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A recent (2017) overview of the status of Arnold's conjecture is given in The number of Hamiltonian fixed points on symplectically aspherical manifolds.

The symplectic counterpart of the fact that a family of commuting diagonalizable matrices is simultaneously diagonalizable is discussed in section 3.1 of On the diagonalizability of a matrix by a symp …

Once you have made the polar decomposition, it is sufficient to find the Haar measure on the compact symplectic group. This can be calculated starting from your favorite parameterization $U(\{\alpha_i …

The established reference (see for example arXiv:0912.0651) to Taubes trick is C.H. Taubes, Seiberg Witten and Gromov invariants for symplectic 4-manifolds (2000). This book collects results from fo …

the $a_i$'s are complex coefficients in the asymptotic expansion $a(h)=\sum_{i=0}^\infty a_i h^i$; a more explicit expression is given on page 275-276 of Guillemin and Sternberg:

An analogue of the Huygens principle (including interference, so more precisely a Huygens-Fresnel principle) in superspace has been formulated by Gomes in arXiv:gr-qc/0602092. The formulation is used …

V.I. Arnold's Mathematical Methods of Classical Mechanics is entirely based on the ideas and methods of symplectic geometry, such as the Birkhoff normal form, the Kolmogorov- Arnold-Moser theorem on t …

The inversion is conveniently described in terms of the Fourier transform $$g(x,p)=\int dy\,e^{-iyp}G(x+y/2,x-y/2).$$ Then the composition $f(x,p)=g(x,p)\star h(x,p)$ is a matrix multiplication [1], $ …

The factor of four and why Goldman has a different factor is explained on page 6 of Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces. A derivation with …

The gravitational or Coulomb potential has a "hidden" symmetry (hidden in the sense that it does not follow from the rotational symmetry). The resulting integral of the motion (the Runge-Lenz vector) …

You'll find an extensive discussion with applications of the "neck stretching" technique in Jonathan Evans' thesis, Symplectic topology of some Stein and rational surfaces (chapter 5).

S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44, 666 (1996). For more info, and the link to Bertini's theorem, see Jonathan Evans' thesis: Symplectic to …

The master thesis Nonholonomic Dynamical Systems by Brett Ryland contains several examples of non-Hamiltonian systems from classical physics: the dynamics of a laser and the evolution of a gas flame ( …

This echos the 2017 comments, but since the question has now been bumped to the front page it might be helpful to give the actual source in Maslov's book [1]. [1] V.P. Maslov, Perturbation Theory …

Symplectic geometry: The natural geometry of economics? (Thomas Russel, 2011). What restrictions does the hypothesis of maximizing behavior place on observed market data? In the context of profi …