The h Bar

fqq

00:58

@Slereah yeah it sort of looks like the spiders/ZX stuff

Evil John Rennie

08:13

It is really happy to see old people come here to help the young ones. Don't take old people for granted. -Not Rennie but Connor

Remember my quotes I am going to get Nobell Prize in physique.

Physics Meta

12:24

0

Q: Undo suggested edits

If I suggest an edit for another's post and immediately realize that I've missed something, I can re-edit it. But can I undo suggested edits (bring the post to the original or previous state) before it is approved?

Ratman

15:35

glS

quick question/opinion fishing: does anyone have a clear idea of if/why/how symplectic geometry is connected to second-order ODEs? The question comes from thinking about Hamiltonian mechanics (and its relation to second order ODEs coming from Newton's eqs) vs symplectic geom.

Ratman

here Weinberg is trying to tell that if I can extend the representations of the connected lorentz group to the non connected parts, than my QFT will conserve P and T, given the fact that is Lorentz invariant. Is this correct?

ACuriousMind

15:53

@glS I'm not sure what you mean by geometry being "connected" to ODEs

e.g. the geodesic equation is a second-order DE - does that mean (pseudo-)Riemannian geometry is "connected" to second-order DEs?

@Ratman yes

glS

@ACuriousMind what I was thinking is, from Newton's eqs (second order DEs), linearising (defining $p=\dot x$) you get a (specific) Hamiltonian system, with $H=p^2/2+V$ and $F=-\partial V$. You then know the form of the equations is preserved by symplectomorphisms/canonical transformations. Which makes me wondering, shouldn't that mean there is some specific class of transformations on the trajectories themselves which corresponds to canonical transformations and preserves Newton's eqs

I mean, obviously there is, but the question is whether there is a "nice" way to characterise such transformations directly on the configuration space rather than on the phase space

ACuriousMind

16:09

@glS What do you mean? Canonical transformations in general do not restrict to transformations on configuration space, since they can "mix" positions and momenta

however, any transformation on configuration space can be extended to a canonical transformation on phase space just by choosing the momentum part of the transformation to be the one that sends the old momentum to the "new momentum" associated to the new positions via Legrendre transform

glS

@ACuriousMind that's exactly my point. You cannot simply do the restriction. However, a canonical transformation does correspond to a transformation... I suppose one should say a transformation of the trajectories? So if I didn't know Hamiltonian mechanics and I wanted to describe the types of coordinates changes corresponding to canonical transformations (or more precisely, "changes of trajectories"?), how would I go about it

ACuriousMind

why would you want to describe canonical transformations if you don't know Hamiltonian mechanics?

their characteristic property is preserving the form of Hamilton's equations, after all

canonical transformations are very explicitly transformations on phase space, not transformations on trajectories or on configuration space

glS

@ACuriousMind true, but if they preserve Hamilton's equations, and those come from second order DEs (which I'm not sure is necessarily the case, but let's suppose it is for the moment), then those transformations also preserve the form of the underlying second order DEs, right? So they'd still be interesting to understand structure/symmetries associated to the dynamics one seeks to describe

ACuriousMind

I don't think that's the case

the second order DE has only N independent variables $Q$, while Hamilton's first-order equations have 2N independent variables $Q,P$ - to me it is obvious that in the latter case you can do more transformations than in the former

in the case where you have just $Q$, it makes no sense to transform $Q$ and $\dot{Q}$ separately

and for $q,p$ you get more transformations because you "bought" additional freedom by "decoupling" $q$ and $p$ from each other

glS

that's what I was trying to get at talking about "changes of coordinates for trajectories". I'm being imprecise because I'm not sure how to formalise this. On the original space you cannot simply have a coordinate transformation $q\mapsto f(q)$, as you point out (I mean, not all phase space transformations correspond to this). It must be something somehow encoding the information in $\dot q\mapsto (...)$.

But then again, if you consider the equations on-shell, the momenta are directly related to the coordinates via $p=\dot q$, at least in simple cases, right? After doing a canonical transf…

ACuriousMind

16:29

canonical transformations are not about on-shell behaviour

and the point of a general canonical transformation being more general than a coordinate transformation is precisely that after the transformation, you may no longer have $p\propto \dot{q}$, even if you started that way

glS

that's a good point I hadn't thought of. Let me try making things a bit more precise. Consider a II order DE, $\ddot q=F(q)$. Let the corresponding Hamiltonian eq be $\dot x=J\partial_x H$, with $x=(q,p)$ and $J$ standard symplectic matrix. Let $\xi=f(x)$ be a canonical transformation, thus s.t. $(\partial f)J(\partial f)^T=J$. The transformed Hamilton eqs are $\dot\xi=J\partial_\xi K$ for some new Hamiltonian $K$.

I can still unravel the equations in the transformed coordinates and get a II order DE, $\ddot\xi=\tilde F(\xi)$, right?

ACuriousMind

what does "unravel" mean?

glS

@ACuriousMind I mean "delinearise". From $\ddot q=F(q)$ I changed variables as $p=\dot q$ and got first order equations on $(q,p)$. One can then also revert the process to go back to the second order DE from the system of first order DEs... unless you can't do it anymore after the tranformation?

But you still have equations of the form $\dot\xi=A(\xi,p_\xi), \dot p_\xi=B(\xi,p_\xi)$... and some $p_\xi=C(\xi,\dot \xi)$ on-shell I think? If the last one was $p_\xi=\dot \xi$ it could be used to de-llinearise, getting back a DE on only $\xi$ and its derivatives. Not sure about the more general situation one would get after transformations.

substituting you should get, on-shell, $\partial_t C(\xi,\dot\xi)=B(\xi,C(\xi,\dot\xi))$, which is a second order DE on $\xi$, which is what I meant with "unravel". So the structure of canonical transformations should somehow be reflected in the form of this equation of motion you get. And because we aren't changing the physics, only our description of it, the new equations of motions should give the same solutions, or at least give solutions that can be remapped to the original ones.

@ACuriousMind now, you say "canonical transformations are not about on-shell behaviour". This might be relevant. I'm not sure how to understand this statement. At the end of the day, all we are doing is looking for "better" ways to solve the equations of motion, which are about on-shell behaviour, no?

ACuriousMind

16:56

@glS what you can do is Legendre transform back to a description in terms of $q,\dot{q}$ to get a Lagrangian, and the E-L equations of that Lagrangian should be equivalent 2nd order DEs

@glS perhaps, but conceptually what we are transforming are not solutions to the equations of motion

we are transforming the phase space

a point in phase space $(q,p)$ is just that, a point, and a canonical transformation just moves it around - $(q,p)\mapsto (Q(q,p), P(q,p))$

glS

@ACuriousMind sure, but those transformations are, at the end of the day, about understanding something about the solutions of the EOMs. Anything you do in the phase space should correspond to some change in the DEs that are the equations of motion.

@ACuriousMind yes, I guess passing through the Lagrangian is a way to do this. Though from this, it doesn't look like you need that machinery, the get the corresponding equation of motions by simple substitution (though the Lagrangian formalism might give a more elegant description, idk)

ACuriousMind

@glS sure - by definition of canonical transformations, a solution $(q(t),p(t))$ of Hamilton's equations is mapped to a solution $(Q(t), P(t))$ of Kamilton's equations

glS

17:11

so let me rephrase again. Phase space and its canonical transformations should correspond to transformations on the second-order DEs that are the equations of motions. There should be a way to reformulate the structure that transpires from canonical transformations in phase space, into transformations of the form of the original equations of motion.

Well, I've got to go now. Thanks for the discussion. I'll think about it some more and see how it goes. Working out some simple example will probably clarify it for me.

Transcript for