
What is Symplectic Geometry? (2016) [pdf] - agronaut
https://www.ams.org/journals/notices/201611/rnoti-p1252.pdf
======
ajkjk
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.)

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

Thanks!

~~~
kkylin
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](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/...](https://mitpress.mit.edu/sites/default/files/titles/content/sicm_edition_2/book.html)

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).

------
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...](https://www.quantamagazine.org/the-fight-to-fix-symplectic-
geometry-20170209)

~~~
sidek
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.)

------
pjbk
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
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.

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

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

------
madrafi
This was quite a treat !

------
tomrod
This seems to have crossover with topological data analysis.

~~~
akimball
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.

~~~
tomrod
Hear here!

