Dynamics of vortices on surfaces

Jair Koiller

FGV/RJ and AGIMB

Santiago de Compostela

23 Junio 2008

OUTLINE (joint work with S.Boatto)

◊ Historical Remarks and backgound

C.C.Lin’s theorems

Ideal hydrodynamics on surfaces

◊ Geometry: Green’s function of Laplace Operator on closed surfaces

◊ Mechanics: dynamics of vortices on surfaces

generalization of C.C.Lin’s theorems

proof of Kimura’s conjecture: vortex dipole describes geodesics

◊ Control: applications in physics, engineering and biologyControl: applications in physics, engineering and biology

◊ Some research suggestions

Vortex pair on a triaxial ellipsoid and Liouville surfaces

Metric symplectic 2-form and expansion of B

Kahler manifolds

USD 106 arXiv: 0802.4313v1

SUMMARY

Brief History of Vortex Dynamics

Euler (1757) E225 E226 E227

Helmholtz (1858) Wirbelbewegungen

Kirchhoff (1876) Vorlesungen

C.C.(Chia-Chiao) Lin (1941) Lin1 Lin2

Recent events/publications

150 years of Helmholtz’s paper Helmholtz 2008

250 years of Euler’s paper

Euler2007(1) Euler2007(2) Euler2007(3) Euler2007(4)

IUTAM http://conf2006.rcd.ru/

K. Moffat’s video

Physica D 237:1—12 (2008) (Tudor Ratiu, editor)

From Descartes’ Celestial Vortices to String Theory

sun in the midst of its

own vortex, packed within

a three-dimensional system

of contiguous vortices

Principia Philosophiae (1644)

Kelvin: On vortex Atoms

Witten

C.C.Lin’s theorems (1941) Lin1 Lin2

Ideal hydrodynamics on surfaces

In order to get the stream function we must solve Poisson’s equation

=

Vorticity:

“sinews and muscles”

Laplace Beltrami operator on S

“ vortex patch dynamics”

Statistical mechanics:

Vorticity concentrates

Desingularization:

Core energy trick

Flucher/Gustafsson

Flucher/Gustafsson

Question:

How to geometrize

The core energy

argument to a

curved surface?

Answer: properties

Of Green’s function

Of Laplace-Beltrami

operator

Green function of the

Laplace-Beltrami Operator

References

KateOkikiolu Kate2 Kate3 womeninmath katepageJeanSteiner Jean2 Jean3

Robin’s function

R(so) = lim G(s,so) – 1/2 log d(s,so) s so

Related themes:

Positive mass conjecture (Shoen and Yau)

Yamabe problem

Zeta functions

Reference: Jean Steiner thesis

Robin’s function

Self motion of a vortex: dso/dt = sgrad R(so)

Proof: core energy argument, geometrized!

Motion of a marker particle:

ds/dt = sgrad G(s,so(t)) 1 ½ dof

R(so) = lim G(s,so) – 1/2 log d(s,so) s so

Extension to vortices with mass is straightforward

Vortices on surfaces: generalization of C.C.Lin’s theorems

Under a conformal change of metric …

Proof

(Okikiolu)

When the total vorticity vanishes: complex structure sufficies

… zero total vorticity, and surface diffeomorphic to the sphere

Taking advantage of the fact that the Green function on the plane is log z

… zero total vorticity, and surface diffeomorphic to the sphere

In more explicit fashion…

Proof of Kimura’s conjecture “a vortex doublet follows a geodesic”

Kimura (1999)

Proof:

If d(s1,s2) = O() then B is O(2)

It is enough to show: T*S <——> S x S center-chord

s–s

( canonical of T*S + perturbation )

NEW!

This function B seems not been used by

The geometric function theory community.

We call it…

Batman’s function

B(s1,s2) = ½ ( R(s1) + R(s2)) - [ G(s1,s2) – (1/2) log(s1,s2) ]

SHORT(ALTERNATIVE) PROOF OF KIMURA’s CONJECTURE

Vortex pair on a surface with symmetry axis k

J seems to be not related to

Clairault.

WHY?

Possible explanation

Some research directions

◊ Control: applications in physics and Control: applications in physics and engineeringengineering

◊ Vortex pair on a triaxial ellipsoid

◊ Metric-symplectic 2-form and expansion of B

◊ Kahler manifolds: “vortons”

◊ prize: $ 106

Research direction 1 - triaxial ellipsoid (Castro Urdiales)

◊ Conformal factor for surfaces of revolution (Gauss)

kindergarden

◊ Jacobi geodesics on the ellipsoid (1838)

Jacobi (1838) Vorlesungen Perelomov (2002) Tabachnikov (2002)

◊ Conformal mapping of the ellipsoid to the plane or sphere

Snyder Schering (1858) 1,2,3 Craig (1880) Muller (1991)

“tour-de-force” using sphero-conical coordinates

◊ Liouville surfaces

Kihohara Miller Bolsinov

Jaocobi used confocal quadrics coordinates to separate the

Kinetic energy. We propose using, alternatively,

Sphero-conical coordinates (also works)

Research direction 2: a lemma needed for the vortex pair problem

On Feb 6, 2008 7:12 AM, Jair Koiller <Jair.Koiller@fgv.br> wrote:

Dear Alan,

gretings from Rio, Carnival just ended, and although wet all the time, I took Luisa to same street parties...

I am finishing with Stefanella that vortex paper, we are thinking of the ellipsoid example for a sequel. And for a sequel, we are writing the vortex equation near the diagonal of SxS as a perturbation of the geodesic equations. I would like some help for the following technical point, involving a deformation of the canonical 2-form in T*S (S is 2-dimensional).

cheers,

Jair

From: hopfish@gmail.com on behalf of Alan WeinsteinSent: Wed 2/6/2008 1:28 PMTo: Jair KoillerSubject: Re: question on a deformation of the canonical 2-form ...

Dear Jair,

Your construction looks like what I did with Claudio Emmrich in our paper on the geometry of Fedosov's quantization. (You can skip a step in your sequence by using the Poisson tensor to go directly from T* to T, rather than in two steps using Leg and J.) There, we used any connection rather than a metric one to construct the geodesics. But we did not construct the deformation term explicitly; it would be nice to know it, especially along the zero section, since it gives an invariant of the metric,symplectic pair.

This also might be related to John Jacob's thesis. If the deformation term vanishes, the images of the fibres of T* must be lagrangian (to some order in epsilon), which means that the geodesic symmetries are symplectic (up to the same order). This would suggest that the deformation term involves in some explicit way the non-invariance of the symplectic form under geodesic symmetries.

I hope that you can check this out!

Best regards,Alan

ps in T*S —> us = Leg-1 ps in TS —> vs = J(us) in TS —> (s1 , s2) in

SxS s1 = exp( - vs), s2 = exp(- vs) , J = /2 rotation

Let be the area form of S. Cmpute the pull back to T*S of

= (s1) - (s2), a symplectic form in SxS,

It is of the form ( o + 2 1 + ...)

Easy to show: o = canonical 2-form of T*S

Can you get the deformation term 1 ?Is it always zero? If not, when does it vanish?

Metric-symplectic 2 form

Research direction 3: Batman’s function

From: boatto@impa.br [mailto:boatto@impa.br]Sent: Mon 6/9/2008 11:36 AM

To: Jair KoillerSubject: Rick Schoen

Ola Jair,

   recebeu os arquivos que enviei para voce?Sabado foi tao corrido no congresso...falei brevementetambem com Rick Schoen (Stanford) que disse que esta

interessado nopreprint e me pediu de envia-lo...acho bom nao e'?

   Abracos,

        Stefanella

Batman’s function

Task: Expand B it in TS center-chord

coordinates, will be O(2)

Are there geometric objets appearing?

B(s1,s2) = ½ ( R(s1) + R(s2)) - [ G(s1,s2) – (1/2) log(s1,s2) ]

Research direction 4: Higher dimensional extension

“Vortons” on Kahler manifolds

[ Vortons in 3d/odd dimensions (not very successful)

The vorton method (E.Novikov, not S.Novikov) ]

Alan Weinstein called our attention: everything here makes sense on a compact Kahler manifold, replacing every two dimensional objects by their higher dimensional analogues. In particular, Calabi-Yau manifolds are very fashionable objects nowadays. The Laplace operator can also be replaced by the Paneitz operator Pn .

Is this formulation related to what theoretical physicists are doing? John Baez page

“Sir Michael Berry proposes that there exists a classical dynamical system, asymmetric with respect to time reversal, the lengths of whose periodic orbits correspond to the rational primes, and whose quantum-mechanical analog has a Hamiltonian with zeros equal to the imaginary parts of the nontrivial zeros of the zeta function. The search for such a dynamical system is one approach to proving the Riemann hypothesis” (Daniel Bump).

http://math.stanford.edu/~bump/

Berry1 Berry2

Research 5 (Homework)

Prove Riemann hypothesis .

Idea: use a quantum vortex problem

(no es cachondeo!)

Hilbert-Polya conjecture

Physics/number theory QM/RH RH