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

In summary, the 2D harmonic oscillator can be represented by a position vector and a momentum vector in R^2, which together can form a single vector in R^4. This representation does not lose any information. There is a relationship between the 2D harmonic oscillator and the group SU(2), where a rotation of a spinor by 4∏ brings it back to its original position. The double rotation of spinors corresponds to the different orbits of the 2D harmonic oscillator, which can also be mapped to the set of two component spinors. This is discussed further in the resources provided.
  • #1
Spinnor
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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.
 
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  • #2
Should we be able to map the different orbits of the 2D H.O. to the set of two component spinors?
 
  • #3
Spinnor said:
...

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
 

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

1. What is 2D H.O.?

2D H.O. stands for two-dimensional harmonic oscillator. It is a mathematical model that describes the motion of a particle in a system with two perpendicular restoring forces.

2. What is phase space in R^2?

Phase space in R^2 refers to a mathematical space where each point represents a unique state of a physical system. In other words, it is a graphical representation of all possible combinations of position and momentum for a given system.

3. How do you graph phase space in R^2?

To graph phase space in R^2, you plot the position of a particle on the x-axis and its momentum on the y-axis. Each point on the graph represents a unique state of the particle in the system.

4. What information can be obtained from a phase space graph in R^2?

A phase space graph in R^2 can provide information about the trajectory of a particle, its energy, and its stability. It can also show how the particle's position and momentum change over time.

5. How is phase space used in analyzing physical systems?

Phase space is a useful tool for analyzing physical systems because it allows for the visualization of all possible states of a system. This can help scientists understand the behavior and dynamics of a system and make predictions about its future states.

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