Symplectic Geometry

 i⋅ i = -1


There is geometry behind that little snipet of code. What appears like an invention to solve quadratic equations turns into a geometric metaphor, for 90^o rotation on a two dimensional euclidean plane.


Now we have:


 i⋅ i = -1 ,  j⋅ j = -1,  k⋅ k = -1



Together with:


 1⋅ 1 = 1, and  i⋅ j = k,  j⋅ k = i,  k⋅ i = j,  j⋅ i = -k,  k⋅ j =-i,   i⋅ k = -j.



We get rotations in four dimensional space. 


These are nice constructs. It happens that:

We have here another code. This is the Symplectic code. This code has a sign change also, but more. It is a direction in a so-called Phase Space. The geometry of this space is the Symplectic geometry. Derivative means change in a certain direction. Two dimensional phase-space for one dimensional motion already has directions, and a sign change. It is the purpose of Hamiltonian Dynamics to understand the geometric consequences of this code.

I start reflexions on these issues with this note.