Lie brackets

Semiclassical

Mar 1, 2017 00:50

Start with Newtonian mechanics, then make the jump to Lagrangian mechanics, and finally to Hamiltonian mechanics.

At which point Poisson brackets show up naturally. That said, this is probably not the route if you're wanting to appreciate them from math directly.

Don't disturb

@ZachHauk I don't do it for me now, I do it more for younger people that think like me and that can be kept unfortunately in a range of very low performance and wrong beliefs about what they actually can do

Ted Shifrin

Intuition is hard to buy cheap!

Semiclassical

Main reason I like Poisson brackets in that perspective is that it connects with the commutator bracket in quantum mechanics.

Ted Shifrin

That's Lie bracket :)

Don't disturb

Some users here should be banned for ever for spreading so much wron belief about how mathemtics should be approached.

BAYMAX

Mar 1, 2017 00:52

rightly said @TedShifrin

yes @Semiclassical

Semiclassical

eh, maybe so. but you'd never bother to say it in physics.

$\{x,p\}=1\rightarrow [\hat{x},\hat{p}]=i\hbar$.

Don't disturb

Be humble, be humble (a profound hypocrite expectation), just as much as you cannot ever get their performance or ever dare to over pass it.

Zach Hauk

I believe Balarka is a pretty humble guy

Ted Shifrin

He's grown up a lot ...

Simple

Anyone explain me what topological conjugate is?

Semiclassical

Mar 1, 2017 00:53

As to why the bracket should be imaginary in quantum mechanics....oh look over there, a convenient distraction.

Ted Shifrin

Topologically conjugate?

BAYMAX

what are $x$ , $p$ i guess continuous functions in $C_{\infty}{M}$ , $M$ is a set @Semiclassical

Zach Hauk

@Ted Now that Balarka's all grown up you have me instead!

Simple

yes

Don't disturb

@ZachHauk You're best as long as you come to them like a begger to share their knowledge with you. But you don't have to do that.

Semiclassical

Mar 1, 2017 00:54

Typically, you'd take $x,p$ to be canonical coordinate and the (conjugate) canonical momentum

Ted Shifrin

Basically, you can make one picture look like the other by a homeomorphic change of coordinates.

Don't disturb

You have the power to perform better than all of them.

Semiclassical

@Simple Context?

Zach Hauk

eh, Balarka was smarter than me at this age

Ted Shifrin

Um, gee, thanks, @Zach.

Semiclassical

Mar 1, 2017 00:55

Comparisons are rarely helpful.

Don't disturb

@ZachHauk He might be 70 or 80.

Akiva Weinberger

Ted still has to deal with Balarka

@Don'tdisturb He's 17

Ted Shifrin

And you, DogAteMy :D

BAYMAX

I have learnt Lagrangian , Hamiltonian 2 years ago , i have to go through them now ... ha ha :) @Semiclassical

Simple

@Semiclassical this is a word on my ode text book

Semiclassical

Mar 1, 2017 00:56

And me, come to think of it.

Oh, nice.

Don't disturb

@ZachHauk that's my whole point. You don't have to be like anybody, you must be like you, the version of you passed through a lot of work.

Semiclassical

I can summarize it pretty well, then.

Ted Shifrin

It's weird having some people on ignore ...

Semiclassical

In Hamiltonian mechanics, you've got the Hamiltonian function $H$ (which in a lot of cases is just $H=p^2/(2m)+V(x)$.)

Akiva Weinberger

@Don'tdisturb I want to star what you're saying but I can't decide which ones. Maybe all of them

Semiclassical

Mar 1, 2017 00:58

The time evolution of the position and momentum is then determined by Hamilton's equations: $\frac{dx}{dt}=\frac{\partial H}{\partial p}$ and $\frac{dp}{dt}=-\frac{\partial H}{\partial x}$.

Akiva Weinberger

@TedShifrin Balarka will have been smarter than me at my age (given that he's smarter than me and younger than me), yes

Semiclassical

(I should probably be doing q not x, but w/e)

BAYMAX

ohh @Semiclassical $q$ , the generalised coordinates .. right or wrong??

Don't disturb

@AkivaWeinberger OK

Semiclassical

Right.

MATH ASKER

Mar 1, 2017 01:00

Find how much gallon of paint it takes to paint a room. If the lenght, height and width is given and also we know that it takes a gallon to paint 350 square feet of the room, why do we need height if width is the height?

Don't disturb

Now I'm really gone.

Ted Shifrin

I was not commenting on silly comparisons, DogAteMy. I was saying I still have to deal with you.

Akiva Weinberger

@MATHASKER 3D room, not 2D

@TedShifrin Ah, I see

Yes, you still have to deal with me :D

MATH ASKER

oh but still why do we do Lenght * height + lenght * height + width * height + width * height

is width in this case thikness?

Semiclassical

Moreover, Hamilton's equations determine the time evolution of any function $f(x,p)$ through the chain rule: $$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial p}\frac{dp}{dt}=\frac{\partial f}{\partial x}\frac{\partial H}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial H}{\partial x}$$

Ted Shifrin

Mar 1, 2017 01:02

No, no. You're painting the ceiling or no?

MATH ASKER

no I don't think we have to paint the ceiling in this case

Akiva Weinberger

@MATHASKER Each of those terms is one of the four walls. Each wall is a rectangle, right?

MATH ASKER

but if we had to would we do lenght * width 2 times?

yes

Ted Shifrin

one time.

Akiva Weinberger

Opposite walls have the same size.

Semiclassical

Mar 1, 2017 01:03

That's a rather interesting looking structure, and in fact that's just the Poisson bracket!

Akiva Weinberger

You never do length*width

MATH ASKER

ya i get why we do it two times

why

Ted Shifrin

Ceiling, DogAteMy

Akiva Weinberger

I thought we weren't painting that

Semiclassical

Namely, $\{f,g\}=\dfrac{\partial f}{\partial x}\dfrac{\partial g}{\partial p}-\dfrac{\partial f}{\partial p}\dfrac{\partial g}{\partial x}$.

Akiva Weinberger

Mar 1, 2017 01:04

Nor the floor

MATH ASKER

ya we don't have to paint floor or the ceiling

Ted Shifrin

Right. There was an if we were in there ...

BAYMAX

Just replaced $H$ by $g$ in the last equation?

@Semiclassical

Ted Shifrin

Anyhow, I'm out.

Akiva Weinberger

One wall is length*height. The next wall is width*height. The next wall is opposite the first one, so it has the same size, so it's length*height again. The last wall is opposite the second one, so it's width*height.

Simple

Mar 1, 2017 01:05

The definition from the book I have is: suppose $X'=AX,X'=BX$ have flow $\Phi^A$ and $\Phi^B$. These two systems are topologically conjugate if there exists a homeomorphism $h:R^2\to R^2$ that satisfies $\Phi^B(t,h(X_0))=h(\Phi^A(t,X_0))$. The homermorphism $h$ is called a conjugacy

Akiva Weinberger

Either draw a picture, or look at the walls of the room you're in…

Semiclassical

Sure. I'm writing it in generality, but in the above we indeed have $\frac{df}{dt}=\{f,H\}$.

BAYMAX

nice @Semiclassical

Simple

what is homeomorphism?

BAYMAX

Is there a way i can save selected messages so i can refer to them later or something like that ??

Akiva Weinberger

Mar 1, 2017 01:07

@Simple A continuous bijection (with a continuous inverse)

Simple

I know it from Wikipedia, not sure how it relate with ode

Akiva Weinberger

The continuous inverse part isn't necessary in this context, though, I don't think

MATH ASKER

why is it width * height, is width in this case the space betweeen the two lines of the wall?

Akiva Weinberger

@Simple Essentially you warp the plane in some way

Semiclassical

I think you can bookmark it if you look at the transcript instead.

Alternatively, you can search the transcript to track down the message if you remember some of it.

MATH ASKER

Mar 1, 2017 01:08

imgur.com/a/z2SST

Simple

it is a transformation?

Akiva Weinberger

@MATHASKER Look at the floor. The floor is a rectangle. One side (edge), the longer side, we'll the length. The other side we'll call the width. The distance between floor and ceiling is the height.

@Simple I suppose, but not necessarily a linear transformation

("Edge" as in, it's one of the twelve edges of the cube-shaped room you're in.

BAYMAX

Do i have to star them but by doing that it will fill the righ side area in the chatroom @Semiclassical

Semiclassical

Nah.

Akiva Weinberger

Well, rectangular prism.)

Simple

Mar 1, 2017 01:10

ok, thanks

Semiclassical

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