UmbralRaptor changed the topic of #kspacademia to: https://gist.github.com/pdn4kd/164b9b85435d87afbec0c3a7e69d3e6d | Dogs are cats. Spiders are cat interferometers. | Космизм сегодня! | Document well, for tomorrow you may get mauled by a ネコバス. | <UmbralRaptor> … one of the other grad students just compared me to nomal O_o | <ferram4> I shall beat my problems to death with an engineer.
<egg|zzz|egg>
rqou: note that if you go from V to W (column vectors) using left-multiplication by the matrix A, you go from W* to V* (row vectors) using right multiplication by the matrix A
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<egg|zzz|egg>
Iskierka: diapsid!
<UmbralRaptor>
This Classical Mechanics class is convincing me I don't actually understand the chain rule.
UmbralRaptor is now known as NomalRaptor
<egg|zzz|egg>
NomalRaptor: Chain's rule?
<NomalRaptor>
d/dx f(g(x)) = g'(x) * f'(g(x))
<egg|zzz|egg>
NomalRaptor: yeah, I was referring to that trubowitz class
<NomalRaptor>
Eggcept it's a bit different when you have a mix of full and partial derivatives.
<egg|zzz|egg>
NomalRaptor: but it's easy, d(fg) = df dg :-p
<NomalRaptor>
egg|zzz|egg: ah
<NomalRaptor>
uh
<egg|zzz|egg>
wait no writing composition that way is a bit evil
<NomalRaptor>
aaaaaaaaaa
<egg|zzz|egg>
d(f∘g) = df dg
<egg|zzz|egg>
here
<egg|zzz|egg>
:D
<NomalRaptor>
ααααα
<egg|zzz|egg>
NomalRaptor: it's obvious though, say you have g:X->Y, f:Y->Z, dg goes from X to linear maps from TX to TY, df goes Y to linear maps from TY to TZ, so the above can only be interpreted as df (at f(x)) * dg (at x)
<egg|zzz|egg>
NomalRaptor: see it's much clearer if you do it all between arbitrary manifolds,
<soundnfury>
egg|zzz|egg visualises it in n dimensions and then sets n to 13
<soundnfury>
whenever I get confused by some analysis, I just stuff in some infinitesimals and try to make it a problem in algebra.
<soundnfury>
Did I mention that I'm not very good at analysis?
<egg|zzz|egg>
NomalRaptor: ?
<soundnfury>
NomalRaptor: you're differentiating... with respect to a derivative? Partially? aaaaaaaaaæ
<egg|zzz|egg>
NomalRaptor: there's no composition there?
<soundnfury>
egg|zzz|egg: that was my thought also
<soundnfury>
but OTOH we don't know what L and q_j may be functions of
<egg|zzz|egg>
soundnfury: also, no need to set n to 13, it helps even when setting n to 1
<soundnfury>
:D
<soundnfury>
NomalRaptor: without knowing what's meant to be kept fixed in that partial derivative, I can't help that much
<NomalRaptor>
L is a Lenergy of quosition and quolocity
<soundnfury>
but I'd be tempted to write all your things as functions of (presumably) t, then mechanically differentiate and use the backwards chain rule
<NomalRaptor>
The above is the first part of Lagrange's equation of motion.
<soundnfury>
oh but mechanically differentiating would give you the _total_ dL/dq_j, hunh.
<soundnfury>
so anyway, it's the time variance of the (linearised) dependence of Lenergy on the jth component of quolocity, yes?
<NomalRaptor>
Yeah
<soundnfury>
presumably evaluated at q_j (t)
<soundnfury>
is grad_{q_j} L anything useful you can reason about? Or is that what you're trying to get to? *shrug*
<soundnfury>
I still don't see where the chain rule comes into any of this
<soundnfury>
Haaaaang on. We may be able to reverse the order of differentiations by adding a correction term.
<soundnfury>
Expand the whole thing out with infinitesimals and fluxions, then rearrange it
<NomalRaptor>
Uh
* soundnfury
gives NomalRaptor a Newton
<soundnfury>
I have no idea if that will produce something useful to you, but then I don't know what you're trying to achieve
* NomalRaptor
gets launched on an escape trajectory out of the galaxy due to a unit conversion error?
<soundnfury>
well, at least you didn't burn up in the atmosphere of Mars…
<SnoopJeDi>
Too soon.
<egg|zzz|egg>
!wpn NomalRaptor
* Qboid
gives NomalRaptor a scandium ? with a һοｍοɡاурһ attachment
<egg|zzz|egg>
NomalRaptor: right so Chain comes in with d/ds' L(q(s), q'(s,s',t), t) and d/ds L(q(s), q'(s,s',t), t)?
<egg|zzz|egg>
s/q\(s\)/q(s,t)/g
<Qboid>
egg|zzz|egg meant to say: NomalRaptor: right so Chain comes in with d/ds' L(q(s,t), q'(s,s',t), t) and d/ds L(q(s,t), q'(s,s',t), t)?
<soundnfury>
NomalRaptor: well, the E-L equation is linear, so doesn't problem 8 just mean that any dF/dt must satisfy the E-L equation? i.e. why does L need to be there at all? /me confus
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<NomalRaptor>
soundnfury: Ask Goldstein?
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<egg|zzz|egg>
NomalRaptor: which are, uh, d/dq' L d/ds' q'(s,s',t) and d/d(q,q') L d/ds (q(s,t),q'(s,s',t)) I guess?
* NomalRaptor
feels a bit short on stack.
* egg|zzz|egg
making up notation as he goes along but hopefully that should make some sense
<egg|zzz|egg>
the latter term being d/dq L d/ds q(s,t) + d/dq' L d/ds' q'(s,s',t) which seems kind of promising?
<soundnfury>
NomalRaptor: uh, so after expanding out into infinitesimals and folding back down, I conclude (quite possibly wrongly) that \frac{d}{dt}(\frac{\partial L}{\partial \dot q_j}) = \frac{\partial}{\partial \dot q_j}(\frac{dL}{dt})
<egg|zzz|egg>
NomalRaptor: (I'm looking at 10. here)
<soundnfury>
(where everything in \dot q_j is evaluated at \dot q_j(t), of course)
<egg|zzz|egg>
NomalRaptor: let's time-differentiate the former term, d/dt(d/dq' L d/ds' q'(s,s',t)) = d/dtdq' L d/ds' q'(s,s',t) + d/dq' L d/dtds' q'(s,s',t)
<soundnfury>
so (for pr.8) you're trying to show that \frac{\partial F}{\partial q_j} - \frac{d}{dt}(\frac{\partial F}{\partial \dot q_j}) is constant
<soundnfury>
hmm, I am suspicious of what exactly \frac{\partial F}{\partial \dot q_j} means, since F is not a function of \dot q_j but only of q_j and t
<soundnfury>
to derivate partially, something must be being held constant
<soundnfury>
and if q_j and t are both held constant, then on the one hand, our derivative vanishes, but on the other hand, \dot q_j can't change either so there's no way to differentiate
<egg|zzz|egg>
partial derivatives are silly, but surely this means the gradient (or maybe just the differential) in those arguments
<egg|zzz|egg>
much like the d/dt isn't particularly impartial, just nobody in their right mind will write it with a rounded d
<soundnfury>
well but isn't that a total derivative then?
<soundnfury>
i.e. ratio of dL/dt and dL/d\dot q_j?
* soundnfury
's head is starting to hurt -_-
<soundnfury>
you can't have a gradient of a function in arguments which are not its arguments
<egg|zzz|egg>
sure
<soundnfury>
and F is specified to be a function of q and t, not of \dot q
<egg|zzz|egg>
it's a function of q' too, just constant :D
<egg|zzz|egg>
everything is a function of everything, and partial derivatives are fake actually :-p
<NomalRaptor>
AAAAAAA
<soundnfury>
so you're saying... \frac{\partial F}{\partial \dot q_j} _is_ zero?
<egg|zzz|egg>
NomalRaptor: which is why d(fg)=df dg is a much better notation!
<soundnfury>
because in that case, as far as I can tell, you're left with (E-L equation of L') = d/dt(ðF/ðq_j)
<soundnfury>
(pretend those eths are rounded ds)
<soundnfury>
and I can't see _any_ sane reason why that expression is necessarily zero
<soundnfury>
so I think some of these partial derivatives _have_ to be along some line other than the obvious one (i.e. some of the other variables are coupled to the 'driven variable' rather than being held constant)
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<egg|zzz|egg>
soundnfury: does q depend on t?
<soundnfury>
egg|zzz|egg: I think so, and I think that's mediated through the L that we have
<egg|zzz|egg>
soundnfury: good, so then d/dt F is a right mess that involves q', and it follows from there?
<soundnfury>
but if our ðF/ðq is mediated through that, then it's really just d(F(t, q(t)) / dq(t), fully a function of t
<soundnfury>
which seems a bit weird and not at all a partial derivative
<soundnfury>
(and F isn't really a function of anything but t either, which means IT'S ALL LIES)
<soundnfury>
aaaaaaaaaah maybe I see it
<egg|zzz|egg>
why F is a function of q too, but the thing we're differentiating is F(q(t),t), not F?
<soundnfury>
I may have been treating that d/dt like a ð/ðt or something
<egg|zzz|egg>
partial derivatives are silly
<soundnfury>
yes but — aaaaargh this is all so evil
<soundnfury>
and this, children, is why after a partial derivative you put a pale and label _exactly_ what you're keeping fixed
<egg|zzz|egg>
using them "properly" only induces more confusion, hence why I stick to straight ds
<soundnfury>
(or, equivalently, put your _fricking arguments_ in the function call on the top)
<egg|zzz|egg>
this ^
<egg|zzz|egg>
soundnfury: the solution of course is to eschew coordinates, and say things like "the chain rule is d(fg)=df dg", which is so much nicer :D
<soundnfury>
that does not solve anything
<soundnfury>
eschew the _j if you like, and say DL/DQ - d/dt DL/D(Q') = 0
<soundnfury>
but still put arguments on those L() and Q() and Q'()!
<egg|zzz|egg>
why yes it does, you end up saying that F is a function from the phase space (including time), and then you're writing d(Fq), which of course is dF dq, and the dq is obvious :D
<soundnfury>
"is obvious" bloody French mathematicians >:(
<G-Mobile>
I just discovered transit-finder.com, october 2nd we're going to have an ISS-moon transit entirely where I am
<UmbralRaptor>
bofh: conifer hedgehogs!
<UmbralRaptor>
G-Mobile: sounds fun to watch?
<G-Mobile>
it'll last 1.34 seconds, so, less watch, more see
<bofh>
reminds me, I need to see if there are any Iridium flares in the area.
<G-Mobile>
one infuriating thing about this site, the date range must be 50 days or less, starting at most 10 days ago and ending at most 10 days from now, entered manually
* Qboid
gives UmbralRaptor a surface-mount snowy fractal
<egg|zzz|egg>
UmbralRaptor: did you get your lagrangians sorted out?
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<egg|work|egg>
UmbralRaptor: 8. seems obvious now that I'm awake and in front of a piece of paper?
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<egg|work|egg>
!wpn whitequark
* Qboid
gives whitequark a thorium box/diffeomorphism hybrid
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* soundnfury
gives egg|work|egg a brachisto-scone
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* UmbralRaptor
gives soundnfury a brachiasaur along a minimum time trajectory.
<UmbralRaptor>
egg|work|egg: I think I ended up needing \frac{\partial L)}{\partial q} to be 0, even with sketchy assumptions to get rid of the other terms.
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<egg|work|egg>
UmbralRaptor: for 8? um
<egg|work|egg>
so the left term is d/dt d/dq' d/dt F(q(t)) = d/dt d/dq' dF/dq dq/dt = d/dt d/dq' dF/dq q' = d/dt dF/dq = d/dq(d/dq(F)) dq/dt = d/dq(d/dq(F)) q'
<egg|work|egg>
UmbralRaptor: and the right term is d/dq d/dt F(q(t)) = d/dq (dF/dq q') = d/dq d/dq F q' + dF/dq dq'/dq?
<egg|work|egg>
UmbralRaptor: with dq'/dq=0 they vanish?
<egg|work|egg>
UmbralRaptor: the question just simplifies to showing that d/dt F(q(t)) satisfies the Euler-Lagrange equation right? (F can depend directly on t but that doesn't matter, just add a coordinate to q which is t and it's the same quesiton)
<egg>
NomalRaptor: there's an eggstraneous dot on the last line, should be dL/dq, right?
<egg>
after the -
<NomalRaptor>
AAAAAAA
<egg>
NomalRaptor: line above seems sane, so probably miscopied?
<NomalRaptor>
… does that count as a floating point error?
<NomalRaptor>
egg: yeah
<egg>
:D
<egg>
NomalRaptor: also, wow doing it with indices and not making t part of q makes things messy
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<soundnfury>
!wpn egg
* Qboid
gives egg a Stern-Gerlach saker
egg is now known as egg|zzz|egg
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<egg|phone|egg>
!wpn whitequark and котя
* Qboid
gives whitequark and котя a diastolic primitive † with an ellipsoid attachment
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<SnoopJeDi>
"Physics Opportunities at an ElecTron-Ion-Collider" --> POETIC
<SnoopJeDi>
?
<bofh>
SnoopJeDi: I see they're challenging the NMR folk for acronym absurdity
<Iskierka>
what's the lowest acronym-letters to actually-relevant-component-word ratio in use?
<bofh>
good question, actually
<Iskierka>
I'd also be curious about the other end. What skips so much that you wonder how on earth that's supposed to identify it
<NomalRaptor>
Iskierka: biochemistry will win dramatically.
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<SnoopJeDi>
So I watched the Ig Nobel ceremony for the first time last night...I'd never realized it was so enjoyable!
<bofh>
heh
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<Iskierka>
can an Ig Nobel be awarded for the same thing as a Nobel a few years later if it turned out way less useful than they claimed? (ex. graphene)
<UmbralRaptor>
Call it the "not everything can be blue LEDs" award?
<Ellied>
I thought graphene was still pretty useful, just not easy to make yet
<SnoopJeDi>
it's pretty h*ckin easy to make
<bofh>
^
<SnoopJeDi>
Iskierka, counterpoint: would you judge the transistor by what it was used for in 1957?
<SnoopJeDi>
Hate the futurists, not the fundamental researchers :P
<SnoopJeDi>
(but *especially* Michio Kaku)
<bofh>
ugh, Michio Kaku. sigh.
<UmbralRaptor>
Wait, hecking is strong enough to need to be censored now?
<SnoopJeDi>
/kick UmbralRaptor watch your language
<SnoopJeDi>
UmbralRaptor, as popularized by @dog_rates
<UmbralRaptor>
hm
<UmbralRaptor>
"What is the only thing graphene can't do?" "Leave the lab."
<SnoopJeDi>
LOL
* UmbralRaptor
heard that at Space Grant a year or 2 back.
<bofh>
ROFL
<FluffyFoxeh>
pfft
<SnoopJeDi>
I batted graphene to my advisor as a potential technology for a Mössbauer-alike for studying neutrinoless double β decay, but it never really went anywhere
<SnoopJeDi>
I barely understand the Mössbauer effect well enough to have contributed anything more than "well graphene is neat and there's a lot of work in that..."