Geometric hydrodynamics via Madelung transform.
basic_science · Level V
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- Record sourced from PubMed, PMID 29844194.
- Also identified by DOI 10.1073/pnas.1719346115 and PMC identifier 6004435.
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Abstract
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.