# classical dynamics on group manifold SU(2)

I am trying to understand how to formulate classical dynamics on group manifold SU(2). This will be an exercise for me to the more advanced subject of path integral on group manifold.

Does someone know of good (pedestrian) lecture notes on classical dynamics on SU(2)? For example, why the free particle Lagrangian look like that: $L=\frac{1}{2}tr\dot{U}\dot{U}^\dagger$? why the trace , whats the role of $U$?

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Please see the following work, by George Chavchanidze, for a full solution of the classical and quantum dynamics of a particle moving freely on SU(2). The solution can be summarized as follows:

At the classical level, the group SU(2), equipped with the invariant metric is isometric to the round three sphere $S^3$. Thus the free particle trajectories (which are geodesics) are great circles on $S^3$.

More importantly, the Noether charges corresponding to the invariance of the dynamics under the action of $SU(2)$ are the angular momentum components. Which are conserved both classically and quantum mechanically.

At the quantum level all matrix element of a representation of SU(2) are solutions of the free Schrodinger equation on the group manifold, this is the Peter-Weyl theorem.

The reason that the Lagrangian takes the given form is because in the canonical coordinates $\{\phi_1, \phi_2, \phi_3 \}$, such that $U=e^{i\sigma_k \phi_k}$ (where $\sigma_k$ are the Pauli matrices), the Lagrangian takes the form

$L= \frac{1}{2} tr(\dot{U} \dot{U}) = \frac{1}{2} g_{ij}\dot{ \phi}_i\dot{ \phi}_j$

where $g_{ij}$ is the invariant metric on $SU(2)$ and the Euler-Lagrange equation corresponding to this metric is the geodesic equation.

Finally, a good review of the classical and quantum dynamics on an arbitrary compact group manifolds by: Marinov and Terentyev (In this review the quantum dynamics is treated by means of the path integral) is available in Prof. Marinov's memorial page.

Update related to the Hydrogen atom:

The dynamics of the electon in the hydrogen atom is not a free motion on a group manifold. However, Lie groups play a very important role in the hydrogen atom dynamics. Classical and quantum dynamics of the hydrogen atom can be constructed geometrically, on phase spaces which are coadjoint orbits (which are certain coset spaces) of the group $SO(3,2)$, please see for example the following work by: Bruno Cordani. (The people of geometric quantization often refer to the hydrogen atom problem by the name of Kepler problem for the obvious reason).

It is important to mention that here the coadjoint orbits constitute of the full phase space, and not of the configuration space, i.e., the directions along the "momenta" are also curved in contrast to the free motion where the momenta are "flat" (motion on a cotangent bundle). Dynamis of this form requires a different type of path integrals sometimes named by "Coherent state path integrals". The research on these subjects is still ongoing.

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Thanks for the explanation. I will look at the two papers especially Marinov's one. Basically, does path integral on SU(2) is the way to calculate path integral for the hydrogen atom? – bill Sep 5 '12 at 14:33
I added an update answering the question about the hydrogen atom – David Bar Moshe Sep 5 '12 at 16:21
I will read more about it, toda – bill Sep 5 '12 at 16:41