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

- Interpret the symplectic area conserved in terms of position and momentum.
- What do inside and outside mean in symplectic (Hamiltionian, Lagrangian) mechanics?

Videos

Readings

- An Introduction to Lie Groups and Symplectic Geometry. Robert Bryant.
- What is a symplectic manifold, really? Ben Webster
- Why symplectic geometry is the natural setting for classical mechanics by Henry Cohn
- What is symplectic geometry? Dusa McDuff
- Lectures on Symplectic Geometry Ana Cannas da Silva
- Lectures on Symplectic Geometry Fraydoun Rezakhanlou
- A Fight to Fix Geometry’s Foundations
- Categorified Symplectic Geometry and the Classical String
- Motivating the conservation of symplectic area... Stack Exchange

Ideas

- Symplectic geometry conserves energy. When kinectic energy (a function of momentum) nears its maximum, then potential energy (a function of position) nears its minimum, and vice versa. Kinectic energy is understood in terms of positive and negative momentum with regard to zero momentum. Potential energy is understood in terms of a continuum stretching from zero to infinity. They are dealing with different dualities, and so symplectic geometry is mediating between these two dualities. This is evidently the source of the anti-symmetry.
- Symplectic form is skew-symmetric - swapping u and v changes sign - so it establishes orientation of surfaces - distinction of inside and outside - duality breaking. And inner product duality no longer holds.
- Trikampis - riba (jausmai) - simplektinė geometrija.
- Swapping position and momentum yields a negative sign because it is like switching from covariant to contravariant.

Ben Webster

- conservation of energy becomes antisymmetry {$\{f,f\}=0$}
- equivariance becomes the Jacobi identity {$\{f,\{g,h\}\}=\{\{f,g\},h\}+\{g,\{f,h\}\}$}.

Quaternions

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} $}

Parsiųstas iš http://www.ms.lt/sodas/Book/SymplecticGeometry

Puslapis paskutinį kartą pakeistas 2019 spalio 13 d., 00:08