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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?
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- 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 {$\{f,\{g,h\}\}=\{\{f,g\},h\}+\{g,\{f,h\}\}$}.