# U(1):?:SU(3) 1D:2D:3D harmonic oscillator.

• Spinnor
In summary, the one dimensional harmonic oscillator is associated with the group U(1), the three dimensional harmonic oscillator is associated with the group SU(3), and the two dimensional harmonic oscillator is associated with the group SU(2). This was confirmed through a discussion about the simply connected special unitary group that is bigger than U(1) but smaller than SU(3). The book "Classical Mechanics" by Goldstein also discusses the SO(4) symmetry of the 1/r potential.
Spinnor
Gold Member
The one dimensional harmonic oscillator is associated with the group U(1) and the three dimensional harmonic oscillator is associated with the group SU(3). Is their a group associated with the two dimensional harmonic oscillator?

Thank you for any thoughts.

I was given the hint by Igor, thoovler@excite.com:

"What simply connected special unitary group is bigger
than U(1), but smaller than SU(3)?"

My guess was SU(2), a Google search seems to confirm this.

Yes, I think you are correct. If you have a copy of Goldstein (Classical Mechanics), they discuss this, as well as the SO(4) symmetry of the 1/r potential.

## 1. What is U(1)?

U(1) is a mathematical group that represents the symmetry of one-dimensional systems. It is used in physics to describe the behavior of particles and fields.

## 2. What is SU(3)?

SU(3) is a mathematical group that represents the symmetry of three-dimensional systems. It is commonly used in particle physics to describe the interactions between quarks and gluons.

## 3. What is a 1D harmonic oscillator?

A 1D harmonic oscillator is a physical system that exhibits periodic motion around an equilibrium point. It is described by the equation of motion x'' + ω^2x = 0, where ω is the angular frequency.

## 4. What is a 2D harmonic oscillator?

A 2D harmonic oscillator is a physical system that exhibits periodic motion in two dimensions. Its equation of motion is given by x'' + y'' + ω^2x = 0, where ω is the angular frequency.

## 5. What is a 3D harmonic oscillator?

A 3D harmonic oscillator is a physical system that exhibits periodic motion in three dimensions. Its equation of motion is given by x'' + y'' + z'' + ω^2x = 0, where ω is the angular frequency.

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