Dynamics of vortices on surfaces

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Dynamics of vortices on surfaces Jair Koiller
FGV/RJ and AGIMB Santiago de
Compostela 23 Junio 2008
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OUTLINE (joint work with S.Boatto) ? Historical
Remarks and backgound C.C.Lins theorems
Ideal hydrodynamics on surfaces ? Geometry
Greens function of Laplace Operator on closed
surfaces ? Mechanics dynamics of vortices on
surfaces generalization of C.C.Lins
theorems proof of Kimuras conjecture
vortex dipole describes geodesics ? Control
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
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SUMMARY
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Brief History of Vortex Dynamics

Euler (1757)
E225
E226 E227 Helmholtz
(1858) Wirbelbewegungen

Kirchhoff (1876)

Vorlesungen C.C.(Chia-Chiao) Lin (1941)
Lin1 Lin2
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Recent events/publications 150 years of
Helmholtzs paper Helmholtz 2008 250 years of
Eulers paper Euler2007(1) Euler2007(2)
Euler2007(3) Euler2007(4) IUTAM
http//conf2006.rcd.ru/ K. Moffats
video Physica D 237112 (2008) (Tudor Ratiu,
editor)
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(No Transcript)
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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
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C.C.Lins theorems (1941) Lin1 Lin2
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Ideal hydrodynamics on surfaces
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In order to get the stream function we must solve
Poissons equation w D y
Vorticity sinews and muscles Laplace
Beltrami operator on S
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vortex patch dynamics Statistical
mechanics Vorticity concentrates
Desingularization Core energy trick Flucher/Gusta
fsson
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Flucher/Gustafsson
Question How to geometrize The core
energy argument to a curved surface? Answer
properties Of Greens function Of
Laplace-Beltrami operator
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Green function of the Laplace-Beltrami Operator
References KateOkikiolu Kate2 Kate3
womeninmath katepage JeanSteiner Jean2
Jean3
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Robins function
R(so) lim G(s,so) 1/2p
log d(s,so) s ? so
Related themes Positive mass conjecture (Shoen
and Yau) Yamabe problem Zeta functions Reference
Jean Steiner thesis
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Robins 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) 1/2p
log d(s,so) s ? so
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Vortices on surfaces generalization of
C.C.Lins theorems
Extension to vortices with mass is
straightforward
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Under a conformal change of metric
Proof
(Okikiolu)
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When the total vorticity vanishes complex
structure sufficies
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zero total vorticity, and surface
diffeomorphic to the sphere
Taking advantage of the fact that the Green
function on the plane is log z
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zero total vorticity, and surface
diffeomorphic to the sphere
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In more explicit fashion
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Proof of Kimuras conjecture
a vortex doublet follows a
geodesic Kimura (1999)
NEW!
Proof If d(s1,s2) O(e) then B is
O(e2) It is enough to show TS ltgt S x S
center-chord W k w(s1) k w(s2) W k (
canonical of TS perturbation )

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This function B seems not been used by The
geometric function theory community. We call
it Batmans function
B(s1,s2) ½ ( R(s1) R(s2)) - G(s1,s2)
(1/2p) log(s1,s2)
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SHORT(ALTERNATIVE) PROOF OF KIMURAs CONJECTURE
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Vortex pair on a surface with symmetry axis k
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J seems to be not related to Clairault. WHY?
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Possible explanation
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Some research directions ? Control
applications in physics and engineering ? Vortex
pair on a triaxial ellipsoid ?
Metric-symplectic 2-form and expansion of B ?
Kahler manifolds vortons ? prize 106
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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
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Jaocobi used confocal quadrics coordinates to
separate the Kinetic energy. We propose using,
alternatively, Sphero-conical coordinates (also
works)
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Research direction 2 a lemma needed for the
vortex pair problem On Feb 6, 2008 712 AM, Jair
Koiller ltJair.Koiller_at_fgv.brgt 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 TS (S is
2-dimensional). cheers, Jair
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From hopfish_at_gmail.com on behalf of Alan
Weinstein Sent Wed 2/6/2008 128 PM To Jair
Koiller Subject 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
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ps in TS gt us Leg-1 ps in TS gt vs
J(us) in TS
gt (s1 , s2) in
SxS s1 exp( - e vs), s2
exp(- e vs) , J p/2
rotation Let w be the area form of S. Cmpute
the pull back to TS of W
w(s1) - w(s2), a symplectic form in SxS, It is
of the form e (
Wo e2 W1 ...) Easy to show Wo
canonical 2-form of TS Can you get
the deformation term W1 ? Is it always zero?
If not, when does it vanish? Metric-symplectic 2
form
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Research direction 3 Batmans function From
boatto_at_impa.br mailtoboatto_at_impa.brSent Mon
6/9/2008 1136 AMTo 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
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Batmans function Task Expand B it in
TS center-chord coordinates, will be O(e2) Are
there geometric objets appearing?
B(s1,s2) ½ ( R(s1) R(s2)) - G(s1,s2)
(1/2p) log(s1,s2)
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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
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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.e
du/bump/ Berry1 Berry2
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Research 5 (Homework) Prove Riemann hypothesis
. Idea use a quantum vortex problem (no es
cachondeo!) Hilbert-Polya conjecture Physics/n
umber theory QM/RH RH