Joint work with

Boris Khesin

University of Toronto

Gerard Misiolek

University of Notre Dame

Arnold's 1966 discovery

\dot u + \nabla_u u = -\nabla p

\dot u + \nabla_u u = -\nabla p

Euler's equations for inviscid incompressible fluid on $M$

Arnold (1966): diffeomorphism $\varphi(t)$ generated by $u(t)$ is geodesic curve on $\mathrm{Diff}_\mu(M)$ w.r.t. $G_\varphi(\dot\varphi,\dot\varphi) = \int_M |\dot\varphi|^2 \mu$

Led to geometric and topological hydrodynamics

$\Rightarrow$ stability results (Arnold and Khesin)

$\Rightarrow$ well-posedness results (Ebin and Marsden)

Newton's equations on $\mathrm{Diff}(M)$

\displaystyle \dot \varphi + \nabla_{\dot\varphi} \dot\varphi +\nabla \frac{\delta \bar U}{\delta\varrho}(\varphi_*\mu)\circ\varphi = 0

\displaystyle \dot \varphi + \nabla_{\dot\varphi} \dot\varphi +\nabla \frac{\delta \bar U}{\delta\varrho}(\varphi_*\mu)\circ\varphi = 0

Aim: extend Arnold's framework

Ingredients:

Riemannian metric $G$ on $\mathrm{Diff}(M)$
Potential function $U:\mathrm{Diff}(M)\to \mathbb{R}$

$\mathrm{Diff}_\mu(M)$ symmetry:

$G_{\varphi\circ\eta}(\dot\varphi\circ\eta,\dot\varphi\circ\eta) = G_\varphi(\dot\varphi,\dot\varphi)\qquad\forall \,\eta\in \mathrm{Diff}_\mu(M)$
$V(\varphi\circ\eta) = V(\varphi)$

Riemannian submersion

\mathrm{Diff}(M)

\mathrm{Diff}(M)

\mathrm{Dens}(M)

\mathrm{Dens}(M)

\mathrm{Id}

\mathrm{Id}

\mu

\mu

\varrho

\varrho

\pi(\varphi)=\varphi_*\mu

\pi(\varphi)=\varphi_*\mu

Moser 1965:

Principal bundle

\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)

\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)

$L^2$ metric on $\mathrm{Diff}(M)$

G_\varphi(\dot\varphi,\dot\varphi) = \int_{M}\left\vert \dot\varphi \right\vert^2 \mu

G_\varphi(\dot\varphi,\dot\varphi) = \int_{M}\left\vert \dot\varphi \right\vert^2 \mu

Induces Otto metric

{\overline{G}}_\varrho(\dot\varrho,\dot\varrho) \Rightarrow d_W^2(\mu,\varrho)

{\overline{G}}_\varrho(\dot\varrho,\dot\varrho) \Rightarrow d_W^2(\mu,\varrho)

\mathrm{Hor}

\mathrm{Hor}

\displaystyle U(\varphi) = \bar U(\varphi_*\mu)\quad \bar U\colon\mathrm{Dens}(M)\to\mathbb{R}

\displaystyle U(\varphi) = \bar U(\varphi_*\mu)\quad \bar U\colon\mathrm{Dens}(M)\to\mathbb{R}

\mathrm{Dens}(M)=\{ \varrho\in\Omega^n(M)\mid \varrho>0, \int_M \varrho = 1\}

\mathrm{Dens}(M)=\{ \varrho\in\Omega^n(M)\mid \varrho>0, \int_M \varrho = 1\}

Smooth probability densities

Induced potential function

Symplectic reduction viewpoint

T^*\mathrm{Diff}(M)

T^*\mathrm{Diff}(M)

T^*\mathrm{Diff}(M)/\mathrm{Diff}_{\mu}(M)\simeq \mathrm{Dens}(M) \times \mathfrak{X}^*(M)

T^*\mathrm{Diff}(M)/\mathrm{Diff}_{\mu}(M)\simeq \mathrm{Dens}(M) \times \mathfrak{X}^*(M)

T^*\mathrm{Diff}(M)//\mathrm{Diff}_{\mu}(M)\simeq T^*\mathrm{Dens}(M)

T^*\mathrm{Diff}(M)//\mathrm{Diff}_{\mu}(M)\simeq T^*\mathrm{Dens}(M)

Poisson reduction

Symplectic reduction

\dot \rho + \mathrm{div}(\rho\nabla \theta) = 0,\; \dot \theta + \frac{1}{2}|\nabla \theta|^2 + \frac{\delta \bar U}{\delta\varrho}(\varrho) = 0

\dot \rho + \mathrm{div}(\rho\nabla \theta) = 0,\; \dot \theta + \frac{1}{2}|\nabla \theta|^2 + \frac{\delta \bar U}{\delta\varrho}(\varrho) = 0

\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot u + \nabla_u u + \nabla\frac{\delta \bar U}{\delta\varrho}(\varrho) = 0

\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot u + \nabla_u u + \nabla\frac{\delta \bar U}{\delta\varrho}(\varrho) = 0

Example: barotropic fluid

\displaystyle\bar U(\varrho) = \int_M e(\rho)\varrho

\displaystyle\bar U(\varrho) = \int_M e(\rho)\varrho

Gives compressible (barotropic) Euler equations

\displaystyle\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot u + \nabla_u u + \frac{1}{\rho}\nabla P(\rho) = 0

\displaystyle\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot u + \nabla_u u + \frac{1}{\rho}\nabla P(\rho) = 0

$\displaystyle P(\rho) = e'(\rho)\rho^2$ is the pressure function

"Potential solutions" $u = \nabla\theta$ $\Rightarrow$ horizontal solutions

Example: Schrödinger equations

\displaystyle\bar U(\varrho) = \int_M F(\rho)\mu + \frac{1}{2}\int_M \frac{|\nabla \rho|^2}{\rho}\mu

\displaystyle\bar U(\varrho) = \int_M F(\rho)\mu + \frac{1}{2}\int_M \frac{|\nabla \rho|^2}{\rho}\mu

Consider horizontal solutions $\Rightarrow$ system on $T^*\mathrm{Dens}(M)$

\displaystyle\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot\theta + \frac{1}{2}|\nabla \theta|^2 + f(\rho) - \frac{2}{\sqrt{\rho}}\nabla\sqrt{\rho} = 0

\displaystyle\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot\theta + \frac{1}{2}|\nabla \theta|^2 + f(\rho) - \frac{2}{\sqrt{\rho}}\nabla\sqrt{\rho} = 0

Fisher functional

From $(\varrho,\theta)\in T^*\mathrm{Dens}(M)$ construct wave function

\displaystyle \psi = \sqrt{\rho\mathrm{e}^{i\theta}}

\displaystyle \psi = \sqrt{\rho\mathrm{e}^{i\theta}}

Theorem (Madelung 1927, von Renesse 2011):

Wave function fulfills (nonlinear) Schrödinger equation $i\dot\psi + \Delta\psi - f(|\psi|^2)\psi=0$

Madelung transform

Madelung transform as symplectomorphism

Theorem: Madelung transform induces symplectomorphism $\Phi\colon T^*\mathrm{Dens}(M)\to PC^\infty(M,\mathbb{C}\backslash\{0\})$ (Fréchet topology of smooth functions)

Geometric quantum mechanics (Kibble 1979):

wave function $\Rightarrow$ element of complex projective space

PC^\infty(M,\mathbb{C})

PC^\infty(M,\mathbb{C})

Kähler manifold

Madelung transform as

Kähler morphism

Theorem: Madelung transform induces Kähler morphism $\Phi\colon T^*\mathrm{Dens}(M)\to PC^\infty(M,\mathbb{C}\backslash\{0\})$ (Fréchet topology of smooth functions)

Is Madelung transform isometry?

Canonical metric on $PC^\infty(M,\mathbb{C})$ : Fubini-Study metric

Fisher-Rao metric on $\mathrm{Dens}(M)$ :

\displaystyle G_\varrho(\dot\varrho,\dot\varrho) = \int_M \frac{\dot\rho^2}{\rho}\mu

\displaystyle G_\varrho(\dot\varrho,\dot\varrho) = \int_M \frac{\dot\rho^2}{\rho}\mu

Sasaki-Fisher-Rao metric

on $T^*\mathrm{Dens}(M)$ :

\displaystyle G_{(\varrho,\theta)}(\dot\varrho,\dot\theta,\dot\varrho,\dot\theta) = \int_M (\frac{\dot\rho^2}{\rho} + \dot\theta^2\rho)\mu

\displaystyle G_{(\varrho,\theta)}(\dot\varrho,\dot\theta,\dot\varrho,\dot\theta) = \int_M (\frac{\dot\rho^2}{\rho} + \dot\theta^2\rho)\mu

Some consequences

Link between classical information geometry, quantum information geometry, and hydrodynamics
New, geometric understanding of several PDE: 2-component Camassa-Holm, vortex filament flow, ...
Geometric and hydrodynamic interpretation of relativistic quantum mechanics:
Klein-Gordon equation as stationary solution of infinite-dimensional Neumann problem on Minkowski space

THANKS!

Geometric hydrodynamics via Madelung transform,

PNAS, 2018

Geometry of the Madelung transform,

arXiv preprint, 2018

Slides available at: slides.com/kmodin

Newton's equations on diffeomorphisms and densities

Klas Modin