2D H.O., graph phase space in R^2?

[Spinnor]

Can I graph the phase space of a 2D harmonic oscillator in R^2 in the following way?

Let one vector in R^2 represent for position of the point mass and let another vector represent momentum. Together these two vectors in R^2 can represent a single vector in R^4? Do we loose any "information" in such a representation?

The 2D harmonic oscillator and the group SU(2) are related. If we rotate a spinor by 4∏ we come back to where we started, What with the 2D harmonic oscillator, if anything, corresponds to the double rotation with spinors?

Thanks for any help!

Edit, to make sense maybe the momentum vector is attached to the end of the position vector.

 

[Spinnor]

Should we be able to map the different orbits of the 2D H.O. to the set of two component spinors?

 

[Spinnor]

...

The 2D harmonic oscillator and the group SU(2) are related. If we rotate a spinor by 4∏ we come back to where we started, What with the 2D harmonic oscillator, if anything, corresponds to the double rotation with spinors?

...

See the first section of,

http://theory.tifr.res.in/~sgupta/courses/qm2008/lec10.pdf

"Symmetries for fun and profit."

and see,

https://www.physicsforums.com/showthread.php?t=548773

SU(N) symmetry in harmonic oscillator