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Adjoint orbits of $\mathfrak{sl}(2,\mathbb{R})$ and their geometry

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.