Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations; it originated as a formalization of the mathematical apparatus of classical mechanics and geometric optics (and the related WKB-method in quantum mechanics and, more generally, the method of stationary phase in harmonic analysis). A wider branch including symplectic geometry is Poisson geometry and a sister branch in odd dimensions is contact geometry. A special and central role in the subject belongs to certain real-like half-dimensional submanifolds, called lagrangian (or Lagrangean) submanifolds, which are in some sense classical points. Symplectic geometry radically changed after the 1985 article of Mikhail Gromov on pseudoholomorphic curves and the subsequent work of Andreas Floer giving birth to symplectic topology or “hard methods” of symplectic geometry.
In its application to physics, symplectic geometry is the fundamental mathematical language for Hamiltonian mechanics, geometric quantization, geometrical optics.
A tremendous amount of insight into higher Lie theory (Lie groupoids, Lie ∞-groupoids, Lie ∞-algebroids) has derived from Alan Weinstein‘s long-term project of understanding the role of symplectic geometry in geometric quantization. See there for more details.
Zoran ?koda?: it is true that large influence of Weinstein’s program can not be overestimated, but it is not the origin of these considerations; it rather builds up on earlier fundamental works of Kirillov, Kostant, Souriau who invented geometric quantization, all of them originally in symplectic context; and the florishing of the subject from mid 1960s till mid 1980-s is related to their work; and other related tracks of Guillemin, Sternberg, Kashiwara, Karasev, Arnold and so on; and vast developments in harmonic analysis and representation theory (Kostant, Auslander, Vogan, Wallach, Stein…), microlocal analysis (Kashiwara, Saito, Hormander, Maslov, Karasev, Duistermaat…), integrable systems/quantum groups (this is more into more general Poisson geometry: Lie-Poisson groups, classical r-matrices, bihamiltonian systems…), and related approaches to quantization (Berezin method, coherent states…).
There is a vertical categorification of symplectic geometry to higher symplectic geometry. This involves multisymplectic geometry and the geometry of symplectic Lie n-algebroids. And their combination.
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
Introductions include
Discussion from the point of view of homological algebra of abelian sheaves is in
The notion of symplectic geometry may be understood as the mathematical structure that underlies the physics of Hamiltonian mechanics. A classical monograph that emphasizes this point of view is
For more on this see Hamiltonian mechanics.
Symplectic geometry is also involved in geometric optics, geometric quantization and the study of oscillatory integrals and microlocal analysis. Books concentrating on these topics include
Victor Guillemin, Shlomo Sternberg, Geometric asymptotics, Amer. Math. Soc. 1977 free online
Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf
J.J. Duistermaat, Fourier integral operators, Progress in Mathematics, Birkhäuser 1995 (and many other references at microlocal analysis).
Alan Weinstein, Symplectic geometry, (survey) Bull. Amer. Math. Soc. 5 (1981), 1-13, doi
N. R. Wallach, Symplectic geometry and Fourier analysis, Math. Sci. Press, Brookline, Mass., 1977.
wikipedia symplectic geometry Application: “Jacobi’s elimination of nodes” is moved to equivariant localization and elimination of nodes.
A proof of the Arnold conjecture in symplectic geometry is in