What is Symplectic Geometry? (2016) [pdf]

https://www.ams.org/journals/notices/201611/rnoti-p1252.pdf

by agronaut • 7 years ago

67 29 7 years ago

pjbk2• 7 years ago

For those into physics, I wholeheartedly recommend Marsden and Ratiu's book, "Introduction to Mechanics and Symmetry", which deals mainly with the different formulations of physics applied to symplectic and associated geometries.

tobmlt• 7 years ago

Thanks for the recommendation! I didn't know this one. I try and lap up everything I can by Marsden, (though more often through the lens of applied researchers: e.g. Desbrun, Hirani, Crane, and others -- much involving computer graphics and/or discrete differential geometry applied to physical simulation. In short, I better say that I am not familiar with the scope of Marsden's work. I am sure much of it is beyond me, but gosh darned it, the exterior calculus is beautiful and these guys write brilliantly readable stuff for an engineer.

Even as a hydrodynamics software guy, I found the computer graphics research community to be the easiest entry-point for, especially, the topology and modern differential geometry. It's especially nice when they do a simulation paper with a high end geometric/analytic approach.

This might be a good place to go in order to have a start at, say, Arnold's ``topological methods in hydrodynamics'' or anything TQFT-esque.

madrafi• 7 years ago

This was quite a treat !

ajkjk5• 7 years ago

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 are a variable and its derivative.

(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.)

pjbk2• 7 years ago

Correct, and it also appears in Electromagnetism and Quantum Physics (in several ways), and in Lie theory that is useful for rotations and differential applications.

danharaj3• 7 years ago

(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).

ajkjk5• 7 years ago

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 symplectic geometry is _necessary_, rather than sufficient, for these connections.

kkylin2• 7 years ago

Don't disagree with the broader point you're making, but nitpick: every symplectic manifold looks locally like R^{2n} with the canonical symplectic structure. Not necessarily so globally.

turnersr• 7 years ago

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

Thanks!

ajkjk5• 7 years ago

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...)

throwlaplace2• 7 years ago

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).

kkylin2• 7 years ago

Generalized coordinates and their "conjugate" momenta. These are the basic dynamical variables used to describe mechanical systems in Hamiltonian mechanics:

https://en.wikipedia.org/wiki/Hamiltonian_mechanics

For certain systems (e.g., particles in euclidean space interacting via conservative forces dependent only on position), the momenta coincide with velocity, but mathematically they are different objects: velocities are vectors, and momenta are covectors. They transform differently under coordinate transformations.

I wish I knew a concise self-contained exposition off the top of my head, but I don't. You can probably find something on-line. I think there's likely a discussion in Structure and Interpretation of Classical Mechanics (Sussman & Wisdom):

https://mitpress.mit.edu/sites/default/files/titles/content/...

Try looking up "cotangent bundles" (mathematical name for the type of spaces appropriate for Hamiltonian mechanics, formulated in terms of generlized coordinates and momenta) and "tangent bundles" (more appropriate for Lagrangian mechanics, formulated in terms of generalized coordinates and velocities).

throwlaplace2• 7 years ago

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.

ajkjk5• 7 years ago

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.

Ceezy• 7 years ago

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.

evanb• 7 years ago

Importantly, the Hamiltonian formulation of classical mechanics has symplectic form, with the conjugate variables (position, momentum) making up the dimensions.

killjoywashere• 7 years ago

@agronaut: most interesting to me: why did you post this? What were you working on that led you to this?

Ragib_Zaman• 7 years ago

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...

messe• 7 years ago

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.

sidek• 7 years ago

As a result, lots of recent work is being done in the algebraic setting (rather than analytic), where the foundations are on much firmer footing.

Being algebraic symplectic is a much stronger condition than analytic symplectic, but is still interesting enough (and, for geometry related to linear algebra problems, as is often relevant in CS, is not a very strong restriction at all.)

tomrod2• 7 years ago

This seems to have crossover with topological data analysis.

akimball• 7 years ago

In particular, for geometrizing semantics. Montague grammar is a tarpit, and pragmatic utility of inference on distributed representations has been abundantly demonstrated in the past decade. Symplectic structure is one of a small class of structures which capture and relate essential features of natural semantics in a metric (read, tractable) representation. This offers a tantalizing prospect for bridging the gap between computation and cognition.

tomrod2• 7 years ago

Hear here!

What is Symplectic Geometry? (2016) [pdf]

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