### Mathematical Physics III - Integrable Systems of Classical Mechanics. Lecture Notes

### Matteo Petrera

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ISBN 978-3-8325-3950-4

190 pages, year of publication: 2015

price: 27.00 €

**Rezension:** "This book is a scholar book which duplicates exactly the lecture of the teacher. As pointet out by the author, this ist not a research book (or a monograph) and its main purpose ist to be a tool to help the student in his (or her) learning process. .... Given its price, this is a bargain. ..." Guy Jumarie (Montréal) In: Zentralblatt MATH, 1330.00029 (2016)

**Inhalt:** These Lecture Notes provide an introduction to the modern theory of classical finite-dimensional integrable systems.

The first chapter focuses on some classical topics of differential geometry. This should help the reader to get acquainted with the required language of smooth manifolds, Lie groups and Lie algebras.

The second chapter is devoted to Poisson and symplectic geometry with special emphasis on the construction of finite-dimensional Hamiltonian systems. Multi-Hamiltonian systems are also considered.

In the third chapter the classical theory of Arnold-Liouville integrability is presented, while chapter four is devoted to a general overview of the modern theory of integrability. Among the topics covered are: Lie-Poisson structures, Lax formalism, double Lie algebras, R-brackets, Adler-Kostant-Symes scheme, Lie bialgebras, r-brackets.

Some examples (Toda system, Garnier system, Gaudin system, Lagrange top) are presented in chapter five. They provide a concrete illustration of the theoretical part.

Finally, the last chapter is devoted to a short overview of the problem of integrable discretization.

Please see as well:

Vol. I:Dynamical Systems and Classical Mechanics

Vol. II:Classical Statistical Mechanics