Boris Khesin - UNIGE · 2021. 4. 25. · Virasoro H_1 Hunter{Saxton (or Dym) equation Di (M) L2 Euler ideal uid Di (M) H1 averaged Euler ow Symp! ... There are suitable functional-analytic

$Page 1: Boris Khesin - UNIGE · 2021. 4. 25. · Virasoro H_1 Hunter{Saxton (or Dym) equation Di (M) L2 Euler ideal uid Di (M) H1 averaged Euler ow Symp! ... There are suitable functional-analytic$
Hamiltonian geometry of compressible fluids

Boris Khesin

(University of Toronto)

(joint with Gerard Misiolek and Klas Modin)

June 11, 2020, Global Poisson Webinar

Boris Khesin Hamiltonian geometry of compressible fluids 1 / 30

Table of contents

1 Euler hydrodynamics

2 Geometry of Diff(M) and optimal transport

3 Madelung transform as a symplectomorphism

4 H1-metrics on Diff(M) and information geometry

5 Madelung transform as a Kahler and momentum map


Arnold’s setting for the Euler equation

M — a Riemannian manifold with volume form µv — velocity field of an inviscid incompressible fluid filling MThe classical Euler equation (1757) on v :

∂tv +∇vv = −∇p .Here div v = 0 and v is tangent to ∂M.∇vv is the Riemannian covariant derivative.

Theorem (Arnold 1966)

The Euler equation is the geodesic flow on the group G = Diffµ(M) ofvolume-preserving diffeomorphisms w.r.t. the right-invariant L2-metricE (v) = 1

2

∫M

(v , v)µ (fluid’s kinetic energy).


Application: Other groups and energies

Group Metric Equation

SO(3) 〈ω,Aω〉 Euler topE(3) = SO(3) n R3 quadratic forms Kirchhoff equation for a body in a fluid

SO(n) Manakov’s metrics n-dimensional topDiff(S1) L2 Hopf (or, inviscid Burgers) equation

Diff(S1) H1/2 Constantin-Lax-Majda-type equationVirasoro L2 KdV equationVirasoro H1 Camassa–Holm equation

Virasoro H1 Hunter–Saxton (or Dym) equationDiffµ(M) L2 Euler ideal fluidDiffµ(M) H1 averaged Euler flow

Sympω(M) L2 symplectic fluidDiff(M) L2 EPDiff equation

Diffµ(M) n Vectµ(M)) L2 ⊕ L2 magnetohydrodynamicsC∞(S1, SO(3)) H−1 Heisenberg magnetic chain

Remark These are Hamiltonian systems on g∗ with the quadraticHamiltonian=kinetic energy for the Lie-Poisson bracket.

There are suitable functional-analytic settings of Sobolev (Hs fors > 1 + n/2) and tame Frechet (C∞) spaces.


Exterior geometry of Diffµ(M) ⊂ Diff(M)

Dens(M) — the space of smooth density functions (“probabilitydensities”) on M:

Dens(M) = ρ ∈ C∞(M) | ρ > 0,

∫M

ρµ = 1

Note:Dens(M) = Diff(M)/Diffµ(M),the space of (left) cosets ofDiffµ(M), with the projectionπ : Diff(M)→ Dens(M).

Fibers are π−1(%)= ϕ ∈ Diff(M) | ϕ∗µ = %.

id

µ %

%

uϕ

ϕ

fibre

fibre

horizontal geodesic

geodesic

π

Diff(M)

Dens(M)


Geometry of Diff(M)

Remark Compare “the dimensions” of the fiber and the base:

dim(M) = 1 2 3 ...

Diffµ(M) ≈ Iso(M) ≈ Ham(M) ≈ Vectµ(M) ≈ Vectµ(M)∧ o ∨ ∨

Dens(M) ≈ C∞(M) C∞(M) C∞(M) C∞(M)

Define an L2-metric on Diff(M) by

Gϕ(ϕ, ϕ) =

∫M

|ϕ|2ϕµ.

For a flat M this is a flat metric on Diff(M).

It is right-invariant for the Diffµ(M)-action (but not Diff(M)-action):Gϕ(ϕ, ϕ) = Gϕη(ϕ η, ϕ η) for η ∈ Diffµ(M).


The Euler geodesic property for a flat M

Let a flow (t, x) 7→ g(t, x) be defined by its velocity field v(t, x):

∂tg(t, x) = v(t, g(t, x)), g(0, x) = x .

The chain rule immediately gives the acceleration

∂2ttg(t, x) = (∂tv +∇vv)(t, g(t, x)).

Geodesics on Diff(M) are straight lines, ∂2ttg(t, x) = 0, which is

equivalent to the Burgers equation

∂tv +∇vv = 0.

The Euler equation ∂tv +∇vv = −∇p is equivalent to

∂2ttg(t, x) = −(∇p)(t, g(t, x)),

which means that the acceleration ∂2ttg ⊥L2 Diffµ(M).

Hence the flow g(t, .) is a geodesic on the submanifoldDiffµ(M) ⊂ Diff(M) for the L2-metric.


Geometry of Diff(M) (cont’d)

Theorem (Otto 2000)

The left coset projection π is a Riemannian submersion with respect tothe L2-metric on Diff(M) and the Kantorovich-Wasserstein metric onDens(M).

Definition of the Kantorovich-Wasserstein (L2) metric

The KW distance between µ, ν ∈ Dens(M):

Dist2(µ, ν) := inf∫M

dist2M(x , ϕ(x))µ | ϕ∗µ = ν .

The corresponding Riemannian metric on Dens(M):

Gρ(ρ, ρ) =

∫M

|∇θ|2ρµ, for ρ+ div(ρ∇θ) = 0,

where ρ ∈ C∞0 (M) is a tangent vector to Dens(M) at the point ρµ.


Hamiltonian view on a Riemannian submersion

Let π : P → B be a principal bundle with the structure group G .

A Riemannian submersion π : P → B preserves lengths of horizontaltangent vectors to P.Geodesics on B can be lifted to horizontal geodesics in P, and the lift isunique for a given initial point in P.

For P/G = B the symplectic reduction (over 0-momentum) isT ∗P//G = T ∗B.If P is equipped with a G -invariant Riemannian metric <,>P it inducesthe metric <,>B on the base B.

Proposition The Riemannian submersion of P to the base B, equippedwith the metrics <,>P and <,>B is the result of the symplecticreduction T ∗P//G = T ∗B with metric identification of T and T ∗.


The Euler equation for barotropic fluids

v — velocity field of a compressible fluidfilling Mρ — density of the fluidThe equations of a compressible(barotropic) fluid (or gas dynamics) are∂tv +∇vv +

1

ρ∇P(ρ) = 0

∂tρ+ div(ρv) = 0,

for the pressure function P(ρ) = e′(ρ)ρ2.

Here e(ρ) is the internal energy depending on fluid’s properties.For an ideal gas P(ρ) = C · ρa with a = 5/3 for monatomic gases (argon,krypton) and a = 7/3 for diatomic gases (such as nitrogen, oxygen, andhence approximately for air).


Barotropic fluid as a Newton’s equation

Theorem (Smolentsev, K.-Misiolek-Modin)

The equations of a compressible barotropic fluid with internal energy e(ρ)are equivalent to Newton’s equations ∇ϕϕ = −∇(δU/δρ) ϕ onϕ ∈ Diff(M) for the potential U(ρ) =

∫Me(ρ)ρµ.

Equivalently, this is the Hamiltoniansystem on T ∗Diff(M) with H = K + U,where U(ϕ) = U(ρ) for ρ = det(Dϕ−1).

For v = ∇θ the equation descends toDens(M).

id

µ %

%

uϕ

ϕ

fibre

fibre

horizontal geodesic

geodesic

π

Diff(M)

Dens(M)


Other Newton’s equations in the L2-geometry

— Classical mechanics: U(ρ) =∫MV (x)ρµ for a smooth potential

function V on M =⇒ Burgers equation with potentialv +∇vv +∇V = 0.

— Shallow water equations: quadratic potential U(ρ) = 12

∫Mρ2µ =⇒

v +∇vv +∇ρ = 0

— Fully compressible fluids: potential U(ρ, σ), smaller symmetry group,larger quotient Dens(M)× Ωn(M) =⇒ v +∇vv + ρ−1∇P(ρ, σ) = 0 andthe continuity equations for ρ and σ

— Compressible MHD: smaller symmetry group Diffµ(M) ∩Diffβ0 (M);potential U =

∫Me(ρ)ρµ+ 1

2

∫Mβ ∧ ?β

— Relativistic Burgers equation: for ϕ : [0, 1]×M → M the action is

S(ϕ) = −∫ 1

0

∫M

c2

√1− 1

c2|ϕ|2 µdt


Alternative approach: semidirect products

Mantra: see the continuity equation =⇒ look for a semidirect productgroup.

Example

For the group S = Diff(M) n C∞(M) with product

(ϕ, f ) · (ψ, g) = (ϕ ψ,ϕ∗g + f ), ϕ∗g = g ϕ−1

define the energy function on s

E (v , %) =

∫M

(1

2(v , v) ρ+ ρ e(ρ)

)µ.

Then the Hamiltonian equation on s∗ gives the baropropic fluid withP(ρ) = ρ2e′(ρ).

Similarly for MHD, a rigid body in a fluid, etc. See F.Dolzhansky,

D.Holm, J.E.Marsden, R.Montgomery, T.Ratiu, A.Weinstein, ...


Hydrodynamics and Quantum Mechanics

Theorem (Madelung, von Renesse)

The (non)linear Schrodinger equation

i∂tψ + ∆ψ + Vψ + f (|ψ|2)ψ = 0

on the wave function ψ : M → C on an n-dim manifold M, whereV : M → R and f : R+ → R, is mapped by the transform ψ =

√ρe iθ to

the equations of a barotropic-type fluid∂tv +∇vv + 2∇(V + f (ρ)−

∆√ρ

√ρ

)= 0

∂tρ+ div(ρv) = 0

for v = ∇θ.

It is regarded as a hydrodynamical form of QM.


Madelung and his paper

E. Schrodinger ”An Undulatory Theory of the Mechanics of Atoms andMolecules” Physical Review, Dec. 1926.

E. Madelung ”Quantentheorie in hydrodynamischer Form” Z. Phys. 1927.


Geometry behind Madelung

The Madelung transform Φ : (ρ, θ) 7→ ψ =√ρe iθ. More precisely:

For (ρ, θ) we have∫ρ = 1, ρ > 0 and [θ] = θ + C | ∀C ∈ R, i.e. we

have (ρ, [θ]) ∈ T ∗Dens(M).

For ψ we have ψ 6= 0, ‖ψ‖2L2 = 1 and [ψ] = ψe iα | ∀α ∈ R, i.e. we

have [ψ] ∈ PC∞(M,C \ 0).

Hence,

Definition

The Madelung transform is

Φ : T ∗Dens(M)→ PC∞(M,C \ 0),

where(ρ, [θ]) 7→ [ψ] for ψ =

√ρe iθ .


Madelung transform as a symplectomorphism

Consider the space of normalized densities Dens(M) and projectivizewave functions PC∞(M,C). Now regard (ρ, [θ]) ∈ T ∗Dens(M).

Theorem (K.-Misiolek-Modin)

The Madelung transform Φ : (ρ, [θ]) 7→ [ψ] for ψ =√ρe iθ induces a

symplectomorphism

Φ: T ∗Dens(M)→ PC∞(M,C\0)

for the canonical symplectic structure of T ∗Dens(M) and the naturalFubini-Study symplectic structure of PC∞(M,C).

The Madelung transform is a symplectic submersion to the unit sphere inL2(M,C) (von Renesse).


Thus the Madelung transform maps Hamiltonian systems to Hamiltonianones: the Hamiltonian

H(ψ) =1

2

∫M

|∇ψ|2µ+1

2

∫M

(V |ψ|2 + F (|ψ|2))µ

of the Schrodinger equation on (the projectivization of) C∞(M,C) forF ′ = f is taken to the Hamiltonian

H(ρ, θ) =1

2

∫M

|∇θ|2ρµ+1

2

∫M

|∇ρ|2

ρµ+ 2

∫M

(V ρ+ F (ρ))µ .

on T ∗Dens(M).


H1-metrics on Diff(M) and information geometry

Example

For M = S1 and right-invariant metrics on Diff(S1):the L2-metric E (v) = 1

2

∫v2 dx =⇒ the Burgers equation

vt + 3vvx = 0;

the H1-metric 12

∫v2 + (v ′)2 dx =⇒ the Camassa–Holm equation

vt + 3vvx − vtxx − 2vxvxx − vvxxx + cvxxx = 0;

the H1-metric 12

∫(v ′)2 dx =⇒ the Hunter–Saxton equation

vxxt + 2vxvxx + vvxxx = 0

For any compact M the (degenerate) H1-metric on Diff(M) is given by(v , v) = 1

4

∫M

(div v)2µ and it descends to Dens(M)

The projection π : Diff(M)→ Dens(M) is ϕ 7→ ρ =√|Det(Dϕ)|.


H1-metrics (cont’d)

What is the induced metric on Dens(M)?

Theorem (K., Lenells, Misiolek, Preston 2010)

There exists an isometry Dens(M) ≈ U ⊂ S∞r , r =√µ(M)

(an open part of an inf-dim sphere).

Corollary

– This is the Fisher-Rao metric onDens(M) used in geometric statistics;– It has constant curvature, explicitdescription of geodesics on Dens(M),their integrability.

id

µ %

%

uϕ

ϕ

fibre

fibre

horizontal geodesic

geodesic

π

Diff(M)

Dens(M)


Summary of two metrics on Dens(M) so far

The Kantorovich-Wasserstein metric:

GKWρ (ρ, ρ) =

∫M

|∇θ|2 ρµ for ρ+ div(ρ∇θ) = 0

(depends on the Riemannian structure on M).

The Fisher-Rao metric:

GFRρ (ρ, ρ) =

∫M

( ρρ

)2

ρµ

(independent of the Riemannian structure on M).


Newton’s Equations for H1-metrics

Step aside: the Neumann problemThe classical (finite-dimensional) Neumann problem is a system on thetangent bundle TSn with the Lagrangian given by

L(q, q) =(q, q)

2− q · Aq, where q ∈ Sn ⊂ Rn+1

and where A is a symmetric positive definite (n + 1)× (n + 1) matrix.This system is related to the geodesic flow on the ellipsoid x ·Ax = 1 andis integrable on T ∗Sn.


Neumann problem (cont’d)

For the unit sphere S∞(M) =f |∫Mf 2µ = 1

⊂ C∞(M)∩ L2(M) take

the quadratic potential V (f ) = 12 〈∇f ,∇f 〉L2 = 1

2

∫M|∇f |2µ .

An infinite-dimensional Neumann problem: Find extremalsf : [0, 1]→ S∞(M) minimizing the action functional

L(f , f ) =1

2〈f , f 〉L2 − 1

2〈∇f ,∇f 〉L2 =

1

2

∫M

(f 2 + f ∆f

)µ.

Consider the Fisher information functional on Dens(M):

I (ρ) =1

2

∫M

|∇ρ|2

ρµ,


Newton’s equations on Dens(M) with respect the Fisher-Rao metric andthe Fisher information potential is equivalent to the infinite-dimensionalNeumann problem, with the map ρ 7→ f =

√ρ establishing the

isomorphism.


Madelung as an isometry and a Kahler map

The Fisher-Rao metric on Dens(M) gives rise tothe Fisher-Rao-Sasaki metric on T ∗Dens(M):

GFRSρ,[θ] ((ρ, θ), (ρ, θ)) :=

∫M

(ρ)2

ρµ +

∫M

(θ)2ρµ


The Madelung transform Φ is an isometry (and hence a Kahler map)between the spaces T ∗Dens(M) equipped with the Fisher-Rao-Sasakimetric and PC∞(M,C\0) equipped with the Fubini-Study metric.

The (infinite-dimesional) Fubini–Study metric on PC∞(M,C) is

GFS(ψ, ψ) :=< ψ, ψ >

‖ψ‖2L2

− < ψ, ψ >< ψ, ψ >

‖ψ‖4L2


Madelung transform as a momentum map [D.Fusca]

Definition

The semidirect product group S = Diff(M) n C∞(M) 3 (ϕ, a) acts onthe space C∞(M,C) 3 ψ of wave functions as follows:

(ϕ, a) ψ =√|Det(Dϕ−1)| e−ia/2(ψ ϕ−1).

(ψ is pushed forward by a diffeomorphism ϕ as a complex-valuedhalf-density, followed by a pointwise phase adjustment by e−ia/2).

This action– descends to the space of cosets [ψ] ∈ PC∞(M,C),– is Hamiltonian.


Madelung transform as a momentum map (cont’d)

Theorem (D.Fusca 2017)

The momentum map

M : C∞(M,C)→ s∗ = Ω1(M)×Dens(M)

for the group S-action on the space of wave functions C∞(M,C) given by

ψ 7→ (m, ρ) =(2 Im(ψ dψ), ψψ

)is the inverse of the Madelung transform (ρ, θ) 7→ ψ =

√ρe iθ, where

ρ > 0, in the following sense: if ψ =√ρe iθ then M(ψ) = (ρ dθ, ρ).

Remark This might resolve T.C.Wallstrom’s critique (1994) ofinequivalence between the Schrodinger equation and its hydrodynamicform, requiring a quantization condition around zeros of ψ:consider the map ψ 7→ (m, ρ) for m = ρ dθ rather than ψ 7→ (dθ, ρ).


Madelung and bouncing droplets?

Corollary: The Madelung transform provides a Kahler map, a strongconnection of QM and hydrodynamics.

Maybe this tighterMadelung connectioncould explain similarity ofbouncing droplets andQM?


Beautiful pictures of pilot-wave hydrodynamics

(the image courtesy of E.Fort, FYFD website, N.Sharp, see also Y.Couderet al, J.Bush et al.)


Several references

E.Madelung 1927 Zeitschrift Phys

V.Arnold 1966 Ann Inst Fourier

N.Smolentsev 1979 Siberian Math J

J.E.Marsden, T.Ratiu, A.Weinstein 1984 Contemp Math

T.C.Wallstrom 1994 Phys. Rev. A

V.Arnold, B.Khesin 1998 Springer

Y.Couder, S.Protiere, E.Fort, A.Boudaoud 2005 Nature

M-K.von Renesse 2012 Canad Math Bull

B.Khesin, J.Lenells, G.Misiolek, S.C.Preston 2013 GAFA

D.Fusca 2017 J Geom Mech

B.Khesin, G.Misiolek, K.Modin 2019 ARMA and arXiv:2001.01143


THANK YOU!


Documents

Boris Khesin - UNIGE · 2021. 4. 25. · Virasoro H_1 Hunter{Saxton (or Dym) equation Di (M) L2 Euler ideal uid Di (M) H1 averaged Euler ow Symp! ... There are suitable functional-analytic