  Let $\mathrm{SL}(n,\mathbb{R})$ be the special linear group and
$\mathfrak{sl}(n,\mathbb{R})$ its Lie algebra. We study geometric properties
associated to the adjoint orbits in the simplest non-trivial case, namely,
those of $\mathfrak{sl}(2,\mathbb{R})$. In particular, we show that just three
possibilities arise: either the adjoint orbit is a one-sheeted hyperboloid, or
a two-sheeted hyperboloid, or else a cone.
  In addition, we introduce a specific potential and study the corresponding
gradient vector field and its dynamics when we restricted to the adjoint orbit.
We conclude by describing the symplectic structure on these adjoint orbits
coming from the well known Kirillov-Kostant-Souriau symplectic form on
coadjoint orbits.
