  In this paper we study geodesics on adjoint orbits of $SL(n,\mathbb{R})$
equipped with $SO(n)$-invariant metrics (maximal compact subgroup). Our main
technique is translate this problem into a geometric problem in the tangent
bundle of certain $SO(n)$-flag manifolds and describe the geodesics equations
with respect to the Sasaki metric on tangent bundle. We also use tools of Lie
Theory in order to obtain some explicit description of families of geodesics.
We deal with the case of $SL(2,\mathbb{R})$ in full details.
