## 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 $i\dot\psi + \Delta\psi - f(|\psi|^2)\psi=0$

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

## 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)

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!

PNAS, 2018

arXiv preprint, 2018

Slides available at: slides.com/kmodin

By Klas Modin

# Newton's equations on diffeomorphisms and densities

Presentation given 2017-11-08 at the GSI'17 conference in Paris.

• 1,630