You'd have to define what you mean by "halt"

damped harmonic oscillations just get smaller and smaller

The amplitude only approaches 0 at infinity.

Right, fair enough, but if you wanted to model the movement of a pendulum in a lab, say. And you wanted to figure out (or predict) how long the pendulum would swing for.

I mean, let's say get sufficiently close to 0 energy / 0 motion left so that we can't see it moving at all anymore.

Whatever that approximation may be, I guess I could figure out a realistic number.

Also we can take the angle to be small, that's fine. I don't want to make it too complicated.

@kylecampbell Well would you be able to plot the amplitude vs time of the theory & then just decide what is "small enough" and call it that?

I suppose, but is there an exact way to solve it without using plots?

Although, if it's a DE that can only be conveniently solved numerically that's okay too. An analytical solution isn't required, though it'd be cool if it existed.

Pretty sure there is analytical forms for small angle versions, as that's how we get phrases like "critically damped" and "under damped" oscillators

sciencedaily.com/releases/2019/04/190415082008.htm

Ideas mix, always

hi

@Secret how neutrinos and dark matter are related?

@Akash.B Not necessarily in any way at all. One possible way they might be related is that heavy sterile neutrinos are a candidate for the dark matter.

@dmckee oh

My sense is that they are considered as part of the second team by most people working in the area.

Not really a long shot, but not the best bet either.

how did the scientist realize the presence of dark matter?

It is easier to look up the history of the idea under the term "Missing mass".

Short short version: galaxies rotate too fast and orbit each other too fast.

Where "too fast" means so fast they would fly apart if the only thing holding them in was the mass we can account for.

This goes back to the middle of the twentieth century.

@Akash.B Neutrinos only feel gravity & the weak force. Dark matter feels gravity, and it (or some of it) may feel the weak force. So it's reasonable to assume some kind of family connection.

@PM2Ring oh new knowledge to me

Because dark matter is mostly oblivious to other forces, apart from gravity, its behaviour is quite different to normal matter. A cloud of normal matter in space will gradually collapse, converting gravitational PE into heat, which can be radiated away. A cloud of DM can't do that, its energy just bounces back & forth between PE & KE indefinitely, apart from possible tiny losses via the weak force.

So DM passing through an extremely active or dense star (including a neutron star) may lose or gain energy via weak interactions. But apart from that, it's pretty intangible. Another option is for it to fall into a black hole. Nothing escapes from those guys. ;) Apart from Hawking radiation, which is very weak for stellar mass (or heavier) BHs, and which may not actually exist.

@kylecampbell Well, it's related to the period of an undamped pendulum. ;) And there's a lovely algorithm for the exact period of an undamped pendulum that doesn't use the small angle approximation. The elliptic integral can be solved using the arithmetic-geometric mean, which converges rapidly.

Gauss wrote about this, but it's been mostly neglected because it doesn't fit nicely with the usual analytic tools, which tend to prefer the ubiquitous power series, and Fourier series. But with a few lines of code, you can do awesome stuff with the AGM. It's probably best known as the basis for a blazingly fast way to calculate pi.

@PM2Ring Interesting... Thank you for that.

In that case, it shouldn't be too hard to set one up for a damped pendulum.

Any recommended sources to get started? I guess I'll start with the undamped version and see how that goes. The thing is, I am ultimately curious about how long it will take for a damped pendulum to come to a halt, and similarly for uncoupled and coupled pendulum waves.

I did find this: geogebra.org/m/dYBDcvVZ

@kylecampbell I agree, although I don't recall anything about that in my admittedly brief exploration of the topic.

When I damped pendulum is making slow small oscillations, I suspect that it gets tricky to model. You have to more accurately model the various sources of friction.

Also, to do really accurate pendulum calculations, you need to take tidal variations of g into account. There's great info about that here: leapsecond.com/hsn2006 That page also has a great intro to the AGM pendulum equation leapsecond.com/hsn2006/pendulum-period-agm.pdf

This page mathoverflow.net/questions/34363/… has an algorithm to compute the elliptic integrals of the 1st & 2nd kind, simultaneously. It also has links to some fairly promising references. I want to check out what Borwein has to say, I bet he covers more than just calculating pi.

Sorry, I don't have any other refs to suggest. I just did a Google search on elliptic integrals arithmetic-geometric mean, and read the stuff that wasn't too complicated for me. ;)

@PM2Ring Thanks!

There's an arxiv titled "The AGM Simple Pendulum" by MB Villarino. I can't give a direct link, since I'm on my phone. IIRC, that article is a popular reference in the articles I read.

Bookmarked that as well.

@kylecampbell the motion of a damped pendulum is a standard undergrad problem.

The eqn of motion is taken as:

$$ m\frac{d^2x}{dt^2} = -A\frac{dx}{dt} - kx $$

You use the ansatz $x = X_0 e^{-(a+ib)t}$

Lots of the articles about elliptic integrals & the AGM use the AGM to derive various power series approximations for the integrals, and for pendulums. Frankly, I don't see much benefit in that, apart from when you want to compare power series, or do polynomial based error analysis. Just treat the AGM as a fundamental function in its own right. It converges so fast, and on a computer is less error-prone than a power series.

i.e. just a sine wave multiplied by an exponential damping factor.

I love how it naturally combines the real & imaginary exponential. It shows the physical connection between those 2 things that can seem weird when you first see that they're mathematically related.

I suppose another important physical thing that does that is normal EM waves vs evanescent waves, which someone here (probably John Rennie) mentioned recently.

@ACuriousMind The MathJax bookmarklet works on my phone, using the Samsung browser, which I suspect is a fork of Android Chrome. It's not ideal, but it makes the page readable. And it also works on the transcript pages.

@M.N.Raia You should try the ChatJax bookmarklets. See math.meta.stackexchange.com/a/3297/207316 I use the top one (IIRC) on my Android phone.

This question inspired me. Imagine a neutron star near the mass limit, eg 95% of the mass needed to collapse into a BH. Now imagine it freefalling towards an antimatter neutron star of the same mass.

Although it sounds awesome, it's probably not significantly more spectacular than if they were both normal matter. They collide at relativistic speed, so the total energy has a large KE component, relative to the rest energy.

@PM2Ring God wants to know yiur location :P

@AbhasKumarSinha Several timezones east of you, and quite a ways south.

To those who think it would be ok to SE Physics to non-mainstream theories, here's today's reason why I respectfully disagree: astronomy.stackexchange.com/users/26681/eric-lee-emerson

^ who think it would be ok to open up SE Physics

@PM2Ring 'by deriving the fine structure constant from first principles' hmm

@PM2Ring "SE Physics" Instead of Physics.SE (which derives from the URL) because.....?

@KyleKanos I'm flexible, I use both forms. :)

Well, IMO, one is backwards & confusing to read/see

But your prerogative to use either, I'm just sayin...

@bolbteppa That sort of thing was tolerated a century ago, when QM & particle physics were still very new, especially if you happened to be a famous scientist, with amazing number skills, like Eddington. These days, it's just sad, almost on the level of attempts to square the circle.

Can this khan academy help me get things done with calculus?cuz it’s out for free ,I don’t know much about calculus (how it goes)so plz tell me is it worth watching khan academy clips on calculus.

I suspect it's much like any source: you get out what you put in.

Never watched any of the videos though

Lemme give you link just go through and tell if it’s worth spending time out there

Yeah, I'm probably not going to watch it because I'm on my mobular device and at work (no wifi for Android devices)

google.co.in/… Here it is

@KyleKanos I shall try to remember to use Physics.SE, but I make no promises. My brain tends to reverse pairs if it doesn't have a strong reason to prefer a particular ordering. Is it the Physics-flavoured Stack Exchange site, or the Stack Exchange-flavoured Physics site? Why not both! :D

@PM2Ring like I said, Physics SE is derived from the URL, so just look up at the site name you are on ;)

@PM2Ring are you a physist

Currently you're on Chat.SE xD

@PM2Ring I'm sure you heard of this fine structure thing recently :p

@Purple No, but I know some physics.

@PM2Ring Hmmm

@bolbteppa Indeed. No (further) comment. ;)

I was a physicist.

@KyleKanos Well, yes, but what do URLs prove? :being silly smilie: And when I'm on my phone, I have to click & scroll things to read URLs.

URLs give the name of the site, don't they?

@Purple What's that supposed to mean. Sorry, I can't respond to your Khan academy question. I've never tried to learn mathematics from a video. I learn well from books, or well-structured online text with a predominantly linear flow. A few animations could be helpful, I suppose.

@Purple yes Khan will be helpful and is worth watching if you don't know any calculus, worth studying systematically to get started

@bolbteppa thanks, can you name some books

How about

amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/…

Yipes! $240 AUD for a calculus text? I'd definitely be buying it 2nd-hand if I needed that book.

I think older editions on there and are very similar and are way way cheaper

@PM2Ring yup

How about this pharaps I can get free pdf and read it on a iPad :/[email protected]

So uh

Back in March (maybe February, I forget) I booked some international flights for the third week of May

Woke up to an email this morning which starts like this:

“Dear Passenger,We are sorry to inform you that due to a schedule change, we have made changes to your booking. You will receive a notification of your ticket cancellation later today. Regrettably, we are unable to automatically rebook you on a new flight. This email is to inform you of the next steps available to you...”

I mean, a month is enough time to arrange for a new flight etc

But I still feel like punching something

Enough time but the prices are already probably higher :\

Yeeeep

To apply a line from an internet video I liked

“Your international airline: You won’t like it, and there’s no other option.”

I’m trying to bear this with humor. But mostly I want to punch something

And them offering a full refund isn’t exactly satisfactory since, as you noted, it doesn’t address the cost of booking a new flight at a closer date

Oh, also: the previous flight was about 10 hours total

The ones I’m seeing now? 13-18 hours total

Ffffff

Actual physics question: Suppose I entangle two spin-1/2 particles in the singlet state. It’s easy enough to show that, if I perform a partial trace over one particle, then the other particle will be in the completely mixed state (ie density matrix is proportional to the identity).

But it seems like that fact is also true if I entangle two spin-s particles in the singlet state

Which I don’t entirely understand. Intuitively it seems right

Seems like I should be able to deduce that fact from the spherical symmetry of the singlet state, ie if I trace out over one particle then the resulting mixed state for the second particle ought to have no preferred direction

Pet peeve: people cross-posting their question from Astronomy to physics after you advise them not to. The original was edited by the answerer astronomy.stackexchange.com/questions/30499/black-hole-density The Physics version is a carbon copy of the original physics.stackexchange.com/questions/473109/…

@PM2Ring Leave comments on both questions pointing out that they're cross-posted. If you think one should be closed, flag appropriately.

If this question ^^^ gets re-opened, I'll answer it. But I'm not going to re-open it unilaterally.

@rob I think I'll vote to reopen. Now that the person has specified that they're talking about special relativity, the question seems clearer (also after your edit).

I love Krasnikov but who uses a thorn as math notation

@vzn "the 2012 discovery of the Higgs boson, predicted in the 1960s"... 'over-reliance on mathematical elegance as a guide...' lol

"As Farmelo recounts, this is given interesting context by studies of Einstein’s notebooks, showing how he later overstated the role of mathematics, and underplayed that of physical insight, in his own breakthrough."

that's actually something the prof I know (who is a pretty well-known Einstein scholar) emphasizes as well

If you compare what Einstein did (in his letters/papers/notebooks etc) vs. his later writings

no google books preview for Farmelo, alas

But I'd be interested to see if he's citing the prof I know on this

Maybe he was overstating the math because he felt GR and EM were so mathematically linked (e.g. via Kaluza-Klein though he wanted an entirely 4D unified theory I think) and wanted to keep ignoring QM (physics)?

hard to say

there's two historical questions here

one is whether Einstein's later writings accurately reflect the process by which he developed GR

with the basic conclusion being that he tends to give too much credit to the math in his later writings

(that's probably a poor way of putting it)

the other historical question is, if he did do that, then why did he do it?

I'd love to see a draft/write-up of what he was working on say in the late 40's early 50's

I think the main reason people have avoided doing that is that it seemingly didn't go anywhere

whether that's a fair characterization is another question, of course, but that's the conventional wisdom

'Deriving all the elementary particles from GR and EM'

@bolbteppa found the relevant paper from the guy I know: pdfs.semanticscholar.org/45b0/…

see section 10, starting at p. 911

"The physics, he felt, had been nothing but a hindrance; he had been saved at the eleventh hour by the mathematics. In his later years Einstein routinely used this version of events to justify the purely mathematical approach in his work in unified field theory"

yeah

i mean, those two passages aren't identical in their attitudes

but they're very similar in terms of the facts they're referring to

"Hardly anybody who has truly understood the theory will be able to avoid coming under its spell. It is a real triumph of the method of the general differential calculus developed by Gauss, Riemann, Christoffel, Ricci, and Levi-Civita"

Agreed, action principles being the most insane version of this

There's a chicken vs. egg issue here of course

Did he view his earlier work as more mathematically-motivated because that's where his work on unified field theory led him, or did his work on unified field theory develop from him taking the role of mathematics as the primary lesson from GR?

And with the way human memory works...well, could be both

Yeah I'd say both and neither at times

yeah

I count myself as soundly ignorant of GR, alas

my knowledge of it is firmly in the hand-waving realm, rather than of technical expertise

It's just amazing he didn't even entertain things like the weak force back then

yeah. "unified field theory" for einstein was solidly EM & GR, with QM being viewed suspiciously

whereas a modern view of "unified field theory" is rather different

@Semiclassical I'm not sure there's a unified view of what constitutes a unified theory today ;)

hah

yeah, i buy that

haha! I got the go-ahead to start investigating generic models built on generic datasets so that I can bypass this no-data problem I currently have. :D

bowchickawowow soon I won't be a data scientist with no data

then you're just a data scientist with no real data :P

so, a generic data scientist? :>

working your way up one step at a time, I see

gotta slowly build up

why does Microsoft Edge not have automatic spelling checking?

prob better for math se

@Secret I think I said something wrong above---it should be the elliptic manifold, not the parabolic manifold; I intend to say a manifold whose curvature is of opposite sign to the hyperbolic manifold, like a Lorentzian manifold and I guess that kind of manifold is called elliptic manifold. Moreover, I have not finished reading that link of question " When does causal separation imply no spacelike separation?" because it contains unfamiliar notations requiring me to read that Wiki article.

@CaptainBohemian people use Edge!? Why?

@rob It looks clear enough to me, especially now that it's been edited. 1 more vote needed to reopen.

It is madness to calculate the Riemann curvature tensor without using forms, e.g. the $G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R$ version requires working out $R$ which is so simple using forms because you can use $\eta_{\mu \nu}$ :\

Can't imagine doing KK reduction without them

@CaptainBohemian Have you read Greg Egan's articles on these topics? He's written novels set in ++++ and in --++ universes, and has relevant technical articles on his website, as well as material about the usual -+++ metric. He also has some articles on topics in QM.