The h Bar

Secret

00:55

Granular modes in Bose Einstein condensates

PM 2Ring

01:09

@LeakyNun A rest frame is like a walking frame, but for people who are too tired to move. ;)

bolbteppa

@AbhasKumarSinha no, your answer here is not right, this step:

$$\require{cancel} \delta S = \int \limits_{t_1}^{t_2} \left(\cancel{L(q, \dot q, t)} + \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot q} \delta \dot q - \cancel{L(q, \dot q, t)} \right)dt = 0 \\ \delta S = \int \limits_{t_1}^{t_2} \left ( \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot q} \dot \delta q
\right)dt = 0 \\ \left [ \frac{\partial L}{\partial q} \delta q(t) \right]_{t_1}^{t_2} + \int \limits_{t_1}^{t_2} L \left(\frac{\partial L}{\partial \dot q} \frac{d}{dt} \delta q(t) \right)dt = 0 \\ (\text{Remember that } \dot q \text{ is de…

Again $\int \limits_{t_1}^{t_2} \frac{\partial L}{\partial q} \delta q dt \neq [ \frac{\partial L}{\partial q} \delta q(t) ]_{t_1}^{t_2}$

Above I quoted 3 lines of equations from your post, on the second line you're not suppose to touch the first half of the equation at all, you're only supposed to do integration by parts on the second term, and the final answer should involve an integral of the first term un-modified. If the iook at the Euler-Lagrange equations:

Noting the second term, the $\frac{\partial L}{\partial q}$ term, actually comes from simply not modifying the part you keep incorrectly modifying

You really need to see an explicit example of this worked out, like a first-principles derivation of the EL equations for an explicit Lagrangian

bolbteppa

03:45

@AbhasKumarSinha wrote a response as an answer with an example

Akash. B

04:43

@bolbteppa what the hell?

@JohnRennie Ewwwww!

bolbteppa

@Akash.B ?

Akash. B

@bolbteppa what those equations say?

bolbteppa

Euler–Lagrange equation

In the calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange in the 1750s. Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks...

Akash. B

John Rennie

05:29

@Akash.B it's simpler than you think. Equations always look frightening the first time you see them because you don't recognise the symbols used. The Euler Lagrange equation is pretty standard and all physicists will learn it as part of the physics degree.

Akash. B

@JohnRennie then can you explain it?

John Rennie

@Akash.B it would take a while to explain ...

Akash. B

okay

@JohnRennie i will wait

enumaris

ahhhh

Akash. B

@JohnRennie

bolbteppa

06:04

Understanding the Euler Lagrange Equation

►

Secret

09:48

Why does the black hole image generate so many memes

Abhas Kumar Sinha

10:13

@Secret Blackholes are secret

Abhas Kumar Sinha

11:10

@bolbteppa So you mean that only this part was wrong, or others have that problem too?

bolbteppa

The way it's written is wrong in

$$\frac{\partial L}{\partial \dot q} \delta q - \int \limits_{t_1}^{t_2} \left( \delta q \frac{d}{dt} \left ( \frac{\partial L}{\partial \dot q} \right) \right)dt = 0 \\ \Rightarrow \int \limits_{t_1}^{t_2} \left (\frac{\partial L}{\partial q}\delta q - \frac{d}{dt} \left ( \frac{\partial L}{\partial \dot q}\delta q \right)\right)dt = 0 \\ $$

$$
\Rightarrow \int \limits_{t_1}^{t_2} \left ( \frac{\partial L}{\partial q} - \frac{d}{dt} \left (\frac{\partial L}{\partial \dot q} \right) \right)\delta q \space dt = 0$$

e.g. the second line

Abhas Kumar Sinha

So, I can't cancel $\partial t$ with $dt$? as I did in second line?

bolbteppa

The first line is wrong, it also does not imply the second line, and the second line does not imply the third line

\begin{align}
\delta S(q) &= \int [\frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \frac{d}{dt} \delta q] dt \\
&= \int [\frac{\partial L}{\partial q} \delta q + \frac{d}{dt} (\frac{\partial L}{\partial \dot{q}} \delta q) - (\frac{d}{dt} \frac{\partial L}{\partial \dot{q}} ) \delta q ] dt \\
&= \int [\frac{d}{dt} (\frac{\partial L}{\partial \dot{q}} \delta q)] dt + \int [\frac{\partial L}{\partial q} \delta q - (\frac{d}{dt} \frac{\partial L}{\partial \dot{q}} ) \delta q ] dt \\

That's the right thing

Why not read a book on calculus of variations before doing mechanics based on calculus of variations without knowing the math it's based on, it would almost be like doing Newtonian mechanics without knowing much calculus beyond basic ideas

amazon.com/Calculus-Variations-Dover-Books-Mathematics/dp/…

Abhas Kumar Sinha

@bolbteppa Will reading about Calculus of Variations will solve my problem?

bolbteppa

@AbhasKumarSinha what is your problem?

Abhas Kumar Sinha

11:20

@bolbteppa Understanding that.

Akash. B

@bolbteppa I didn't understand part after he had started using letter "d"

bolbteppa

@AbhasKumarSinha in your question you're asking about $\int [ f(x + \delta x) - f(x)]dx$ but the mechanics problem writing $\int L(t,q + \delta q,\dot{q} + \delta \dot{q}) dt$ is treating $\delta q$ as a function of $t$ in other words this is a real misunderstanding of what's going on, this is a non-trivial subject more advanced than calculus basically

Abhas Kumar Sinha

Okay!

@bolbteppa Are there any other easy books other than Landau and Lifshitz's Mechanics?

bolbteppa

In calculus you are treating numbers $x$ as your variables and working with functions $f$ of numbers, $f(x)$, in calculus of variations you are treating functions $f$ as your variables and working with functions $S(f)$ of functions (where the functions are real-valued functions of vector variables)

There are books like

amazon.com/Classical-Dynamics-Particles-Systems-Thornton/dp/…

Akash. B

@bolbteppa

bolbteppa

11:25

which do Newtonian mechanics and introduce basic calculus of variations midway through it so you learn it in line with learning advanced mechanics

@Akash.B ?

Akash. B

5 mins ago, by Akash. B

@bolbteppa I didn't understand part after he had started using letter "d"

bolbteppa

This nptel course

youtube.com/playlist?list=PLB368471AD70B8A6B

also introduces calculus of variations in mechanics in a very easy way with lots of explanations

@Akash.B you can use these resources as well to learn what's going on :p

Akash. B

@bolbteppa which?

bolbteppa

The youtube videos I just sent

Abhas Kumar Sinha

Prof. Balakrishnan's Lecture: youtube.com/watch?v=sCZ80l6UarM is suitable for beginners, but it doesn't go very depth (Suitable for those who know).

Akash. B

11:28

5 hours ago, by bolbteppa

Understanding the Euler Lagrange Equation

►

@bolbteppa you mean this?

Abhas Kumar Sinha

@Akash.B You mean that $d$? it's derivative

Akash. B

okay

bolbteppa

@Akash.B both that and the one I just sent

@AbhasKumarSinha Balakrishnan is explaining the thing you're having trouble with in that video around minute 13

Abhas Kumar Sinha

@bolbteppa oh okay, let me see that again

Abhas Kumar Sinha

14:08

What's the difference between the Lagrangian mechanics and Hamiltonian mechanics? Is the Principle of Least actions just hamiltonian mechanics?

enumaris

14:43

They work in different spaces

Lagrangian mechanics works with coordinates and velocities, Hamiltonian mechanics works in phase space (coordinates and momenta)

Principle of Lease (stationary) action appears in both

fun times

John Rennie

16:58

@DanielSank independent.co.uk/life-style/gadgets-and-tech/news/…

Semiclassical

17:09

@JohnRennie i guess i shouldn't be surprised at that kind of crap---the internet is full of a**holes, who'd-a-thunk-it---but yet I somehow am

bolbteppa

17:22

That is unbelievable tbh

With Gamergate etc... this stuff has been going on for ages

Emilio Pisanty

@DanielSank I'm more interested in this instance as a case study of the precarization of academic work

why was the primary responsibility for that reconstruction given to a PhD student instead of to a staff scientist?

could it be because the PhD student was cheaper, and we're so used to the student churn machine using students as cheap scientific labour that we don't even blink an eye anymore?

Semiclassical

Sounds about right.

The reference point I use for this kind of thing nowadays is something that came up during the US budget haggling last fall (or was it the year before that? losing track)

Emilio Pisanty

@Semiclassical what, charging students extra tax?

that sounds like a 2017 thing

Semiclassical

yeah, some GOP rep put some language in that would've made graduate student tuition benefits count as taxable income

vzn

in theory salon, 14 secs ago, by vzn

How Katie Bouman Accidentally Became the Face of the Black Hole Project / NYT

Emilio Pisanty

17:27

@Semiclassical yeah, I remember

vzn

> While she led the development of an algorithm to take a picture of a black hole, an effort that was the subject of a TED Talk she gave in 2016, her colleagues said that technique was not ultimately used to create this particular image.

Semiclassical

And I was always a bit ambivalent about that. On the one hand, it was clearly intended as a means of hitting public universities in a vulnerable spot

and doing so by targeting grad students, who are rather vulnerable

so I found that pretty despicable.

On the other hand...well, who created that system, and why does it exist?

Emilio Pisanty

@Semiclassical there was a particularly interesting interview with some Republican lawmaker that basically said "well, you don't complain enough and you don't vote enough, so we're just going to keep hitting you", as I recall

Semiclassical

Yuck

Emilio Pisanty

it'd be worth finding that one and adding it to the annals

Semiclassical

17:29

It's a system which ensures that the universities have access to cheap grad student labor

in exchange for a "tuition benefit"

Emilio Pisanty

@Semiclassical I think that's too simplistic a reading

the cheap-grad-student-labor is a thing everywhere

over a wide range of tuition-costs structures

Semiclassical

as in, not just in the US context?

Emilio Pisanty

though it is true that it's considerably worse in the US than it is in broad stretches of Europe, particularly when you look at years-to-graduation statistics

in the US it is much easier and much more common for PIs to keep students around for ages and milk them for papers before allowing to graduate

up by a factor of two, and occasionally closer to three, with respect to e.g. France

Semiclassical

I guess my main feeling there was: If the US universities are vulnerable on this point, it's due to their reliance on cheap grad student labor

Emilio Pisanty

@Semiclassical that sounds like a too-easy scapegoat on that particular issue tbh

Semiclassical

17:34

eh. I in turn think it's too easy to scapegoat the GOP for putting forward that proposal

Emilio Pisanty

@Semiclassical any system you care to build will have that type of 'vulnerability' to a hostile lawmaker

Semiclassical

it's despicable to target grad students like that, but I also don't care for the university systems putting the grad students in that position in the first place

Emilio Pisanty

the precise details will change from country to country, but if you want to tax grad students, you can always figure out a way

@Semiclassical what position?

having tuition-waiver systems in place?

I don't think there are any real alternatives

you can call them something else

but it is a fact that infrastructure and mentors' salaries cost money

in any system you go to, PhD-studentship funding will include a component where money goes to the university to pay for those things

the specific accounting system in place in the US was not "designed" by anyone, it just developed organically over time

if the possibility of that specific attack had been on the table when it developed, then funding agencies and universities would have re-designed the individual components of the system so that the accounting for that went strictly between the funding agencies and the universities

but why would anyone have thought that this type of stupid tax bill would ever be in the offing

no, this falls squarely on the GOP.

Semiclassical

Saying that it developed "organically" seems too kind a reading of the situation

after all, the US tax code and tuition-benefit system didn't come from the sky. it was developed by people

and while I think one can certainly go too far in reading those intentions---I don't see a point in conspiracy theories

I think it's entirely too charitable to suppose that the system we've inherited in the US was inevitable

Emilio Pisanty

@Semiclassical key word being 'people', as opposed to 'person'

many different design decisions spread out over decades and over hundreds of separate institutions

@Semiclassical I'm not saying this was "inevitable", in any way.

I'm saying it wasn't caused by any one single decision

it's a system

systems are complex

Emilio Pisanty

17:46

ah

here we go

3

> Jesse recalls a conversation he once had with a senior staff member of the Senate Health, Education, Labor and Pensions Committee:

Semiclassical

yeah, I see it

and I buy it

Emilio Pisanty

> "It's easier to get money from grad students than going through the appropriations committee

> Grad students are an easy population to get money from, you guys are too busy to notice

> until you speak up, we're going to keep coming for you."

Semiclassical

As I said, I have no illusions about -why- the GOP would pursue that tactic

moving to an uncontroversial point, I posted this integral at the start of the year:

18

Q: What is the surface area of the 3-dimensional elliptope?

The $n$-elliptope is defined as the set of $n$-by-$n$ correlation matrices; that is, the set of $n$-by-$n$ symmetric positive-definite matrices with ones on the diagonal. Such matrices are parametrized by their $n(n-1)/2$ upper off-diagonal elements. In the case of $n=3$, this yields the 3-ellipt...

two bounties later, and -still- no response.

Emilio Pisanty

18:05

@Semiclassical ooof

Semiclassical

the weird thing to me is the difficulty of the integral vs. the simplicity of the numerical result

like, why the heck should that complicated integral be $5\pi$

Emilio Pisanty

there's no shortage of over-complicated integrals and series that integrate to simple factors of $\pi$ and $e$

Semiclassical

true enough

there's a bunch of ways to make stuff like that complicated

John Rennie

@EmilioPisanty I suspect you're over analysing this.

Emilio Pisanty

@JohnRennie am I?

there's 200 astronomers in that collaboration

John Rennie

18:10

Dr Bouman clearly loves the work and I suspect she just happened to be in the right place at the right time when people were looking ofr some human interest angle on the story.

Emilio Pisanty

what fraction are students?

John Rennie

No-one is suggesting Dr. Bouman herself has behaved improperly.

She's not a student.

Emilio Pisanty

@JohnRennie .... including me

@JohnRennie that's the impression I got from the earlier press, but it may be wrong

was she a student at the time the work was done?

John Rennie

She was a grad student when the work started, but then that was three years ago.

Emilio Pisanty

@JohnRennie so, why wasn't that work assigned to a staff scientist?

I mean, aside from the fact that the figure of staff scientist has been all but extinguished in favour of a steady churn of grad students

John Rennie

18:13

@EmilioPisanty Huh? If I'd been a student there I would have eaten my own kidneys to get that work.

Emilio Pisanty

@JohnRennie it's a systemic issue

John Rennie

@EmilioPisanty What, the fact that enthusiastic grads will kill to be be allowed to work on leading projects no matter how arduous the work is?

Emilio Pisanty

like all systemic issues, in any particular instance there are always other factors, and the systemic issues are only ever contributing factors, but that doesn't mean they're not there

John Rennie

That's kind of what we do ("did" in my case).

Emilio Pisanty

@JohnRennie if that's the way you want to see it, then flip the script

if I'd been a staff scientist there I would have eaten my own kidneys to get that work

John Rennie

18:16

Well yes. I don't know how she got lucky. Maybe she just did it anyway and did it better than anyone else. Who knows?

Semiclassical

I think it's difficult to extract a lesson like that from a particular case.

John Rennie

But then it isn't clear precisely what her input was. In amongst all the noise it's going to be hard to find out.

Emilio Pisanty

@JohnRennie As I said, it's a systemic issue.

Semiclassical

(Whereas I'm far more willing to draw conclusions about systematic issues with regard to grad student tuition waivers, given how universal that is)

John Rennie

What is the issue? Unfairly assigning tedious work to grad students?

Emilio Pisanty

18:18

All across the board you have the staffing of scientific labs shifting from staff-scientist positions, with reasonable job security and a more robust position against the various forms of abuse that can come down, over towards a configuration where the bulk of the research is done by students who are cheaper and in much more precarious positions

it's not an issue of who is "allowed" to do the research

it's an issue of the amount of job security that's afforded to them

John Rennie

OK. I can't comment.

Emilio Pisanty

It's not just me saying it

This has been extensively written about, at the very least, for well over a decade (but I do think that the effects have become stronger in the past decade)

@JohnRennie fair enough

if you want to read more on the subject, "academic precarity" is a good starting search term

I found this piece particularly good (though it focuses on the dead-tenure-track side of academic precarity, rather than the grad-student side)

Emilio Pisanty

18:58

@Semiclassical It looks to me like the $\alpha$ integral may be susceptible to attacks along the lines of Gradshteyn & Ryzhik 3.679(1)

or maybe 3.679(3).2. ?

hmmmmm

or maybe not

boy, you do have one messed up problem there

anakhro

19:15

Is there a popular average for the mass increase for main sequence stars?

Like 1.27 solar masses per Gyr or something.

I am trying to do a problem where you are given an initial mass function, and that stars life time roughly follow $10m^{-2.5}$, m in solar masses. Then you are asked to find the mass of the most massive of main sequence stars after 1Gyr.

Semiclassical

@EmilioPisanty ikr

Emilio Pisanty

@anakhro that makes no sense

anakhro

The issue is that after 1Gyr, the one I'd expect (before 1Gyr passes, it would be those of mass 10^(2/5) that are just about to die) the mass to increase.

Emilio Pisanty

masses don't 'increase'

Semiclassical

The fact that it seemingly has a closed-form despite that is what interests me

anakhro

19:19

@EmilioPisanty what do you mean?

Emilio Pisanty

you're just calculating attrition, essentially

Semiclassical

Had the number come out as unfamiliar, I’d have just shrugged and moved on

HDE 226868

@anakhro Why not just calculate the mass of a star with a lifetime of ~1 Gyr?

Emilio Pisanty

after 1Gyr the more massive stars will have died off, and they're asking for the most massive ones behind that

anakhro

@HDE226868 I do, but then I get the wrong answer.

Emilio Pisanty

19:20

(but HDE is much better placed than me to answer this.)

@Semiclassical yeah, I agree

anakhro

The answer should be 3.2 solar masses. I get 2.5 solar masses.

HDE 226868

What's the precise mass-lifetime approximation you're using?

Also, are you neglecting pre-main sequence evolution?

anakhro

Well it gives: "Assume that the main sequence lifetime of a $1 M_\odot$ star is 10 Gyr, and main sequence lifetime scales with mass as $M^{-2.5}$."

err

HDE 226868

Plugging in 3.2 gives me about 0.97 Gyr. I suspect it's maybe an algebraic mistake.

anakhro

Sorry, it should have been $M^{-2.5}$

Oh.

I think I see what my problem is.

Heh

Serves me right.

I am using an earlier edition of the textbook, and a friend gave me the new version's numbers. But I am comparing it to the old textbook answers.

2.5 solar masses is the correct answer.

Now I feel like an idiot.

HDE 226868

19:25

~~Ah, so using the -2 scaling gives 3.2, but the -2.5 scaling gives 2.5? That's amusing. I wonder if they changed the exponent based on new data.~~ Probably not, actually; that power law is usually derived from a standard mass-luminosity relationship. But depending on your choice of that law, both -2 and -2.5 are reasonable.

anakhro

Yeah, I would assume that's maybe the case.

Do you like astrophysics, @HDE226868?

PM 2Ring

The discussion about the exploitation of grad students reminds me of Privatdozent. "The title, Privatdozent, as such does not imply a salaried appointment; it merely denotes permission to teach and examine independently at the conferring faculty without a professorial appointment.

The title has its origins in German-speaking countries in Europe before 1800. It referred to a lecturer who received fees from his students rather than a university salary. [...] In 2012 more than 5000 honorarium Privatdozenten worked at German universities without a salary."

Semiclassical

one could also easily fold in a discussion about adjunct profs

PM 2Ring

Einstein was a privatdozent "Because he was only being paid out of his students' fees, Einstein continued working full- time at the patent office."

HDE 226868

@anakhro I'm an undergraduate astrophysics major, so yes. :-)

anakhro

19:30

@HDE226868 would you mind helping me with another question if you had time?

HDE 226868

@anakhro Sure.

anakhro

Okay let me type it up.

Actually, a screenshot might be faster.

PM 2Ring

Several famous mathematicians were privatdozents. "in 1907 under the supervision of Paul Gordan, [Emmy Noether] worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. [...]

In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent."

anakhro

@HDE226868 they include a hint to: " Calculate the Keplerian velocity of the accreted material a moment before it hits the neutron star surface, and use it to derive the angular momentum per unit mass of this material, $J/m$."

I think I calculated that and got $v = \sqrt{GM/R^3}$

I don't quite know how to use this to derive $J/m$

HDE 226868

@anakhro That seems to have the wrong units.

anakhro

19:36

Hmmm.

Oh. I think it would be just $\sqrt{GM/R}$?

Let me calculate it again.

HDE 226868

I'd expect $v=\sqrt{GM/R}$, assuming Newtonian gravity.

anakhro

Yeah.

Yeah that's it, sorry.

HDE 226868

Do you know the expression for orbital angular momentum?

anakhro

That is $\omega$, isn't it? So $2\pi/P$?

The hint then says to use the derivation for $J/m$ in the equation $\frac{d}{dt}(I\omega) = \dot M\frac{J}{m}$, where $I = (2/5)MR^2$ is the moment of inertia.

Or $\omega = v/R$ is maybe what you were asking for.

HDE 226868

@anakhro Close; it's $L=I\omega$.

anakhro

19:41

Ah, I see.

HDE 226868

From that, you can find $L/m$ (angular momentum per unit mass), and then follow the hint to get the change in the neutron star's angular momentum.

anakhro

So calculate $L/m = I\omega/m = (2/5)R^2\omega$? Or am I mincing $M$ and $m$?

HDE 226868

@anakhro So, there are technically two moments of inertia - one for the accreted matter, one for the neutron star. That's the inertia of the neutron star.

A better expression for the orbital angular momentum is $L_orb = pr = mvr$.

That makes L/m easier to find, because you already figured out v.

anakhro

In that case, $J/m = vR,$ where the $v$ is as we calculated?

Oh sorry

HDE 226868

Yup.

anakhro

19:48

So that's for the accreted mass.

So toss that into my formula and let's see what I get.

So this gives the equation $\frac25\frac{d}{dt}(MR^2\omega) = \dot MvR$

They now say to "neglect changes in the neutron star’s mass and radius"

I don't know why they say this, because it makes it sound like they want me to treat them as constants.

But I want to solve for $\dot\omega$

HDE 226868

That sounds right. They assume the neutron star's inertia is constant, so $\dot{L}=\frac{d}{dt}(I\omega)=I\dot{\omega}$.

Because accretion rates are low.

anakhro

So $\dot M$ is small, close to 0?

But it's okay to treat it as 0 on the left, but not on the right?

HDE 226868

$\dot M\neq0$, but over small timescales, it's small enough that $\dot{I}\approx0$.

It's a weird assumption, but it works.

anakhro

Ah, I see.

So we have $\dot\omega = \frac{5\dot Mv}{2MR}$

Now I use $P = 2\pi/\omega$ from before.

$\implies \dot P = -2\pi\dot\omega/\omega^2$

HDE 226868

@anakhro Yep.

anakhro

19:55

So that's my differential equation.

Plugging in $\omega = 2\pi/P$ and $\dot\omega$

HDE 226868

Yes, I was just going to suggest rewriting $\omega$ in terms of $P$.

anakhro

Great!

Thank you!

I will try the next part (solving it) on my own for now.

But I appreciate you taking the time to help!

HDE 226868

@anakhro You're welcome.

PM 2Ring

20:13

@anakhro See how it parallels the linear version, $p = mv$ ? When you learn a new formula, it can be helpful to do dimensional analysis on it, to make sure it makes sense. And it can help you to understand how the formula was originally discovered / invented. That also applies to formulae you derive yourself.

anakhro

Yeah, dimensional analysis is interesting that way.

PM 2Ring

Especially when deriving complicated formulae, where it's easy to mess up an exponent somewhere. Like accidentally getting $R^3$ from $R^2$, instead of getting the proper result of $R$. ;)

Semiclassical

doing dimensional analysis is also pretty easy if you're doing kg/m/s

it gets more annoying when you starting throwing in electromagnetism units

PM 2Ring

Or you're using geometric units with $c = G = 1$. Those units make the equations look less cluttered, and simplify calculations, but it's like climbing a wobbly staircase with no handrails. :)

Semiclassical

yep. good once you've got a sense of balance but

liable to send you sprawling if you don't :P

Lozansky

20:53

Are there any rules in antenna theory? It doesn't seem to be governed by the laws of physics nor math

Semiclassical

What book are you using?

Lozansky

My professor's handouts

Semiclassical

The most obvious reference text is Jackson, with all the agony that entails

Yikes

user10535

Howdy; I got a friction problem. I have rolled a ball down a ramp

Lozansky

So in an exercise, you are given the dipoles $p_1 = p_0\sin(\omega t), p_2 = p_0\sin(\omega t+\pi/2)$

user10535

20:57

And measured the distance that the ball moved a square box

Semiclassical

Is that one dipole or two?

Lozansky

And he's like, let's make the ansatz $p_1 = p_0 e^{i\omega t}, p_2 = p_0e^{i(\omega t + \pi/2)}$

Two dipoles

Semiclassical

I'm also guessing you need some $i$'s

(or j's, depending on what convention you use)

Lozansky

Yeah

Semiclassical

$e^{i \pi/2}$

Lozansky

20:58

I'm pretty tired

Semiclassical

gotcha

usually the idea is not so much "use this ansatz" but "use a complex-valued source, and take the real part at the end"

Lozansky

Yeah

I understand why

Since it simplifies stuff like Poynting

Semiclassical

Yeah. Gotta be veeeery careful about using formulas, of course

Lozansky

But I feel it should at least be mentioned what you are doing

Semiclassical

Yeah, the standards of presentation aren't always the best

Lozansky

21:01

Anyway, in this exercise I am only interested in the radiation function so it doesn't matter I suppose

Semiclassical

I mean, it's definitely a useful procedure. For instance, the (complex) net dipole there would be $p=p_0+p_1 = p_0 (1+i)e^{i\omega t}$

Of course, you could also write that as $p=p_0\sqrt{2}e^{i \pi/4}e^{i \omega t}$

in which case the real part gives you $p=p_0\sqrt{2}\cos(\omega t+\pi/4)$

Lozansky

Sure

Semiclassical

So evidently you can think of this as a single dipole

Lozansky

For the fields, you'd need to adjust the exponential in accordance to their positions though

Semiclassical

hmm

I'm not sure that's true. In the far-field limit, at least, I'd imagine only the actual dipole moment matters

not the particular details of how that dipole moment is created

Lozansky

21:06

Yeah but the fields are out of phase due to the retarded time?

Semiclassical

hmm

could not tell you there

Lozansky

$E(r,t) \propto \sum \ddot{p_n}(t-(r-\hat{r}\cdot \mathbb{w_n})/c)$

I mean this thingy

user10535

21:24

Ok, I gave in and restarted

And I measured the angle

If I want to find the height of the ball from a certain horizontal distance on the ruler, do I need to multiply the distance by the sine or by the cosine of the angle from the ruler from the ground?

PM 2Ring

22:07

@user10535 Hint: consider what happens if the ramp angle $\alpha$ is very small. The ball will roll down the ramp very slowly. For small angles (measured in radians), $\sin \alpha \approx \alpha$ and $\cos \alpha \approx 1$.

PM 2Ring

23:27

I bet if there were a video on YouTube of Dr Katie ~~Bowman~~ Bouman looking at the M87* image on a computer monitor and saying "My God, it's full of stars!", it would go viral. :D

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