Periodic geodesics in special unitary groups Looking at the solution of the equation of motion for the action using a bi-invariant metric on a Lie group. If I take the lie group as $SU(2)$ the geodesics are always periodic because $SU(2)$ is the three-sphere, how can I prove this more formally?
Besides, I don't see clear how can I say something about periodic solutions in $SU(3)$ or in general in $SU(n)$? 

The answer at https://mathoverflow.net/questions/126006/is-it-true-that-the-geodesics-on-son-and-sun-are-closed and the comments below it completely answer your question.
Edit: you seem to be under the impression that geodesics in a compact group are the same as one parameter subgroups.  Unfortunately this is usually not the case.  See here:
https://mathoverflow.net/questions/81590/one-parameter-subgroup-and-geodesics-on-lie-group
