After reading about and trying hard to understand even dimensional symplectic geometry, it was a surprise for me, not being able to find publications regarding the most simple case of two-dimensional symplectic geometry (in the plane).
I understand that this case is by far less interesting for physicists and mathematicians in contrast to challenging higher even-dimensional spaces.
From an engineers point of view I am especially interested in a comparison of Euclidean geometry and symplectic geometry and highly appreciate some pointers to papers I am possibly not aware of ... Thanks.
Dear Stefan Goessner
I think to study the fundamental forms in the plane reduced to the classical calculus ( mainly: Greens and stokes' Theorems with elementary derivations), that's why the advanced studies in differential geometry focus on the higher even dimensions.
The following is an article available online as a pdf file.
Best regards
Dear Stefan Goessner If you are interested in symplectic linear algebra, then there is not much to say in dimension 2: lines are lagrangian. If you are interested in symplectic geometry/topology then there is a lot to say: any closed orientable surface is a 2D symplectic space and one can study it's fundamental group (called surface groups) and their representations.
For symplectic linear algebra:
Chapter Euclidean and Symplectic Vector Spaces
For surface groups and their representations (advanced):
https://www.mathi.uni-heidelberg.de/~wienhard/aim_intro_final.pdf
https://hal.archives-ouvertes.fr/hal-00627916/document
https://people.math.ethz.ch/~acannas/Papers/lsg.pdf
https://projecteuclid.org/download/pdf_1/euclid.bams/1183548217
https://arxiv.org/pdf/1503.00238.pdf
@ Mohamed: You are right, there is not so much about symplectic geometry in dimension 2.
@ Muhammad: I do know the first two papers.
Thank you all for providing me with additional and interesting reads.
I am currently writing down some few aspects with symplectic geometry in R^2.
Thanks again
Dear Stefan,
since the symplectic Form in 2d reduces to a volume Form, symplectic geometry in 2d is just volume preserving geometry (and you will find interesting stuff probably under the latter Name).
Martin
Hi Stefan,
It is better to look for volume- preserving instead of symplectctic. For instance, for the Arnold conjecture about the fixed points of Hamiltonian diffeo see:
C.P. Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics.
Eliashberg, Y. (1979). Estimates on the number of fixed points of area preserving transformations. Syktyvkar University Preprint
Also S. Smale paper on the group of volume-preserving diffo of the 2- sphere.
Sincerely,
Andrés
@ Martin & Andres,
thanks for pointers to volume-preserving geometry. I found some some additional interesting papers.
So according to 'Embedding Problems in Symplectic Geometry':
'In dimension 2, "symplectic" and "volume-preserving" is the same. In higher dimensions however, the difference between them turns out to be huge ...'
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