boris khesin - unige · 2021. 4. 25. · virasoro h_1 hunter{saxton (or dym) equation di (m) l2...
TRANSCRIPT
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
Boris Khesin Hamiltonian geometry of compressible fluids 2 / 30
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).
Boris Khesin Hamiltonian geometry of compressible fluids 3 / 30
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.
Boris Khesin Hamiltonian geometry of compressible fluids 4 / 30
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)
Boris Khesin Hamiltonian geometry of compressible fluids 5 / 30
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).
Boris Khesin Hamiltonian geometry of compressible fluids 6 / 30
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.
Boris Khesin Hamiltonian geometry of compressible fluids 7 / 30
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.
Boris Khesin Hamiltonian geometry of compressible fluids 8 / 30
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 ∗.
Boris Khesin Hamiltonian geometry of compressible fluids 9 / 30
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).
Boris Khesin Hamiltonian geometry of compressible fluids 10 / 30
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)
Boris Khesin Hamiltonian geometry of compressible fluids 11 / 30
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
Boris Khesin Hamiltonian geometry of compressible fluids 12 / 30
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, ...
Boris Khesin Hamiltonian geometry of compressible fluids 13 / 30
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.
Boris Khesin Hamiltonian geometry of compressible fluids 14 / 30
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.
Boris Khesin Hamiltonian geometry of compressible fluids 15 / 30
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 .
Boris Khesin Hamiltonian geometry of compressible fluids 16 / 30
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).
Boris Khesin Hamiltonian geometry of compressible fluids 17 / 30
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).
Boris Khesin Hamiltonian geometry of compressible fluids 18 / 30
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)|.
Boris Khesin Hamiltonian geometry of compressible fluids 19 / 30
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)
Boris Khesin Hamiltonian geometry of compressible fluids 20 / 30
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).
Boris Khesin Hamiltonian geometry of compressible fluids 21 / 30
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.
Boris Khesin Hamiltonian geometry of compressible fluids 22 / 30
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,
Theorem (K.-Misiolek-Modin)
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.
Boris Khesin Hamiltonian geometry of compressible fluids 23 / 30
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
Theorem (K.-Misiolek-Modin)
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
Boris Khesin Hamiltonian geometry of compressible fluids 24 / 30
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.
Boris Khesin Hamiltonian geometry of compressible fluids 25 / 30
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).
Boris Khesin Hamiltonian geometry of compressible fluids 26 / 30
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?
Boris Khesin Hamiltonian geometry of compressible fluids 27 / 30
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.)
Boris Khesin Hamiltonian geometry of compressible fluids 28 / 30
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
Boris Khesin Hamiltonian geometry of compressible fluids 29 / 30
THANK YOU!
Boris Khesin Hamiltonian geometry of compressible fluids 30 / 30