Lectures on Fluid Dynamics: A Particle Theorist’s View of Supersymmetric, Non-Abelian, Noncommutative Fluid Mechanics and d-Branes

120 Pages·2002·8.86 MB·English

by  Roman Jackiw (auth.)

CRM Series in Mathematical Physics
Springer Science+Business Media, LLC
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Roman Jackiw
Lectures on
Fluid Dynamics
A Particle Theorist's View of
Supersymmetric, Non-Abelian,
Noncommutative Fluid Mechanics
and d-Branes
Springer
Roman Jackiw
Center for Theoretical Physics
Massachusetts Institute of Technology
Cambridge, MA 02139
USA
[email protected]
Editorial Board
Joel S. Feldman Duong H. Phong
Department of Mathematics Department of Mathematics
University of British Columbia Columbia University
Vancouver, British Columbia V6T 1Z2 New York, NY 10027-0029
Canada USA
[email protected] [email protected]
Yvan Saint-Aubin Luc Vinet
Departement de Mathematiques Departement de Mathematiques
et de Statistique et de Statistique
Universite de Montreal CRM, Universite de Montreal
c.P. 6128, Succursale Centre-ville C.P. 6128, Succursale Centre-ville
Montreal, Quebec H3C 317 Montreal, Quebec H3C 317
Canada Canada
[email protected] [email protected]
Library of Congress Cataloging-in-Publication Data
Jackiw, Roman W.
Lectures on fluid dynamics : a partide theorist's view of supersyrnmetric, non-Abelian,
Noncornmutative fluid mechanics and d-branes I Roman Jackiw.
p. cm. - (The CRM series in mathematical physics)
IncJudes bibliographical references and index.
ISBN 978-1-4419-2993-8 ISBN 978-1-4757-3665-6 (eBook)
DOI 10.1007/978-1-4757-3665-6
1. Fluid dynamics. 2. Supersymmetry. 3. Fluid mechanics. 4. String models. 1. Title.
II. CRM series in mathematical physics.
QA911 .133 2002
532" .05-dc21 2001059794
Printed on acid-free paper.
© 2002 Springer Science+Business Media New York
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Series Preface
The Centre de recherches mathematiques (CRM) was created
in 1968 by the Universite de Montreal to promote research in
the mathematical sciences. It is now a national institute that
hosts several groups and holds special theme years, summer
schools, workshops, and a post-doctoral program. The focus of
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The CRM Series in Mathematical Physics includes mono
graphs, lecture notes, and proceedings based on research pur
sued and on events held at the CRM.
Yvan Saint-Aubin
Montreal
Preface
This monograph is derived from a series of six lectures which
were given at the Centre de recherches mathematiques (CRM)
in Montreal, in March and June 2000, while the author was
holder of the Aisenstadt Chair.
Precis
During the March 2000 meeting of the Workshop on Strings, Du
ality, and Geometry in Montreal, Canada, three lectures were
delivered on topics in fluid mechanics, while the author was
holder of the Aisenstadt Chair. Three more lectures were pre
sented in June 2000, during the Montreal Workshop on Inte
grable Models in Condensed Matter and N onequilibrium Phys
ics. The following are brief descriptive remarks on the content
of these lectures.
1. Introduction. The motivation for the research is explained.
2. Classical Equations. The classical theory is reviewed, but
in a manner different from text book discussions.
2.1. Equations of motion. Summary of conservation and Eu
ler equations.
2.2. A word on canonical formulations. An advertisement of
the method for finding the canonical structure for the
above (developed with L.D. Faddeev).
2.3. The irrotational case. C. Eckart's Lagrangian and a rel
ativistic generalization for vortex-free motion.
2.4. Nonvanishing vorticity and the Clebsch parametrization.
In the presence of vorticity, the velocity Chern-Simons
term (kinetic helicity) provides an obstruction to the
construction of a Lagrangian for the motion. C.C. Lin's
method overcomes this obstruction, and leads to the
Clebsch parametrization for the velocity vector.
2.5. Some further remarks on the Clebsch parametrization.
The properties and peculiarities of this presentation for
a 3-vector.
x Precis
3. Specific Models. Nonrelativistic and relativistic fluid me
chanics in spatial dimensions greater than one.
3.1. Galileo-invariant nonrelativistic model. The Chaplygin
gas [negative pressure, inversely proportional to density]
is studied, selected solutions are presented, and unex
pected symmetries are identified.
3.2. Lorentz-invariant relativistic model. The scalar Born
Infeld model is found to be the relativistic generalization
of the Chaplygin gas, and shares with it unexpected
symmetries.
3.3. Some remarks on relativistic fluid mechanics. The dy
namics for isentropic relativistic fluids is given a La
grangian formulation, and the Born-Infeld model is fit
ted into that framework.
4. Common Ancestry: The N ambu-Goto Action. Both
the Chaplygin gas and the Born-Infeld model devolve from
the parametrization-invariant Nambu-Goto action, when
specific parametrization is made.
4.1. Light-cone parametrization. Chaplygin gas is derived.
4.2. Cartesian parametrization. The Born-Infeld model is
derived.
4.3. Hodographic transformation. Chaplygin gas is derived
(again).
4.4. Interrelations. The Chaplygin gas and the Born-Infeld
model are related because: (1) the former is the nonrel
ativistic limit of the latter; and (2) both descend from
the same Na mbu-Goto action.
5. Supersymmetric Generalization. Fluid mechanics en
hanced by supersymmetry.
5.1. Chaplygin gas with Grassmann variables. Vorticity is
parametrized by Grassmann variables, which act like
Gaussian potentials of the Clebsch parametrization.
Precis xi
5.2. Supersymmetry. Supercharges, transformations gener
ated by them, and their algebra.
5.3. Supermembrane connection. Supermembrane Lagrang
ian in three spatial dimensions.
5.4. Hodographic transformation. Supersymmetric Chaply
gin gas in two spatial dimensions is derived.
5.5. Light-cone parametrization. Supersymmetric Chaplygin
gas in two spatial dimensions is derived (again).
5.6. Further consequences of the supermembrane connection.
Hidden symmetries of the supersymmetric model.
6. One-Dimensional Case. The previous models in one spa
tial dimension are completely integrable.
6.1. Specific solutions for the Chaplygin gas on a line. Some
special solutions are presented; an infinite number of
constants of motion is identified; Riemann coordinates
are introduced and the fluid equations, as well as con
stants of motion, are expressed in terms of them.
6.2. Aside on the integrability of the cubic potential in one di
mension. The one-dimensional problem with pressure ex
(density)3 possesses the SO(2, 1) "Schrodinger symme
try" and the equations of motion, in Riemann form,
become free.
6.3. General solution for the Chaplygin gas on a line. Solu
tion obtained by linearization.
6.4. Born-Infeld model on a line. When formulated in terms
of its Riemann coordinates, it becomes trivially equiva
lent to the Chaplygin gas.
6.5. General solution of the Nambu-Goto theory for a
(d = 1) -brane (string) in two spatial dimensions (on a
plane). The explicit string solution is transformed by a
hodographic transformation to the Chaplygin gas solu
tion, and a relation is established between this solution
and the one found by linearization.
Precis
Xll
6.6. Supersymmetric Chaplygin gas on a line. The supersym
metric lineal Chaplygin gas, which descends from a su
perstring on a plane, is completely integrated, and the
algebra of its conserved charges is described.
1. Toward a Non-Abelian Fluid Mechanics. Motivation
for this theory is given.
7.1. Proposal for non-Abelian fluid mechanics. A Lagrang
ian is proposed; it involves a non-Abelian auxiliary field
whose Chern-Simons density should be a total deriva
tive.
7.2. Non-Abelian Clebsch parametrization (or, casting the
non-Abelian Chern-Simons density into total derivative
form). Total derivative form for the non-Abelian Chern
Simons density is found, thereby generalizing the Abe
lian Clebsch parametrization, which achieves a total de
rivative form for the Abelian density.
7.3. Proposal for non-Abelian magnetohydrodynamics. Our
proposal, which generalizes the proposal in Section 7.1
to include a dynamical non-Abelian gauge field, reduces
in the Abelian limit to conventional magnetohydrody
namlCS.
8. Noncommutative Fluid Mechanics. Noncommutative
field theories are widely studied owing to their relation with
string theory. But it is possible to present these ideas in the
more mundane setting of fluid mechanics in the presence of
a strong magnetic field.
8.1 Noncommuting particles. As an introduction to the fluid
mechanical construction, we review the problem of a
charged particle in a strong magnetic field and show that,
after projection of the motion onto the lowest Landau
level, the particle's coordinates cease to (Poisson) com
mute.