So, internal force can cause a change in a KE that must be accounted for which means that internal forces must be included.

I am unreasonably pleased with myself for that exposition and will write it up in detail at some point.

@dmckee Beautiful examples.

Goes over my head. Oh well.

@dmckee The work-energy theorem doesn't already explicitly state that rotational kinetic energy counts?

@0celo7 If you are doing springs now you should have had rotation already or will get there soon.

@dmckee I took AP physics C

@HDE226868 It was alleged by other users that it did not. Goldstein says it does, but I was worried there might be two ways to construct the thing.

So this is all review

wonder if work energy theorem is valuid on a general manifold

hmm

Arnold!

@dmckee Interesting.

Now I believe that I've shown there is only one correct way.

Uhm, so, the phrase "work-energy theorem" does not exist in German, and I spent the last minutes trying to figure out what it is, but I fail to see how it is not just conservation of energy.

energy is conserved in Lagrangian mechanics...somehow

eh, to busy to remember

@0celo7 ?

@ACuriousMind I'm wondering what the work energy theorem in the Lagrangian scheme is

@0celo7 AFAICS, it's conservation of energy

And that's Noether for time translation

right

idk, mechanics is too hard

@dmckee congrats if your students understand what you're talking about

I don't

@0celo7 I think the upper division students would get it. Most of the intro-year ones ... probably not.

I'm still confused by what the problem is

@dmckee Of course...

@dmckee: Can you explain to me the difference between the "work-energy theorem" and conservation of energy?

work energy: $W=\Delta K=-\Delta U$ (usually the first)

Or is it just the name for conservation in Newtonian mechanics?

conservation is the second

@dmckee Right

Ok I see now.

There is a close relationship between the W-E theorem and the conserv. of energy but they are not them unless all forces are conservative.

bah, nonconservative forces

I like my cows smooth, spherical and in a vacuum

The work-energy theorem relates the net work done to an object or system (including that from all forces) to the change in it's kinetic energy.

$W_{net} = \Delta T_{system}$

what does friction in the Lagrangian scheme look like?

@dmckee Well, but if forces are not conservative, then energy is not conserved

To get the conservation of energy you start pealing the net work apart into contribtution from various forces, converting the conservative force into potentials and moving them to the energy side.

Why does the UV index break UV levels into a discrete set of -low-, -mod-, -high-, -very high-, -extreme-? Is there continuous data this is broken up from, or are multiple pieces of data weighted and combined(and scaled) in some way to obtain this?

@ACuriousMind Yep, but the W-E theorem works fine with non-conservative forces.

And it is exactly not conserved by the formula for the work of the non-conservative forces

Ah

@ACuriousMind Yep, again. That is after you've moved all the conservative force to the energy side in the form of potential energies.

So taking gravity as an example

$W_{net} = W_\text{everything but gravity} + W_\text{gravity}$

$W_{net} = W_\text{everything but gravity} + -mg\Delta h$

Then pull the gravitational potential energy term across

The W-E theorem is the more general statement of what the charge for time translation is, even if it is not conserved.

$W_\text{everything but gravity} = \Delta T + \Delta U$

continue with other conservative forces until you have

::shakes yellow book:: damn you Arnold for not covering nonconservative forces

@ACuriousMind The Noether charge?

@0celo7: Non-conservative forces do not fit into the Lagrangian formalism easily

$W_\text{non-conservative forces} = \Delta T + \sum_{\text{force } i} \Delta U_i = \Delta E$

If no non-conservative forces are present this reduces to

$\Delta E = 0$

@ACuriousMind Arnold completely avoids them

So I have no good grasp of what they really mean

@dmckee Yes, I think I see now why this has its own name

@0celo7 Friction is the quintesential example

yes, of course

hence why I asked about it above

@0celo7 Yeah, because they don't make for a nice mathematical formalism.

example?

@ACuriousMind I've been told a couple of times there is a way to manage friction, but no one has shown it to me.

is it possible to insert friction terms in Lagrangian mechanics at all?

I speculate that you simply establish an extra energy term for heat, but I don't know where you go from there.

journals.aps.org/pre/abstract/10.1103/PhysRevE.53.1890

>fractional derivatives

lol

no wonder he doesn't bother

@0celo7 Not really. Search for papers on it, you'll see that you either have to modify the Euler-Lagrange equation or take some weird actions.

Ah. Excellent. I wonder if I will understand it?

dang, being a research university is nice

I can access random papers :D

@0celo7 I tunnel my web browsing through FNAL for that. Secure shell is capable of wonders too numerous to count.

@dmckee Huh?

I'm at a rinkydink little place that doesn't subscribe to any real journals to speak of. But Fermilab does. So I make a tunnel over a secure shell connection to the lab and the journal thinks I'm coming from there and gives me what I want.

Only now I need to remember my fermi kerberos password...

If anyone cared about my question above, I found the answer: Shorter wavelengths do more damage to skin, so there is a weighted model that is integrated over (weighting shorter wavelengths higher starting at roughly 295nm and ending at roughly 380nm) and then they just divide by $25\,\, mW/m^2$, and then the values are discretized

@dmckee :(

Springer access is soooo nice

In my last year here I'm gonna order like 20 MyCopy books just in case I want to read them :D

2-4x cheaper than Amazon

@dmckee You mind explaining what the issue with the work energy theorem and internal stuff is?

@0celo7 Someone told me I was wrong about something I thought I understood, and that was distressing so I spent a long time thinking about it.

well what is the thing you supposedly didn't understand

And I learned that I had understood it, but not very deeply. In the course of my thinking I advanced another level into it.

@0celo7 What part the work-energy theorem plays in basic Newtonian mechanics.

The thing at question was whether you needed to account for forces between parts of the system in computing the net work or not.

is it not just $\Delta T+\Delta U=-\Delta \text{other}$?

and the work is an expression for $\Delta T$

The plain work energy theorem is $W_{net} = \Delta T$.

But the interpretation of "net" was at question.

see my second sentence

Did it mean only external forces or did it include internal ones.

ah

all

I gave a couple of examples of systems where internal work changes the total kinetic energy, but another user wanted interpret the $\Delta T$ as "the kinetic energy of the center of mass" (thus exclusing rotational KE, for instance).

Arnold makes it fairly clear

Although tbh he's a mathematician writing a book on geometry, so while the book is good, I had to hunt for the result

He certainly doesn't advertise it

I looked it up in Goldstein and he includes the internal KE (rotational and so on), but I couldn't rule out ther being two way to construct the understanding at first.

well Arnold defines force as external + interactions

and then shows sum of works = integrals of force = integrals of (external + interactions)

@dmckee I recommend Arnold, it's a fun little book

If you don't mind "conservation of energy" being called a "procedure for finding a first integral of the Euler-Lagrange equations" :D

@0celo7 LOL. I may have to look at it. I feel like I sometimes miss the big, structural mathematics because I'm staring at the individual minus signs fretting.

@dmckee the second exercise justifies the mathematical setting (n-tuple of reals) of mechanics very elegantly

"All Galilean spaces are isomorphic"

I remember that one

And since $\mathbb{R}^{3N}\times\mathbb{R}$ is Galilean, we can specialize without sacrificing generality!

And I haven't read the book :P

@ACuriousMind you also know random HE theorems that I couldn't figure out, etc.

amazon.com/…

I think this is the most complicated classical mechanics book.

Unless @ACuriousMind knows some category theory book?

I'm sure classical mechanics is sheaf cohomology somehow.

I know no books

lecture notes, papers, whatever

amazon.com/An-Invitation-Morse-Theory-Universitext/dp/…

How nice, an invitation!

@ACuriousMind Does "Erlangen program" mean anything to you?

Something about groups and geometry, but no specifics

reading list ++

sigh

This is just getting stupid :(

@ACuriousMind Book that introduces manifolds and geometry using the language of category theory

@0celo7 Why would you read that?

You should read Milnor's book, obviously :P

@0celo7 Here, this is the text of Klein's lecture. Basically, Klein was the first to (publicly) announce that geometry is essentially the study of properties invariant under a certain group of transformations.

This view unifies all kinds of geometry known at that point and perhaps more importantly yields a hierarchy among different types of geometry, based on viewing certain transformation groups as subgroups of other, larger ones. Consequently, one can say that the branch of geometry associated with the subgroup is less fundamental than the other.

hey

Hello

@0celo7 you should be more concerned with Langlands program

ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/…

So I think basically a criterion for CTCs in Pauli Fierz would be the mass density going imaginary, but then again that would not be enough

Basically any spacetime with the light cone wider than Minkowski or not in the same direction would have that

I think it may be interesting to see what the stress energy tensor of the $h$ field is like!

Wait a moment

Hm

I wonder if you can twist it so much that it might pass in the past light cone

I don't think so though

Although I guess that if it can go from timelike to spacelike, there might not be anything preventing it from going to past timelike

Hey @Qmechanic

@Slereah : Hey

Do you happen to know what this fellow meant :

Is there a theorem about the Cauchy problem or causality or whatever linked to the principal part of a linear PDE

@Slereah : That's more or less the same question as here and here. I'm sure people have developed a general theory for this, although I have not looked for any references.

I know, I asked one of these question :p

And was somewhat responsible for the other

I asked Moretti but he did not seem to notice!

@Danu I'm not...just thought the title was interesting.

I'm looking into PDE theorems but damn it's not easy

Why does this user try to send us to area51 or superuser?

@Slereah Give me a few years, I'll help you then

TO THE TIME MACHINE

@ACuriousMind To make you think. Work out the gray matter.

@Slereah You like saying that, don't you?

Who's that?

@0celo7 Really?

@0celo7 : You young people

No culture at all!

He has two watches

@ACuriousMind What kind of question is that? I don't know who that is.

That's shocking

That is Dr. Emmet Lathorp Brown

Who?

@ACuriousMind He can't be too important if you're not telling me who he is.

He is the real hero of Back to the Future

You damn kids with your twerking and your twitter and your ebola

@Slereah Never seen those movies.

Go see them right now

@Qmechanic I hope you don't mind me asking you a question about one of your answers! In the following post, how do you arrive at equation (7)? Working from the line previous I arrive at the result in the attached picture, seemingly the Poisson brackets cancel out leaving only the bare charge. physics.stackexchange.com/questions/69271/…

If I reverse one of the Poisson brackets I generate the time derivative of Q as desired but still have the other one!

@AngusTheMan : Which equality in eq. (7) are you talking about?

@Qmechanic the final result

@AngusTheMan You used Hamiltons equations of motion in your version. The calculation in Qmechanic's post is supposed to be off-shell.

@ACuriousMind : Agree.

@ACuriousMind Ah thats very true !

@Slereah No?

@ACuriousMind I just realized: Arnold never makes clear that the Hamiltonian generates time translation. I don't think he ever writes something like $\dot f=\{f,H\}$ :o

@0celo7 If he says somewhere that time evolution is the flow associated to the vector field associated to $H$, that is the same.

@ACuriousMind Well, he might...how are they the same?

@0celo7 You have to think about what $\dot{f}$ means. $f$ is a function of $q$ and $p$, so what is $\dot{f}$ supposed to be?

@ACuriousMind Chain rule?

Yes, but how? I.e. how are $q(t),p(t)$ defined?

They're defined as the action of the Hamiltonian flow on the initial data

Exactly. Thus, $\dot{f} = \{f,H\}$ follows from saying "time evolution is the flow of the Hamiltonian vector field".

wat

What, using the chain rule?

Yes, and then the Hamiltonian equations of motion

Well, no sh*t

I knew that

But is there some abstract derivation using the Poincare form?

That's what I'm asking

What is the Poincare form?

derivative of the soldering form

/tautological one form

You mean the symplectic 2-form with "Poincare form"?

si

@0celo7 Well, $\{f,g\} = \omega(X_f,X_g)$ by definition. Since $\omega(X_H,X_f) = \mathrm{d}H(f) = \mathcal{L}_H f$, this is $\dot{f}$, since the Lie derivative of a function can be evaluated by evolving the function along the flow and then taking the derivative.

Ugh symplectic form :p

Sorry, should be $\mathrm{d}H(X_f)$ up there.

And I also mixed the order up, should be $\omega(X_f,X_H) = \mathrm{d}f(X_H) = \mathcal{L}_{X_H} f$.

Still, the argument goes through abstractly, although this is just a very convoluted way to state the "chain rule + Hamilton's e.o.m." from earlier.

@ACuriousMind Thanks

you've infected me, this nice coordinate free stuff is much nicer than "chain rule + eom"

@ACuriousMind did you see the horrible notation Arnold uses for Hamiltonian vectors? $I\mathrm{d}f:=X_f$

ugh

@Danu hey

o/

\o

did you just come in or is chat doing something strange again

it does strange things

So anyway

now that you know all about hamiltonian mechanics

Do you know what are the conditions for the ability to define it on a spacetime manifold :p

I still don't know the answer!

...why wouldn't you be able to define it?

well, now that I think about it

hmm

you need to separate out time somehow

Well yes, but there is still

THE THING

Carlip's book does the 2+1D Schwarzschild spacetime using the ADM formalism

But that spacetime has CTCs

I thought you needed to have Cauchy surfaces to have Hamiltonian mechanics

CTCs a bunch of bullshiteeet

Is he abusing the formalism or can you still do it

Well yes, because movement is in space and not in spacetime

@Slereah I dunno

And the gravitational field is inhomogeneous speed of light

I learned classical hamiltonian mechanics

and dirac belt etc

@Slereah You need a (nice) foliation of the spacetime into space-like surfaces.

so you just "pick" a Hamiltonian vector?

and that gives you the time direction

Can you have a foliation in the case of the 2+1D black hole, though

or you looks at the flow of geodesics?

or what

(the vacuum one, not BTZ)

It has CTCs up the ass

Since it has a weird spacetime identification

>black hole

>up the ass

I see what you did there

Like what is the exact causality condition for the hamiltonian to be well defined

@Slereah Existence of the spacelike foliation. It's equivalent to global hyperbolicity, see MO

Hm

so, if you've got CTCs, using a Hamiltonian formalism is a big no-no from a rigorous standpoint

Then was Carlip abusing notation

mb I should send him an email

IF HE ANSWERS IT

(I am refering to you, Matt Visser!)

(Answer my mail!)

@Slereah He must have been. You need the foliation to define the $g_{ij}(t,\vec x)$ that are the dynamical variables of the Hamiltonian formulation.

Damn physicists!

Always abusing formalism!

What happens if I just pick some random timelike vector field for that, though?

So

@Slereah What do you mean?

No CTCs $\implies$ globally hyperbolic $\implies$ $M\cong \mathbb{R}\times\Sigma$

Well for instance take Minkowski space's definition of the hamiltonian

So we just do Hamiltonian mechanics on $\mathbb{R}\times T^*\Sigma$

And then compare it to the $T^4$ flat spacetime

?

@0celo7 : Not exactly

You can have a non-globally hyperbolic spacetime without CTCs

damn, that's right

the arrow is the other way around!

Like $(\mathbb{R}^3 - \{0\}) \times \mathbb{R}$

yes yes

YES

oh for the sake of duck my TeX'd homework won't print

"pdf corrupted"

pls it works just fine

@Slereah: I don't think an everywhere timelike vector field exists if there are CTCs.

@ACuriousMind huh?

An everywhere timelike vector field always exists for any spacetime!

never heard that one before afaik

That's like the one criterion for a Lorentzian manifold

Ah, yes. Silly me

exactly, that's how you get a Lorentz metric

What do you want to do with it, though?

Well what prevents toroidal flat spacetime from having a well defined hamiltonian, for instance?

It has the same timelike vector field as Minkowski space

Up to some identification

the Hamiltonian has nothing to do with the everywhere timelike vector field, I think

hamiltonian generates time translation

You have to be able to define the canonical variables $\gamma_{ij}(t,\vec x)$

so the hamiltonian vector field has to be everywhere timelike

Well you can define it in such a spacetime, too

@0celo7: It generates the time translation on the phase space. This makes no statement about any vector field living on the configuration space/spacetime

I mean you can define all the achronal spacelike hypersurfaces you want

It just won't be a Cauchy surface

Or even a partial one

@Slereah Ah, well, no one demanded they be Cauchy surfaces. You need a smooth spacelike foliation, not Cauchy surfaces.

@ACuriousMind true

Well plenty of CTC spacetimes have a foliation, though!

since when did ACM become a GR expert o.o

QFT curved spacetime?

Maybe the theorem is just one way

Maybe a globally hyperbolic spacetime guarantees the hamiltonian

But not the other way around

Hm

@Slereah Yeah, the thing from MO I quoted was for spacelike Cauchy surfaces. The Hamiltonian formulation makes no statement about the leaves of the foliation being Cauchy.

So you just need spacelike foliations, which exist for more than just the globally hyperbolic ones

Argh

Hm

Why are you so obsessed with CTCs, though?

Are there any spacetimes that have no foliation I wonder?

why aren't you?

It's an interesting topic!

Maybe that is a nice SE question!

Examples of spacetimes with no foliation

ask PO or MO

Wait a moment, though

I'm not sure the Hamiltonian formulation evolves stuff properly if the foliation is not Cauchy

Possibly

does Wald not talk about this in his last chapter

It might induce some discontinuous data if the initial data doesn't line up propertly

It could be that if you feed the initial conditions into it, it evolves them into the globally hyperbolic manifold associated to the leaf you gave the conditions on, and doesn't evolve them on the actual manifold you foliated.

Hm true

They need to be Cauchy

cf. Wald

I recall a paper about spacetimes with CTCs having the possibility to evolve into several different spacetimes from an initial slice

what part of Wald

the appendix on Hamiltonian mechanics -.-

He doesn't seem to explain the choice, though

well

yeah, he doesn't

Skimming, I don't find any place where the property of the surfaces being Cauchy is used

I mean I can see why there might be a problem

Since CTC developments for a surface doesn't necessarily exist or is unique

But that depends on both the metric and the field

Plenty of CTCs are benign for some fields

Hm wait, is it

Uniqueness is okay in some cases but I'm not so sure about existence

Except for some very contrived examples

"Einstein, Evidence, Causality, CTCs, PDEs and Shit"

There's a great book title

Did Einstein ever know about CTCs, I wonder

The first CTC paper was before his death, but IIRC Godel's paper doesn't draw attention to them

Let's see

The earliest paper to really discuss CTCs that I can remember is Tipler, but that was in the 70's

Let's see the biblio

"Global Structure of the Kerr Family of Gravitational Fields" is a bit earlier, 68

Oh great

That paper references an unpublished paper as an earlier source

You are a monster Carter

Carter?

The author is B. Carter

Lil Wayne?

Or Jay-Z?