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What is Symplectic Geometry? (2016) [pdf]

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Re: What is Symplectic Geometry? (2016) [pdf]

#11

Post by ajkjk »

turnersr wrote: post #9
ajkjk wrote: post #4

This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…

What does "paired-off coordinates, like (x,v) in mechanics" mean? What is x and v here? Thanks!

Those are position and velocity. In analytical mechanics they are treated as being on 'equal footing', in the sense that a system's initial conditions are (usually) its position and velocity, and time evolution is governed by the equations dx = v and m dv = F. (more generally, https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equatio...)

Re: What is Symplectic Geometry? (2016) [pdf]

#12

Post by throwlaplace »

turnersr wrote: post #9
ajkjk wrote: post #4

This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…

What does "paired-off coordinates, like (x,v) in mechanics" mean? What is x and v here? Thanks!

There's no inherent pairing. The example op is alluding is number 4 in the bulleted list where the coordinates (p,q) are in a completely different space (called either phase space by physicists or the cotangent bundle by mathematicians).

Re: What is Symplectic Geometry? (2016) [pdf]

#13

Post by ajkjk »

throwlaplace wrote: post #10
ajkjk wrote: post #4

This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…

Symplectic geometry is a collection of facts having to do with symplectic manifolds. Just like euclidean geometry is a collection of facts having to do euclidean manifolds. That's the way in which terms like "____ geometry" are defined and understood.

I understand that well, but I don't like it. Math for its own sake with no collection of reality alienates a huge category of people who might be interested in it if it was better explained.

Re: What is Symplectic Geometry? (2016) [pdf]

#14

Post by danharaj »

ajkjk wrote: post #8
danharaj wrote: post #7

Earlier quoted context omitted.

> (In classic mechanics, particularly, there is an unusual symmetry to position and velocity, such that the laws of mechanics look roughly a rotation x -> v, v -> -x, which is why this works so well.) In fact, every symplectic manifold locally looks like this (Darboux's theorem).

Yeah, but I'm interested in understanding things the other direction: if there is not some major class of problems that are best described by symplectic manifolds, then why care about Darboux's theorem at all? If there are, why isn't that front-and-center? This article conspicuously avoids motivating symplectic geometry _at all_, which is so frustrating. It mentions connections to subjects, but it doesn't mention why…

Is "any hamiltonian dynamics on any phase space of a physical system" not a major enough class of problems for you? This is one way mathematicians study physics. Are you asking for applications of symplectic geometry? If so, here, I've been reading a cluster of papers in this area for a few weeks now:

https://www.math.colostate.edu/~clayton/research/papers/fram...

Re: What is Symplectic Geometry? (2016) [pdf]

#15

Post by Koshkin »

ajkjk wrote: post #13
throwlaplace wrote: post #10

Earlier quoted context omitted.

Symplectic geometry is a collection of facts having to do with symplectic manifolds. Just like euclidean geometry is a collection of facts having to do euclidean manifolds. That's the way in which terms like "____ geometry" are defined and understood.

I understand that well, but I don't like it. Math for its own sake with no collection of reality alienates a huge category of people who might be interested in it if it was better explained.

> a huge category

And I thought category theory was "math for its own sake"!

Re: What is Symplectic Geometry? (2016) [pdf]

#16

Post by adyavanapalli »

ajkjk wrote: post #13
throwlaplace wrote: post #10

Earlier quoted context omitted.

Symplectic geometry is a collection of facts having to do with symplectic manifolds. Just like euclidean geometry is a collection of facts having to do euclidean manifolds. That's the way in which terms like "____ geometry" are defined and understood.

I understand that well, but I don't like it. Math for its own sake with no collection of reality alienates a huge category of people who might be interested in it if it was better explained.

The intended audience of AMS's Graduate Student Section is mathematics graduate students. Nevertheless, it's hard to say whether symplectic manifolds has any easily explained analogy for lay audiences. The subject originally arose from more abstract formulations of classical mechanics.

Re: What is Symplectic Geometry? (2016) [pdf]

#18

Post by Ceezy »

ajkjk wrote: post #4

This really doesn't make it clear what symplectic geometry... is, or why I should care about it. I have eventually figured out an answer that was satisfactory to me, after much frustration: it is math on a manifold that has a concept of paired-off coordinates, like (x,v) in mechanics. Typically this is interesting because it is an alternate characterization of the mathematics of a space where the relevant quantities…

True, the main reason is to understand differential equations. The first vector is supposed to be the speed and the second the acceleration. And the area represented by the dqdp is invariant through time in a conservative system.

Re: What is Symplectic Geometry? (2016) [pdf]

#19

Post by Ragib_Zaman »

Quite interestingly, Symplectic Geometry is currently under review/investigation for some of the foundational papers in the field having serious gaps and outright errors after closer inspection. These concerns were always spoken of in hush hush tones and only in recent times have people stated their concerns publically. Some of the original authors refuse to retract their papers despite being assured their academic positions (which realistically, came through the reputation built up by these papers) are secure. Here's a quanta article about this fiasco: https://www.quantamagazine.org/the-fight-to-fix-symplectic-g...

Re: What is Symplectic Geometry? (2016) [pdf]

#20

Post by messe »

Ragib_Zaman wrote: post #19

Quite interestingly, Symplectic Geometry is currently under review/investigation for some of the foundational papers in the field having serious gaps and outright errors after closer inspection. These concerns were always spoken of in hush hush tones and only in recent times have people stated their concerns publically. Some of the original authors refuse to retract their papers despite being assured their academic p…

That said, it's worth noting that the parts that are of interest to someone new to Symplectic Geometry, tend to be those related to Hamiltonian Mechanics or similar, and are likely not under review. Most of Symplectic Geometry is "probably" (I'm using quotes and italics to hedge my bets) "fine"-ish.
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