  We study the geometry of geodesics on $\mathsf{SL}(n)$, equipped with the
Hilbert-Schmidt metric which makes it a Riemannian manifold. These geodesics
are known to be related to affine motions of incompressible ideal fluids. The
$n = 2$ case is special and completely integrable, and the geodesic motion was
completely described by Roberts, Shkoller, and Sideris; when $n = 3$, Sideris
demonstrated some interesting features of the dynamics and analyzed several
classes of explicit solutions. Our analysis shows that the geodesics in higher
dimensions exhibit much more complex dynamics. We generalize the
Virial-identity-based criterion of unboundedness of geodesic given by Sideris,
and use it to give an alternative proof of the classification of geodesics in
2D obtained by Roberts--Shkoller--Sideris. We study several explicit families
of solutions in general dimensions that generalize those found by Sideris in
3D. We additionally classify all "exponential type" geodesics, and use it to
demonstrate the existence of purely rotational solutions whenever $n$ is even.
Finally, we study "block diagonal" solutions using a new formulation of the
geodesic equation in first order form, that isolates the conserved angular
momenta. This reveals the existence of a rich family of bounded geodesic
motions in even dimension $n \geq 4$. This in particular allows us to conclude
that the generalization of the swirling and shear flows of Sideris to even
dimensions $n \geq 4$ are in fact dynamically unstable.
