Geometric Hydrodynamics and Infinite-dimensional Newton's Equations

Klas Modin

Joint work with

Boris Khesin

University of Toronto

Gerard Misiolek

University of Notre Dame

Recent survey:

Geometric hydrodynamics and infinite-dimensional Newton’s equations, BAMS, 2021

 

Geometric hydrodynamics

(Arnold, 1966)

Optimal transport

Probability theory

Shape analysis

Information geometry

Quantum mechanics

\mathrm{Diff}_\mu(M)
\mathfrak{X}_\mu(M)
\iff

Lagrangian coordinates

\(\varphi(t)\) geodesic on \(\mathrm{Diff}_\mu(M)\)

\[\nabla_{\dot\varphi}\dot\varphi = -\nabla p\circ\varphi\]

Eulerian coordinates

\(v(t)=\dot\varphi(t)\circ\varphi(t)^{-1}\) fulfills Euler's equations

\[\dot v + \nabla_v v = -\nabla p\]

v(t)
\varphi(t)

Geometric hydrodynamics

(Arnold, 1966)

Optimal transport

Probability theory

Shape analysis

Information geometry

Quantum mechanics

Newton's equations on Diff(M)

\displaystyle \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 on \(\mathrm{Diff}(M)\)
  • Potential function \(U:\mathrm{Diff}(M)\to \mathbb{R}\)

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

  • \(\langle\dot\varphi\circ\eta,\dot\varphi\circ\eta\rangle_{\varphi\circ\eta} = \langle\dot\varphi,\dot\varphi\rangle_\varphi\qquad\forall \,\eta\in \mathrm{Diff}_\mu(M)\)
  • \(U(\varphi\circ\eta) = U(\varphi)\) \(\quad\Rightarrow\quad\) \(U(\varphi) = \bar U(\varphi_*\mu)\)

Riemannian submersion

\mathrm{Diff}(M)
\mathrm{Dens}(M)
\mathrm{Id}
\mu
\varrho
\pi(\varphi)=\varphi_*\mu

Moser 1965:

Principal bundle

\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

Induces Otto metric

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

Smooth probability densities

Induced potential function

Form of equations

\mathrm{Diff}(M)
\mathrm{Dens}(M)

On \(\mathrm{Diff}(M)\)

On \(\mathfrak{X}^*(M)\times\mathrm{Dens}(M)\)

On \(T^*\mathrm{Dens}(M)\)

\mathfrak{X}(M)
\displaystyle \nabla_{\dot\varphi} \dot\varphi +\nabla \frac{\delta \bar U}{\delta\varrho}(\varphi_*\mu)\circ\varphi = 0
\displaystyle \dot m + \mathcal{L}_v m - \rho \nabla \frac{\delta\bar U}{\delta\rho}
\displaystyle \dot \rho + \mathrm{div}(\rho v) = 0
\displaystyle \dot \rho + \mathrm{div}(\rho v) = 0
\displaystyle \dot \theta + \frac{1}{2}|\nabla\theta|^2 + \frac{\delta \bar U}{\delta\rho}=0

Form of equations

\mathrm{Diff}(M)
\mathrm{Dens}(M)

On \(\mathrm{Diff}(M)\)

On \(\mathfrak{X}^*(M)\times\mathrm{Dens}(M)\)

On \(T^*\mathrm{Dens}(M)\)

\mathfrak{X}(M)
\displaystyle \nabla_{\dot\varphi} \dot\varphi +\nabla \frac{\delta \bar U}{\delta\varrho}(\varphi_*\mu)\circ\varphi = 0
\displaystyle \dot m + \mathcal{L}_v m - \rho \nabla \frac{\delta\bar U}{\delta\rho}
\displaystyle \dot \rho + \mathrm{div}(\rho v) = 0
\displaystyle \dot \rho + \mathrm{div}(\rho v) = 0
\displaystyle \dot \theta + \frac{1}{2}|\nabla\theta|^2 + \frac{\delta \bar U}{\delta\rho}=0

Form of equations

\mathrm{Diff}(M)
\mathrm{Dens}(M)

On \(\mathrm{Diff}(M)\)

On \(\mathfrak{X}^*(M)\times\mathrm{Dens}(M)\)

On \(T^*\mathrm{Dens}(M)\)

\mathfrak{X}(M)
\displaystyle \nabla_{\dot\varphi} \dot\varphi +\nabla \frac{\delta \bar U}{\delta\varrho}(\varphi_*\mu)\circ\varphi = 0
\displaystyle \dot m + \mathcal{L}_v m - \rho \nabla \frac{\delta\bar U}{\delta\rho}
\displaystyle \dot \rho + \mathrm{div}(\rho v) = 0
\displaystyle \dot \rho + \mathrm{div}(\rho v) = 0
\displaystyle \dot \theta + \frac{1}{2}|\nabla\theta|^2 + \frac{\delta \bar U}{\delta\rho}=0

Examples

  • Shallow water equations
  • Inviscid Burgers
  • Hamilton-Jacobi
  • Barotropic fluid equations
  • Compressible magnetohydrodynamics (MHD)
  • Fully compressible fluid equations
  • Relativistic fluid equations
     
  • \(\mu\)-Camassa-Holm equation
  • Infinite-dimensional Neumann problem
  • Klein-Gordon equation
     
  • Linear and non-linear Schrödinger equations
  • Heat flow
  • Incompressible Schrödinger equation (a.k.a. Schrödinger's smoke)
  • 2-component Hunter-Saxton equation
  • ...

Wasserstein-Otto

metric

Fisher-Rao

metric

Using Madelung

transform

Case study

Lagrangian

\displaystyle L(v,\rho) = \frac{1}{2}\int_M |v|^2\rho

\(\Rightarrow\) Burgers equation, horizontal solutions = OMT

\dot v + \nabla_v v = 0

Case study

Lagrangian

\displaystyle L(v,\rho) = \frac{1}{2}\int_M |v|^2\rho

\(\Rightarrow\) horizontal solutions = Schrödinger equation

\displaystyle i\hbar \dot\psi = -\frac{\hbar^2}{2}\Delta\psi
\displaystyle - \frac{\hbar^2}{8}\int_M \frac{|\nabla\rho|^2}{\rho}

Fisher functional

Case study

Lagrangian

\displaystyle L(v,\rho) = \frac{1}{2}\int_M |v|^2\rho

\(\Rightarrow\) horizontal solutions = double Heat flows

\displaystyle \dot\psi^+ = \epsilon\Delta\psi^+
\displaystyle + \frac{\epsilon^2}{8}\int_M \frac{|\nabla\rho|^2}{\rho}

 \(\hbar=\pm i\epsilon\)

\displaystyle \dot\psi^- = -\epsilon\Delta\psi^-
\rho = \psi^+\psi^-

Schrödinger bridge ( = entropic regularization of OMT)

\psi^+_0\mathrm{e}^{\epsilon\Delta}\psi^-_1 = \rho_0 \qquad \psi^-_1\mathrm{e}^{\epsilon\Delta}\psi^+_0 = \rho_1

THANKS!

Geometric Hydrodynamics and Infinite-dimensional Newton's Equations

By Klas Modin

Private

Geometric Hydrodynamics and Infinite-dimensional Newton's Equations

Presentation given at the 5th Conference on Geometric Science of Information (GSI'21), Paris, Sorbonne University, 21-23 July 2021

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