Iš Gvildenu svetainės

Book: SymplecticGeometry

See: Study math

Grasp the key concepts of symplectic geometry and their origins in the symplectic Lie algebras and groups.





Ben Webster

Symplectic geometry

spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional lengths and angles that are familiar from Euclidean and Riemannian geometry. It is naturally associated with the field of complex rather than real numbers. However, it is not as rigid as complex geometry: one of its most intriguing aspects is its curious mixture of rigidity (structure) and flabbiness (lack of structure). What is Symplectic Geometry? by Dusa McDuff

mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a falling apple. The trajectory of such a system is determined if one knows its position and velocity (speed and direction of motion) at any one time. Thus for an object of unit mass moving in a given straight line one needs two pieces of information, the position q and velocity (or more correctly momentum) p:= ̇q. This pair of real numbers (x1,x2) := (p,q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp∧dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action pdq round the boundary ∂S.

single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld quantization principle says that the area of this region is quantized, i.e. it has to be an integral multiple of a number called Planck’s constant. Thus one can think of the symplectic area as a measure of the entanglement of position and velocity.


A complex number {$c=aI+bJ$} is built up from real numbers {$a$} and {$b$}.

{$a\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} + b\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} $}

A quaternion {$q=cI+dJ$} is built up from complex numbers {$c$} and {$d$}.

{$c\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} + d\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} $}


the threesome.

Parsiųstas iš http://www.ms.lt/sodas/Book/SymplecticGeometry
Puslapis paskutinį kartą pakeistas 2020 gegužės 24 d., 09:23