# Book: SymplecticGeometry

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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?
• How does the geometric product in a Clifford Algebra model angular momentum, the basis for symplectic geometry, which is otherwise typically described by the cross product?

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.
• Cotangent space The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.

Ben Webster

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

Symplectic geometry

• Symplectic geometry is an even dimensional geometry. It lives on even dimensional

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

• McDuff: First of all, what is a symplectic structure? The concept arose in the study of classical

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.

• momentum x position is angular momentum
• McDuff: This might seem a rather arbitrary measurement. However, mathematicians in the nineteenth century proved that it is preserved under time evolution. In other words, if a set of particles have positions and velocities in the region S1 at the time t1 then at any later time t2 their positions and velocities will form a region S2 with the same area. Area also has an interpretation in modern particle (i.e. quantum) physics. Heisenberg’s Uncertainty Principle says that we can no longer know both position and velocity to an arbitrary degree of accuracy. Thus we should not think of a particle as occupying a

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.

• Symplectic area is orientable.
• Area (volume) is a pseudoscalar such as the scalar triple product.
• Symplectic geometry is naturally related to time because it is swept out (in one dimension) in time. And so the time (one-)dimension thereby "defines" the geometry of the area.
• Symplectic "sweep" is related to equivalence (for example, natural transformation) relevant for arguments of equality by continuity (for example, the Fundamental Theorem of Calculus, integration).
• Symplectic geometry relates a point and its line, that is, it treats the moving point as a line with an origin, and relates the relative distance between the origins and the relative momentum between the origins. Thus it is a relation between two dimensions. And the boundary of the curve can be fuzzy, as in quantum mechanics and the Heisenberg principle.

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 {$d$}.

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

Notes

• Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry.
• Symplectic geometry defines slack. It defines motion as oriented area.
• Brouwer Fixed Point Theorem holds on a disk with boundary. He also showed that a reversible T which preserves area on the disk without boundary has a fixed point. (Conjugated through a translation.) (So area preservation is equivalent to having a boundary.) This relates perspectives and symplectic geometry.
• Symplectic maps map loops to loops with the same area. Area of a closed curve is given by differential forms. There has to be an energy in the background, the Hamiltonian. Symplectic space can associate area to a small loop (small triangle).
• Consider how simplexes (in differential geometry and Stokes theorem) relate to symplectic geometry. What is the relevant polytope for symplectic geometry? Coordinate systems? Or cross-polytopes?
• Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example).
• Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors?
• Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions.
• Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside?
• Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position.

the threesome.

• Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing.
• Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum).
• What is the connection between symplectic geometry and homology? See Morse theory. See Floer theory.
• Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"?
• Benet linkage - keturgrandinis - lygiagretainis, antilygiagretainis
• Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry.
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Puslapis paskutinį kartą pakeistas 2020 gegužės 24 d., 09:23