The h Bar

I wouldn't say only scholars would understand it -- it's probably something we've come across in high school at some point in the US. Along with things like deja vu and other phrases from other languages we've adopted.

How many remember it or use it often is what I don't know.

I also can't speak for non-American English speakers, native or otherwise

Captain Bohemian

@tpg2114 our English curriculum in high school never taught us deja vu.

tpg2114

We definitely had some lessons on loanwords at some point. I didn't take English classes after high school, so I know they came somewhere in the primary school system

Oh neat, this is a pretty cool Wikipedia page to get lost in

Captain Bohemian

3:39 PM

@tpg2114 We didn't have English classes in primary school.

our English classes started from junior high school.

Slereah

Raison d'être is in English but nobody pronounces it like French

bc they are awful at French

tpg2114

There's some French-named towns near me, Versailles and Bellefontaine. The locals who live there pronounce them like "Ver-sails" and "Bell-fountain"

About as un-French as you can get

tpg2114

6:17 PM

I've got an incredibly ill-formed question... if I have governing equations for a deterministic dynamic system, what math tricks do I need to convert it into governing equations for the probability density of the system state?

I'm thinking something like the Burgers' equation

Slereah

6:32 PM

Errr do you mean the evolution of the probability density with time?

tpg2114

Yeah

Slereah

Same as everything

Poisson bracket of the Hamiltonian

tpg2114

So for example, if it was deterministic, the probability density at the start would just be a delta function at whatever value and it should stay as a delta function

"Same as everything" -- I'm an engineer who does CFD, so this is all outside of my experience :)

Slereah

Write your initial probability density in terms of your canonical variables, and then do its time evolution

Never done Hamiltonian mechanics?

tpg2114

Maybe once in a dynamics of vibrations class

But not for fluid dynamics

But it sounds like that's my starting point

Slereah

6:37 PM

For any variable $f(x(t), p(t))$ in classical mechanics, you can write its time evolution as

$$\frac{d}{dt} f(x,p) = \{ f, H\} (x,p)$$

With $H$ the Hamiltonian and $\{,\}$ the Poisson brackets

This is the Liouville theorem, if you need more details :

Liouville's theorem (Hamiltonian)

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.There are related mathematical results in symplectic topology and ergodic theory. There are extensions of...

tpg2114

Okay -- I'll see what I can dig up. Thanks!

I've come at this from a different perspective, where I assume my deterministic variables are stochastic functions of a random variable, and then project onto a polynomial chaos expansion

user434058

7:25 PM

Yes! This exactly was my doubt that night! (Funny thing, I had ~~starred~~ bookmarked it for referencing it in the future quite a time ago, though I forgot it when I was struggling with the transition from Lagrangian mechanics to Hamiltonian mechanics :-))

user434058

15

Q: Why exactly do we say $L = L(q, \dot{q})$ and $H = H(q, p)$?

In classical mechanics, we perform a Legendre transform to switch from $L(q, \dot{q})$ to $H(q, p)$. This has always been confusing to me, because we can always write $L$ in terms of $q$ and $p$ by just taking the expression for $\dot{q}(q, p)$ and stuffing it in. In thermodynamics, we say $U$ ...

ACuriousMind

@FakeMod I pointed you to my answer there about a month ago :P

Charlie

You mention that the configuration space doesn't have a metric, what kind of object is it?

Oh I just found the answer nvm lol

is the topology of the configuration space ever of much interest or is it just "there" for the sake of rigour?

ACuriousMind

@Charlie I'm not sure what you mean by it just being "there" - the configuration space is the space of your generalized positions. If these are really just positions then it's a boring $\mathbb{R}^n$ but if they are e.g. Euler angles because you're describing orientation in space then it's $S^3$.

user434058

@ACuriousMind That's when I bookmarked it. BTW, I had upvoted your answer then, but TBH, I couldn't make up more than 30% of what you said there :P

user434058

7:57 PM

Heh, funny. Folks have upvoted my comment linking my (related) answer under another question, however my orginal answer hasn't recieved an upvote. Probably the comment was upvoted because of its relevance rather than my answer's quality, probably...

Jake Rose

Howdy

Mentioned this the other day, but getting quite interested in geometry. Is there any good books that start off broad then develop links to GR and QFT? (Or any other ohysics theories)

Nike Dattani

10:37 PM

Does anyone know why this was off-topic? physics.stackexchange.com/q/439184/134583

ACuriousMind

10:51 PM

@NikeDattani It's closed as HW-like/check-my-work

(I still don't get why the close notice is hidden from low-rep users nowadays :P)

Nike Dattani

@ACuriousMind It says closed as "off-topic"

Can it not be closed as "community-specific reason: HW" ?

I'm only asking because this user recently asked a question about the Berry phase on Matter Modeling SE.

So I checked out her previous network posts and found that one.

ACuriousMind

It is closed as that, but the closing notice is only visible to users with the close/reopen privilege

For some reason SE decided to hide the specific close reasons from low-rep users with the last redesign of the notices. We've already complained about that at the time on meta, but to no avail.

Cf. meta.stackexchange.com/a/339733/263383

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