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.)
> (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 symplectic geometry is _necessary_, rather than sufficient, for these connections.
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:
No, I'm plenty aware of the applications. My complaint is that this article hardly mentioned them! What's the point of an article explaining a theory without motivating why it exist?
Yes, too many mathematicians write about areas of mathematics that could be useful to others, without giving any motivation, or any hand-waving explanations, or the critical distinctions that make it a useful approach.
There is a barrier of communication that (most) mathematicians don't want to take any time to overcome. Often, the opening is enticing, then it's straight into lemmas, formal language, and citing of famous results by name. Their slight inclination to explain it to others dissipates in the first paragraph, then it's off to the races to impress their peers.
This is easy to see on Wikipedia - most mathematics articles are utterly useless to any non-mathematician who wants to get a general appreciation of an approach to see if it could shed some light on their problem.
In almost all cases, it is better to read domain generalists coming the other way into the higher mathematics.
The blunt reason is that this article was written for mathematics grad students who already know that symplectic manifolds are a hot topic via diffusion.
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.
Generalized coordinates and their "conjugate" momenta. These are the basic dynamical variables used to describe mechanical systems in 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):
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).
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...)
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).
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.
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.
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.
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...
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.)
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.
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.
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.
Importantly, the Hamiltonian formulation of classical mechanics has symplectic form, with the conjugate variables (position, momentum) making up the dimensions.
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.
(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.)