PT - JOURNAL ARTICLE
AU - Khesin, Boris
AU - Misiolek, Gerard
AU - Modin, Klas
TI - Geometric hydrodynamics via Madelung transform
DP - 2018 Jun 12
TA - Proceedings of the National Academy of Sciences
PG - 6165--6170
VI - 115
IP - 24
4099 - http://www.pnas.org/content/115/24/6165.short
4100 - http://www.pnas.org/content/115/24/6165.full
SO - Proc Natl Acad Sci USA2018 Jun 12; 115
AB - Geometry has always played a fundamental role in theoretical physics via symmetries and conservation laws. We present a geometric framework revealing a closer link between hydrodynamics and quantum mechanics than previously recognized. Newton’s equations, generalized to infinite-dimensional spaces of fluid flow maps (diffeomorphisms), are used to develop a unified setting and uncover new connections between many equations of mathematical physics. These include equations of compressible fluids, motion of particles on spheres in quadratic potentials, and the Klein–Gordon and nonlinear Schrödinger equations, as well as their relation to information geometry and optimal mass transport. This work contributes toward a better understanding of geometric structures arising in hydrodynamics and quantum mechanics.We introduce a geometric framework to study Newton’s equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important partial differential equations of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schrödinger equation and Newton’s equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a Kähler map when the space of densities is equipped with the Fisher–Rao information metric. We describe several dynamical applications of these results.