Newton's equations on diffeomorphisms and densities

Klas Modin

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
u˙+∇uu=−∇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
φ˙+∇φ˙φ˙+∇δUˉδϱ(φ∗μ)∘φ=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)
Diff(M)\mathrm{Diff}(M)
\mathrm{Dens}(M)
Dens(M)\mathrm{Dens}(M)
\mathrm{Id}
Id\mathrm{Id}
\mu
μ\mu
\varrho
ϱ\varrho
\pi(\varphi)=\varphi_*\mu
π(φ)=φ∗μ\pi(\varphi)=\varphi_*\mu

Moser 1965:

Principal bundle

\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)
Diff(M)/Diffμ0(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
Gφ(φ˙,φ˙)=∫M∣φ˙∣2μG_\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)
G‾ϱ(ϱ˙,ϱ˙)⇒dW2(μ,ϱ){\overline{G}}_\varrho(\dot\varrho,\dot\varrho) \Rightarrow d_W^2(\mu,\varrho)
\mathrm{Hor}
Hor\mathrm{Hor}
\displaystyle U(\varphi) = \bar U(\varphi_*\mu)\quad \bar U\colon\mathrm{Dens}(M)\to\mathbb{R}
U(φ)=Uˉ(φ∗μ)Uˉ​:Dens(M)→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\}
Dens(M)={ϱ∈Ωn(M)∣ϱ>0,∫Mϱ=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∗Diff(M)T^*\mathrm{Diff}(M)
T^*\mathrm{Diff}(M)/\mathrm{Diff}_{\mu}(M)\simeq \mathrm{Dens}(M) \times \mathfrak{X}^*(M)
T∗Diff(M)/Diffμ(M)≃Dens(M)×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∗Diff(M)//Diffμ(M)≃T∗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
ρ˙+div(ρ∇θ)=0,  θ˙+12∣∇θ∣2+δUˉδϱ(ϱ)=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
ρ˙+div(ρu)=0,  u˙+∇uu+∇δUˉδϱ(ϱ)=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
Uˉ(ϱ)=∫Me(ρ)ϱ\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
ρ˙+div(ρu)=0,  u˙+∇uu+1ρ∇P(ρ)=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
Uˉ(ϱ)=∫MF(ρ)μ+12∫M∣∇ρ∣2ρμ\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
ρ˙+div(ρu)=0,  θ˙+12∣∇θ∣2+f(ρ)−2ρ∇ρ=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}}
ψ=ρeiθ\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∞(M,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
Gϱ(ϱ˙,ϱ˙)=∫Mρ˙2ρμ\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
G(ϱ,θ)(ϱ˙,θ˙,ϱ˙,θ˙)=∫M(ρ˙2ρ+θ˙2ρ)μ\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