SYMPLECTIC GEOMETRY? – Over the next few days, I'll be developing an introduction to here. Don't be afraid, we will be taking the easy route...

is the geometry of !

1/

Symplectic geometry is the modern destillate of an idea going back to Leonard Euler and Joseph Louis Lagrange.

Wikipedia:
> The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.

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> Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.

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Read about other alternate formulations of classical mechanics here:

mastodon.cloud/@RefurioAnachro

Little symplectic timeline:

Newton's "Principia", 1687.
The Euler-Lagrange equation, 1750's.
Lagrangian mechanics, 1788.
Hamiltonian mechanics, 1833.
Poincare defines what will later be known as symplectic geometry, 1912
Nöther's theorem, 1915.
Weyl coins the term "Symplectic geometry", 1939.
Arnol'd invents symplectomorphisms and symplectic topology, 1965,
Arnol'd's conjecture, 1974.
Gromov non-squeezing theorem 1985.

4/

Symplectic geometry is tailored for doing physics and works well for:

+ geometrical optics
+ classical mechanics
+ relativity
+ quantum mechanics

...and many other areas! On the other hand, examples for areas where symplectic geometry doesn't 'just work' include:

- statistical mechanics and thermodynamics
- dissipative systems
- noisy or lossy systems

Tobias Osborne – Symplectic geometry & classical mechanics – 1/21

5/

While we're at it, here's a nice and short exposition:
Dusa McDuff – Symplectic geometry

And a classic book in a translation from 2001:
V. I. Arnol'd, A. B. Givental, translation by G.Wassermann – Symplectic Geometry
maths.ed.ac.uk/~v1ranick/paper

I wonder, how old is the russian orgiginal?

/6

I meant to link to Dusa McDuff's paper "What is Symplectic Geometry?" in /6 above. It is a fun paper, and you should read it even if you don't usually read math papers:

math.barnard.edu/sites/default

Somehow the video for 11/ came out here when I tried to paste the link. Apologies to my early readers.

In Euclidean geometry you get lengths and angles as basic tools. Projective geometry loses angles, and replaces the concept of length by one of proportional length. In topology you get rid of both these tools, and this makes it into what is probably the most flabby kind of geometry. Symplectic geometry gets rid of both, length and angle, but introduces a new formalism to measure area instead... So, it is about as much related to geometry as a tomato is to a frog...

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Symplectic geometry is the ultimate generalization of Hamiltonian mechanics! The basic idea of Hamiltonian mechanics is to write down an expression (called Hamiltonian) for a conserved quantity (like energy) as depending on an unknown function p(t) (e.g. position), and one related to its first derivative q(t) (say, momentum). You then demand that the Hamiltonian is stationary, that it doesn't change, which is cleverly formalized by setting its derivative to zero, and solve for p!

8/

On g+, +Beat Toedtli noticed that I had swapped the meaning of p and q! In the literature you'll find p used in Hamiltonians to refer to momentum, and q to the "generalized coordinate". I can't change my earlier posts here, but I will try to stick with the popular nomenclature from now on.

The announcement on g+ is here:

The earlier Lagrangian mechanics involved an extra parameter for the time t, and it always uses the derivative of the first kind of coordinates as the second kind of coordinates.

en.wikipedia.org/wiki/Lagrangi

The extra parameter t amounts to another coordinate dimension, making the resulting spaces odd-dimensional. This is called contact geometry and it is closely related to symplectic geometry.

en.wikipedia.org/wiki/contact%

9/

en.wikipedia.org/wiki/Hamilton

Now you know what a Hamiltonian H(p,q) is, first notice that p and q are simply coordinates. The position p(t) may be on a manifold that isn't standard Euclidean space. The momentum q(t) however, is always to be understood in the space that is tangent to the manifold at the position p(t). So a symplectic space is always even-dimensional, and smooth enough for one degree of infinitesimals.

10/

In the literature you might find the labels p and q used the other way around. See my annotation below 8/

A paper on the history of symplectic topology:

Michele Audin – Vladimir Igorevich Arnold and the Invention of Symplectic Topology
www-irma.u-strasbg.fr/~maudin/

Expository talks on symplectic geometry (videos):

Dusa McDuff – Symplectic geometry

Helmut Hofer – First Steps in Symplectic Dynamics

Dusa McDuff – Symplectic Topology Today

/11

There was a bit of a kerfuffle regarding the foundation of symplectic geometry. In 1996 Kenji Fukaya and Kaoru Ono published a paper on counting fixed points. Everybody referred to it but only more than a decade later did someone notice that it was difficult to follow, because the explanation was incomplete.

When Dusa McDuff joined the party she decided to fix it. Read more about it here:

quantamagazine.org/the-fight-t

13/

I have removed the post labeled 12/ necause it was the same as 10/. I'm sorry for the confusion.

Andreas Floer came up with the idea to use homology to attack symplectic geometry. At that time, much of the relevant literature was only available in russian, and a bit in french.

en.wikipedia.org/wiki/Andreas_

en.wikipedia.org/wiki/Floer_ho

Another tragic mathematician.

14/

A differential 2-form w on a (real) manifold M is a gadget that, at any point p ∈ M, eats two tangent vectors and spits out a real number in a skew-symmetric, bilinear way:

w_p: T_p M x T_p M -> R

((( T_p M is the tangent space at a point p ∈ M. If we consider all these spaces for all p it's called a tangent bundle TM. )))

For every v ∈ T_p M there is a symplectic buddy u ∈ T_p M such that

w(v,u) = 1

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> Linear "Darboux Theorem". Any two symplectic spaces of the same dimension are symplectically isomorphic, i.e. there exists a linear isomorphism between them which preserves the skew-scalar product.

> Corollary. A symplectic structure on a 2n-dimensional linar space has the form p1^q1 + ... + pn ^ qn in suitable coordinates (p1,...,pn,q1,...,qn).

> Such coordinates are called Darboux coordinates, and the space R^2n with this skew-scalar product is called the standard symplectic space.

16/

Having defined tangent bundles, we can now situate Langangians as a map L: TM -> R for which we can then solve for level sets.

Remember, TM is the tangent bundle for our manifold M. A point of TM is then a point p on M together with a vector in T_pM, a vector tangent to M at p.

17/

@RefurioAnachro
This is God's work.

I will be reading your toots, surely.

Thanks, @Breakfastisready! I should be able to deliver a bunch of nice and handy intuitions soon! That 500 char limit makes me feel like I'm spreading candy with a shotgun... So if, when, I succeed to find a nice angle¹, there should be loads of factlets around that picture that inspires. Upshot is, I can do this in real-time!

Enjoy!

¹: or a nice area, like dp_i ^ dq_i. Everyone is welcome as long as you follow our code of conduct! Thank you. Mastodon.cloud is maintained by Sujitech, LLC.