November 12, 2021
Geodesics on SL(n) with the Hilbert-Schmidt metric, and one application to fluids
One way to analyze the motion of rigid bodies is to study the corresponding trajectory in SO(n); the physical features of the body give rise to a Riemannian metric. If we were to replace rigid bodies with incompressible fluids, the same procedure leads to Arnol’d’s equations on the volume-preserving diffeomorphism group, into which SL(n) sits as a subgroup. For some fluid bodies, this subgroup is totally geodesic; in this case the induced Riemannian metric on SL(n) is the Hilbert-Schmidt metric. Extending previous works by Sideris and Roberts-Shkoller-Sideris, we analyze the geodesic motion in this setting. Results are obtained in collaboration with my REU students Audrey Rosevear (Amherst) and Samuel Sottile (MSU).