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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) …

To build intuition for the Planck constant $\hbar$, which I understand is the purpose of the OP, I would start by noting that $\hbar$ is not a dimensionless number: it has dimensions of energy $\times …

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 …

An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively. [For t …

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 …

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 …

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 …

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

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).

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 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 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 ( …

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 …

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 …

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 …