
for which it is correct, I think?
15:26
@0celo7 Well, that's not true - it's simply mathematically false that the stress-energy is symmetric for all Lagrangians.
It's not physics that's wrong, it's your derivation because it leads to a purely mathematical contradiction. The specific error is rather elusive, though.
@ACuriousMind exactly
I mean is there a contradiction in it though?
I'm not sure there is, given the hypothesis
If you assume that the fields are scalar fields and $L$ is invariant, I think the stress energy tensor is symmetric
@Slereah He isn't assuming the fields are scalar fields. He simply doesn't care whether they are and defines his variation as if they were.
Is there much of a difference?
@Slereah Yes. Your argument would work if there was something like $\phi_\ast u$ in there and then 0celo7 purposefully replaced it by the scalar expression, but that's not the case. He's just defining $w$ and not choosing to define it in the "obvious" way for a tensor field.
And since indeed the final result is covariant, objecting that this is not coordinate-invariant doesn't work
15:35
But if you change coordinates for a collection of scalar fields, the result is also coordinate-invariant
the variation is a choice, so I am free to make the choice.
what "should" happen is that it results in no conservation law
but somehow it does here, but an incorrect one
@0celo7 I am beginning to think it must be that (2) does not imply (1).
@ACuriousMind That was my conclusion before I gave up yesterday
(2) is weird, as you correctly observed that holds trivially for all $L$ indepedent of $x$, so it would mean you get all these conserved quantities for basically every theory ever.
@boides you can check this page out, it's quite good and helpful
15:37
yep
Kifflom.
@ACuriousMind do you agree that Theorem 1 follows from (1)?
so writing (2) for a solution gives $$L(x,u(x),Du(x))=L(\phi(x),u(x),Du(x))$$
and we agree this is true if $\partial L/\partial x=0$
@0celo7 I think something is wrong here in the setup - you act as if $w$ is wholly independent from $y$. But it's not - the flow acting on $x$ already induces a change in the $u$, in the case where the flow is generated by a vector field $X$ this is of course the Lie derivative. Now, you can protest again that you treat $w$ as indepedent from $y$, but you don't - you assume that the whole thingy obeys the E-L equations as we flow along $\tau$,
so you need to respect that the fields need to change as the Lie derivative w.r.t. $X$
Hmm, where do I treat $w$ as independent of $y$?
In that you claim you're free to pick it in the proof of corollary 2.
When, in fact, it is already determined by you picking $y$ as being generated by $X$
Only if you choose that way (1) is implied by (2)
15:47
(1) has no connection between w and y
in (2) I pick a specific w and y
@0celo7 Only the implicit one that by saying that everything obeys the E-L equation, the part of $w$ (or rather $m$) that you actually can pick is $m - \mathcal{L}_X$ for a $y$ generated by $X$.
No, (1) does not require a solution of the EL equations
Now, in Corollary 2, you basically consign yourself to not being allowed to pick that part at all, i.e. the independent variation of the fields is zero. Otherwise, I don't see how (2) would imply (1) with an arbitrary choice of $w$.
remember, Noether is about the Lagrangian symmetries, not solutions
one can also write theorem 1 for $u$'s that don't solve the equations
@0celo7 Yeah, forget my rambling about the E-L equations, I'm working this out as I go along. So, how is (1) supposed to follow from (2) for an arbitrary choice of $w$ that is not the one induced by the variation $y$?
15:51
@ACuriousMind well unless I'm severely wrong here, let's jut interpret (2) as saying there is no explicit $x$ dependence for $L$
now we take some random $u:\Omega\to\Bbb V$ and the associated variation $w$ of $u$ and $y$ of $x$, where $y$ is generated by the flow and $w=u\circ y$
these satisfy the axioms laid out in the first paragraph, agreed?
and now we check that (1) holds
How do we check that?
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
there's where the pullback comes in
but a translation is an isometry so you just move the arguments in the integral
wait, rotations are isometries too
yeah, so $\phi(\omega)$ is the isometric image of $\omega$
thus we expect $\int_{\omega}L=\int_{\phi(\omega)}L\circ\phi^{-1}$
evaluated on $w$
now $w\circ\phi^{-1}=u$
so we get $$\int_\omega L(x,w,Dw)=\int_{\phi(\omega)}L(y,u,Du)$$
and then you just move the $y$ to $x$ and you're done
something like that
@ACuriousMind is that at all convincing?
We have $L(x,w,Dw)=L(y,w,Dw)=L(x,u,Du)\circ\phi$, so it follows from the change of variables formula
The issue must be that we really need a pullback and not a composition
I feel there's something going on here with you sometimes writing arguments for the second and third component of $L$ and sometimes not.
$L\circ \phi$ is really just $L(\phi(x), u(x),Du(x))$ isn't it?
16:03
Well, note that for $u$ fixed, we have a map $F:\Omega\to\Bbb R$ defined by $x\mapsto L(x,u(x),Du(x))$ and I really mean $F\circ\phi$
Or, rather, $L\circ \phi$ doesn't really make sense because the domains don't match
Ah, yes.
Why do you have $L = L \circ \phi$
I don't
Well, depends on what you mean by that
I am NOT claiming that $F=F\circ\phi$
So wait, at what point in your derivation do we actually use (2) at all, @0celo7?
@ACuriousMind I just want (1) to hold
I figured (2) was reasonable and implies (1)
now I'm not so sure it does
16:07
I can't rly contribute much because I'm not quite sure what all the various quantities are
so the RHS of (1) is just $$\int_{\phi(\omega)}F(x)\, dx$$
this equals $$\int_{\phi}\phi^*(F(x)\, dx)$$
Anonymous
If I'm asking a question w.r.t to a research paper (which is under a paywall), do I just link to the site with a paywall, or upload it to something like google drive and then share the link to the pdf? I'm talking about the main site. (BTW are such questions allowed on the main site?)
assuming $\phi$ is killing, we have $\phi^*dx=dx$
so this is $$\int_\omega \phi^*F(x)\, dx$$
but $F$ is a scalar, so $\phi^*F(x)=F(\phi(x))$
@ACuriousMind Ah. I found the error
Anonymous
@ACuriousMind You around? Could you please have a look at my query above ^
$D(w)=D(u\circ\phi)\ne (Du)\circ\phi$
But it is so for translations
thx
Anonymous
16:11
I think linking to sci-hub or something would be unethical...so that option is out
inb4 it's true for rotations (I don't think it is)
@Blue It's likely that any question that requires the answerers to actually read a specific paper is not a good fit for the site. You can link to the paper however you want, but the essentials necessary for an expert to answer the question should be in the question itself, not in a link
@Blue you link to the abstract
@0celo7 Heh, well, I didn't do much more than ask you to flesh out how (1) comes about in the end... ;)
Anonymous
@ACuriousMind Oh, got it. Thanks for the advice
16:13
@ACuriousMind Can you confirm that this $\ne $ is true for rotations?
It has to be false
Anonymous
I'll try to make the question self-sufficient
What is $w$ and $u$
rotations are a position dependent transformation so it must be so!
I know
What I mean is
What does it correspond to
16:14
@0celo7 Yes
in non-crazy notation
@Slereah The field and the field variation.
@ACuriousMind other way around
I.e. what the usual physicist will write as $\phi$ and $\phi'$
16:14
Wait, is it the field variation or the field derivative
@0celo7 No, $\delta \phi = m$ :P
oh, right
@Slereah $u=\phi$, $w=\phi'$ (variation), $m=\delta\phi$ (infinitesimal variation)
also called a multiplier in math circles
what's $Du$ then?
16:16
$\partial_\mu\phi$
What do you mean by "a variation" then
look at the definition...
@ACuriousMind So that was a pretty subtle thing, no?
@0celo7 I think it would have been much easier to catch if you had written more detail than "(2) implies (1)" ;)
Ah! Eureka! @ACuriousMind Weinberg's Belinfante-Rosenfeld tensor is $T+$ derivatives of $L$ wrt. the derivatives of the field.
So it is the derivative terms that couple to the Lorentz transformations
And the Christodoulou-Noether theorem has corrections involving $\nabla_\mu X^\nu$...
It all makes sense now...
and here you were accusing all physics of being wrong
16:20
I still stand by that statement
I'm still not sure what the problem or solution was but good that it is solved
Well, if you don't have kinematic terms there's no spin stuff
Probably
$\partial \Lambda A = \Lambda \partial A + A \partial \Lambda$ and all that
@Slereah I am slowly making sense of the details of ADM
hopefully it will be real
Otherwise we'll have to throw out all those books
16:45
@0celo7 You're into PUBG now? :P
> When I see the quality of bread in supermarkets, it is impossible not to get angry. The bread is frozen, some of it comes from Romania or who knows where, nothing is carried out in accordance of the rules of the art.
@BalarkaSen why do you say that?
Man, the French really hate Romanian bread, eh? :P
@0celo7 discord lets me see what game you are playing
17:05
@BalarkaSen why is it amusing?
I didn't say anything about amusement
17:23
@ACuriousMind bread bigots?
The upper crust usually are :P
17:39
hiiiiiiiiya
i.imgur.com/6SC0RRa.png for this diagram, would the internal torque at any point x be T_a or T_b?
Anonymous
17:57
@Johnathon Did you have a look at the diagram you posted? :P
Anonymous
Anonymous
I see this ^
whoops my bad
Anonymous
Find net torque using $(T_{A}x-T_{B}(d-x))\hat{k}$, $\hat{k}$ being the unit vector pointing out of the plane of the paper.
18:04
Sorry I should have made it more clear, T_a and T_b are both torques acting in the same direction
$T_a$ $T_b$ mathjax test
Anonymous
Are they sliding torques (generated due to a couple)? They should be, otherwise a rigid body like that hinged at two points would be fixed. In that case just add up the torques in the same direction.
@Blue It's a shaft sorry, ishould have stated all this at the beginning, my bad
A shaft rotating at constant speed and has uniform diameter
Anonymous
What are the forces generating the torques?
@BalarkaSen I just called the Ugandan meme people in PUBG racist @Phase
They called me really mean things
looooool
get rekt boi
I don't know how to feel about that meme really. It's so cancerous, but simultaneously too popular
I think the right retaliation is just to come up with a better meme which are not shouted around in game servers by 12 year olds...
Anonymous
18:13
@BalarkaSen There's a paragraph in Nash Sen which I'm having difficulty in understanding. Will you be here for a while?
Anonymous
Okaies:
which paragraph? I have the book
@Blue I tried to simplify it from this main question: i.imgur.com/dvqCvn8.png what I don't understand is why between 1000 and 2000mm that the internal torque is the same as the torque at D2
Anonymous
@BalarkaSen Page 3. Example 2
Anonymous
18:14
I'm having trouble understanding the deformation of $a_i$ stuff
Anonymous
They're choosing $a_i^{\epsilon}=\epsilon a_i$
Anonymous
$i=0,...,q-1$
Anonymous
Okay so far
Anonymous
But
Anonymous
How is $n_0(a_0^{\epsilon},...,a_{q-1})=n_0(a_0,...,a_{q-1})$ ?
Anonymous
18:16
They just say "evidently"....it's not really evident to me as to what they're doing
it's a typo. They mean $n_0(a_0^\epsilon, \cdots, a_{q-1}^\epsilon) = n_0(a_0, \cdots, a_{q-1})$
i.e., the value of the integral doesn't change if you perturb the coefficients
Anonymous
@BalarkaSen Ok. But then how is $n_0(a_0^\epsilon, \cdots, a_{q-1}^\epsilon) = n_0(a_0, \cdots, a_{q-1})$, unless $\epsilon$ is close to $1$ ?
Anonymous
We have to perturb it continuously, no?
Anonymous
They don't mention $\epsilon$ is close to $1$
$n_0(a_0, \cdots, a_{q-1})$ is a 1) continuous function 2) takes integer values always.
That forces it to be constant.
Anonymous
18:20
@BalarkaSen Yeah. I got that much. But say $n_0(a_0^3, \cdots, a_{q-1}^3)$ may be $2$ (an integer), while $n_0(a_0, \cdots, a_{q-1})$ is $1$
That can't happen, is what I mean by "constant"
$f(z) = n_0(a_0^z, \cdots, a_{q-1}^z)$ is a function which takes integer values for all $z$, and is continuous. If $f(1) = 1$ and $f(3) = 2$, it has to take a non-integer value somewhere in $(1, 3)$ by intermediate value theorem.
I.e,., there must be a $c \in (1, 3)$ such that $1 < f(c) < 2$
Contradiction, because $f(c)$ must be an integer...
Anonymous
Oooo
Anonymous
Understood
Anonymous
Thanks !
18:22
I feel like it's such a defeat to accept you just have to solve Hermite etc using series first and then getting the rest of the craziness like Rodrigues, generating functions etc as the easiest route :(
No problem
@Blue $\displaystyle \frac1{2\pi i} \int_C \frac{f'(z)}{f(z)} dz = n_0$ is a beautiful theorem in complex analysis.
It's called the "argument principle". Do you know the geometric interpretation?
Anonymous
@BalarkaSen Yep! I know that proof of that one
Anonymous
It's $n_0-n_p$ though
Anonymous
In case there are some poles
18:29
True, I took a holomorphic function.
Anonymous
Basically we need the fact that $\frac{f'(z)}{f(z)}=\frac{d}{dz}\log f(z)$
Oo, careful. $\log$ is a very dodgy function in complex analysis.
Anonymous
Now, if $f(z)$ has multiplicity $m$ at $z=a$
That's good for an intuitive picture but you're better off knowing an actual proof!
(In fact that's what my geometric interpretation is)
Anonymous
@BalarkaSen Oh, do you have some other proof?
Anonymous
18:30
I only know this one
Anonymous
If $g(z)$ is holomorphic and non-vanishing over a simply connected set in $\Bbb{C}$
Anonymous
Then $\log g(z)$ is holomorphic
Anonymous
(I'm using the fact that you can then write $f(z)$ as $(z-a)^{m} g(z)$ such that $g(z)$ is holomorphic and non-vanishing)
Ah I see your proof will end up being equivalent to mine then.
But you don't need to write log to do this
Anonymous
@BalarkaSen Okay?
Anonymous
18:35
Then how
If $f(z) = (z - a)^m g(z)$, $f'(z) = m(z - a)^{m-1} g(z) + (z -a)^m g'(z)$. So $f'(z)/f(z) = m/(z - a) + h(z)$ where $h(z) = g'(z)/g(z)$ is something holomorphic inside $C$.
So that integrates to $2\pi i m$ over $C$
Then add up the multiplicities each of the zeroes contribute
Anonymous
Oh. You're directly finding $f'(z)/f(z)$
Anonymous
That's a good idea
Anonymous
And also there's no problem of zeros in denominator since we're choosing $g(z)$ carefully
Anonymous
18:37
Gotcha
Right.
Anonymous
Nice
But the geometric interpretation goes something like so:
The integral, as you observed, is $\frac1{2\pi i} \int_C d\log f(z)$
Anonymous
There's a $d/dz$ in there
Thanks, edited
What I wrote is a dodgy expression, but let's pretend it's right
Anonymous
18:39
okey
log is a multivalued function which changes value by $+2\pi i$ if you loop counterclockwise around $z = 0$, and changes value by $-2\pi i$ if you loop clockwise around $z = 0$
So for each zero $z = z_0$, $f(z_0) = 0$, of multiplicity $m_0$, $C$ loops around that zero of multiplicity $m_0$
That contributes to a $+\frac1{2\pi i} \cdot 2\pi i m_0 = +m_0$ in the integral
Anonymous
Oh, yes. Like for n loops in calculus of residues or contour integration
Anonymous
Understood :)
And similarly for each pole $z = z_\infty$, $1/f(z_\infty) = 0$, of multiplicity $m_\infty$, $C$ loops around $\infty$ (clockwise!)
Which contributes similarly to a $-m_\infty$ in the integral
@Blue Right
It's really a rather deep topological fact
Anonymous
@BalarkaSen Yup, cool observation. I was thinking of something like that too :D
18:43
Mhm
Anonymous
@BalarkaSen Sounds interesting
Anonymous
I'm at just page 3 and it's already so interesting :P
I like the approach of this book
Not hyper-mathematical of course, but neat
vzn
vzn
19:04
> My 70y old step dad finally managed to get his paper published. It allegedly proves that spooky action at a distance does not exist and that Bell's theorem is incorrect. He'd be thrilled to get your feedback!
your grandfather's probably a crank
Maybe it's Duffield
Anonymous
Duffield is 70yo?
Anonymous
:p
20:12
Allo
Anonymous
@CooperCape ola
The Gauss law has vanishing Poisson bracket with the Hamiltonian constraint.
Anonymous
Reading Gallian and Axler?
Yeaaaahhh
Haven’t really looked much through gallian
Anonymous
Cool. I just got through the first 2 chapters of Gallian
Anonymous
20:14
It's nice
Anonymous
Also reading a topo book
Oooh neat... it’s all group theory at the beginning right?
Anonymous
@CooperCape Yep
I’ll leave topology etc. For when I am better at maths...
(Hah nevverrr)
Anonymous
@CooperCape Before starting topo I suggest mastering at least basic complex analysis, first
20:15
Okay nokay
I got enough maths to do rn anyways
Anonymous
@CooperCape Heh...high school is broring XD
Anonymous
(P.S: Even college is boring)
Depends... on the middle of exams currently eek
Oh... oh no...
College was my getaway ;)
Anonymous
If you will take physics in first year they'll again teach those basic newtonian mech, sound, light, etc along with some programming I guess
Anonymous
And some basic math like multivariable calc and linear algebra (more basic level than Axler)
20:18
Sounds about right (from talking to friends etc.)
I really dislike mechanics tho ;p
Anonymous
The way it is taught in most schools makes it boring
@CooperCape Note: what you think you like and dislike before uni $\neq$ what you actually like and dislike :P
Statics of rigid bodies is duhhhhhhhg
@Mithrandir24601 Good good.... maybe mechanics shall be my ting
Anonymous
It would be nice if they showed more visual simulations and real examples. (Newtonian) Mechanics has some very counter-intuitive results too...at times
Yeah it’s all very rod-like at the mo
20:26
can someone actually explain to me why the 1st and 3rd laws exist
Surely conservation of momentum and $F = \frac{d}{dt}p$ do it all, no?
Maybe he just wanted a few extra laws to his name.
Having just 1 is a bit basic.
Anonymous
@Phase There are some issues with the definition of inertial frame. Using second law you cannot prove the existence of an inertial frame, which is spoken about in the first law.
How so though, doesn't $\frac{d}{dt}p \iff a = 0$?
or am I missing something
Anonymous
14
Q: Why is Newton's first law necessary?

Murtuza VadhariaNewton's second law says $F=ma$. Now if we put $F=0$ we get $a=0$ which is Newton's first law. So why do we need Newton's first law ? Before asking I did some searching and I got this: Newtons first law is necessary to define inertial reference frame on which the second law can be applied. But...

Anonymous
12
A: Why is Newton's first law necessary?

ACuriousMindNewton's first law postulates that there is (at least) one inertial reference frame for every object, in which said object will continue in uniform motion unless acted upon by a force. Newton second law states that, within the inertial reference frame for any object, $F = ma$. Without the first...

Anonymous
20:33
ACM also wrote an answer
yeah I saw the wild ACM
appearing from the long grass
Was it shiny?
is that a new thing?
Back in my day Pokémon were just Pokémon
I swear you could always get shiny Pokemon?
They were rare af tho
really?
20:35
I only played on DS tho was too late to be a game boy child
i dont remember there being any in 1st generation
Ahh they were gen 2
And up
back in my day we played Duel Monsters
with large stone tablets
and the loser got banished
Anonymous
There was something called Duel Masters
Anonymous
I forget
Anonymous
20:38
Ah, it was an anime
sounds like a yugioh rip off
Damn...
I was mario Pokemon and wii sports
Duel Masters was first serialised 3 years after YGO
that sounds like damn plagiarism to me
Hummmmm
Anonymous
20:42
Best Answer: Duel Masters is a parody of Yu-Gi-Oh.
I have never seen so much salt from so many permavirgins
Permavirgins....
Now that’s a saying...
@Phase I feel like a human!
@0celo7 you're nearly human!
Just a bit more evolution and you'll be there buddy
21:05
> Yield: 11 cups
my god
2.8 L soup
are you kidding me
Thai ginger soup
oh hell yes
21:55
@Blue Wasn't that a YuGiOh! thing?
@DanielSank that's duel monsters, the in-universe name for the card game
@0celo7 Oh, right.

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