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

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 its scientific activities ranges from pure to applied mathematics and includes statistics, theoretical computer science, mathemat­ ical methods in biology and life sciences, and mathematical and theoretical physics. The CRM also promotes collaboration be­ tween mathematicians and industry. It is subsidized by the Nat­ ural Sciences and Engineering Research Council of Canada, the Fonds FCAR of the Province of Quebec, the Canadian Institute for Advanced Research, and has private endowments. Current activities, fellowships, and annual reports can be found on the CRM Web page at www. CRM. UMontreal. CA. 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.

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This book contains lectures on fluid dynamics given in Montreal, while the author held the Aisenstadt Chair, as well as other lectures on the same topics. It begins by explaining the motivation and reviewing the classical theory, but in a manner different from textbook discussions. Among the topics discussed are the conservation laws and Euler equations, and a method for finding their canonical structure; C. Eckart's Lagrangian and a relativistic generalization for vortex-free motion; nonvanishing vorticity and the Clebsch parameterization for the velocity vector. Jackiw then discusses some specific models for nonrelativistic and relativistic fluid mechanics in spatial dimensions greater than one, including the Chaplygin gas (whose negative pressure is inversely proportional to density), and the scalar Born-Infeld model. He shows how both the Chaplygin gas and the Born-Infeld model devolve from the parameterization-invariant Nambu-Goto action, when specific parameterization is made. As in particle physics, Jackiw shows, fluid mechanics enhanced by supersymmetry. For one-dimensional cases, the models discussed above are completely integrable, and Jackiw gives the General solution of the Chaplygin gas and the Born-Infeld model on a line. General solution of the Nambu- Goto theory for a 1-brane (string) in two spatial dimensions. Jackiw discusses the need for a non-Abelian fluid mechanics and proposes a Lagrangian, which involves a non-Abelian auxiliary field whose Chern-Simons density should be a total derivative. The generalization to magnetohydrodynamics, which results from including a dynamical non-Abelian gauge field, reduces in the Abelian limit to conventional magnetohydrodynamics.

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This book contains lectures on fluid dynamics given in Montreal, while the author held the Aisenstadt Chair, as well as other lectures on the same topics. It begins by explaining the motivation and reviewing the classical theory, but in a manner different from textbook discussions. Among the topics discussed are the conservation laws and Euler equations, and a method for finding their canonical structure; C. Eckart's Lagrangian and a relativistic generalization for vortex-free motion; nonvanishing vorticity and the Clebsch parameterization for the velocity vector. Jackiw then discusses some specific models for nonrelativistic and relativistic fluid mechanics in spatial dimensions greater than one, including the Chaplygin gas (whose negative pressure is inversely proportional to density), and the scalar Born-Infeld model. He shows how both the Chaplygin gas and the Born-Infeld model devolve from the parameterization-invariant Nambu-Goto action, when specific parameterization is made. As in particle physics, Jackiw shows, fluid mechanics enhanced by supersymmetry. For one-dimensional cases, the models discussed above are completely integrable, and Jackiw gives the General solution of the Chaplygin gas and the Born-Infeld model on a line. General solution of the Nambu- Goto theory for a 1-brane (string) in two spatial dimensions. Jackiw discusses the need for a non-Abelian fluid mechanics and proposes a Lagrangian, which involves a non-Abelian auxiliary field whose Chern-Simons density should be a total derivative. The generalization to magnetohydrodynamics, which results from including a dynamical non-Abelian gauge field, reduces in the Abelian limit to conventional magnetohydrodynamics.
Content:
Front Matter....Pages i-xvi
Introduction....Pages 1-2
Classical Equations....Pages 3-21
Specific Models....Pages 23-39
Common Ancestry: The Nambu—Goto Action....Pages 41-47
Supersymmetric Generalization....Pages 49-60
One-Dimensional Case....Pages 61-80
Toward a Non-Abelian Fluid Mechanics....Pages 81-89
Noncommutative Fluid Mechanics....Pages 91-99
Back Matter....Pages 101-114

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This book contains lectures on fluid dynamics given in Montreal, while the author held the Aisenstadt Chair, as well as other lectures on the same topics. It begins by explaining the motivation and reviewing the classical theory, but in a manner different from textbook discussions. Among the topics discussed are the conservation laws and Euler equations, and a method for finding their canonical structure; C. Eckart's Lagrangian and a relativistic generalization for vortex-free motion; nonvanishing vorticity and the Clebsch parameterization for the velocity vector. Jackiw then discusses some specific models for nonrelativistic and relativistic fluid mechanics in spatial dimensions greater than one, including the Chaplygin gas (whose negative pressure is inversely proportional to density), and the scalar Born-Infeld model. He shows how both the Chaplygin gas and the Born-Infeld model devolve from the parameterization-invariant Nambu-Goto action, when specific parameterization is made. As in particle physics, Jackiw shows, fluid mechanics enhanced by supersymmetry. For one-dimensional cases, the models discussed above are completely integrable, and Jackiw gives the General solution of the Chaplygin gas and the Born-Infeld model on a line. General solution of the Nambu- Goto theory for a 1-brane (string) in two spatial dimensions. Jackiw discusses the need for a non-Abelian fluid mechanics and proposes a Lagrangian, which involves a non-Abelian auxiliary field whose Chern-Simons density should be a total derivative. The generalization to magnetohydrodynamics, which results from including a dynamical non-Abelian gauge field, reduces in the Abelian limit to conventional magnetohydrodynamics.
Content:
Front Matter....Pages i-xvi
Introduction....Pages 1-2
Classical Equations....Pages 3-21
Specific Models....Pages 23-39
Common Ancestry: The Nambu—Goto Action....Pages 41-47
Supersymmetric Generalization....Pages 49-60
One-Dimensional Case....Pages 61-80
Toward a Non-Abelian Fluid Mechanics....Pages 81-89
Noncommutative Fluid Mechanics....Pages 91-99
Back Matter....Pages 101-114
....