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Topological and Symplectic methods in singular Lagrangian systems
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Topological and Symplectic methods in singular Lagrangian systems
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Academic year 2018/2019
- Teacher
- Prof. Alessandro Portaluri (Lecturer)
- Type
- Basic
- Delivery
- Formal authority
- Language
- Italian
- Attendance
- Obligatory
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Sommario del corso
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Course objectives
The aim of this course is to provide some basic notions in Symplectic Topology and in Calculus of Variations in order to penetrate the intricate dynamics of singular Lagrangian systems. More in detail, the course will be divided into three main parts.
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Course delivery
TERM: September-December
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Learning assessment methods
METHOD OF ASSESSMENT: Presentation
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Program
In the first part we will introduce we'll discuss the weak sequential lower semi-continuity, coercivity, convexity of some classical singular Lagrangian systems. In particular various classical examples (central force problems, intermolecular force problem: the Lennard-Jones potential and the N-body problem), will be treated in details in order to highlight different phenomena and drawbacks. The main applications are the classical results of Gordon on conservative dynamical systems involving strong and weak forces.
The second part is essentially devoted to introduce the students to the notion of Conley-Zehnder index (through the Maslov intersection theory). Some final applications will be given to the classification of the solutions of the Kepler problem through the Conley-Zehnder index and to provide some sufficient instability and hyperbolicity criteria for closed geodesics and more generally for periodic orbits of (non)autonomous Lagrangian systems.Suggested readings and bibliography
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BOOKS TO BE USED:
H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations
A. Ambrosetti - V. Coti-Zelati: Periodic Solutions of Singular Lagrangian Systems
A. Ambrosetti - G. Prodi: Primer in Nonlinear Analysis
Y. Long: Index Theory for Symplectic Paths with Applications
E. Fadell - S. Husseini: Geometry and Topology of Configuration spaces
PAPERS
from: Gordon, Palais, Portaluri, Rabinowitz, Robbin, Salamon, Smale, Zehnder
TOPICS/CHAPTERS TO BE COVERED:
Direct method in Calculus of Variations. Lagrangian action functional, Fréchet and Gateaux differentiability of the Lagrangian action functional. Variational formulation of 1D boundary value problems. Linear Symplectic geometry, the differential structure of the Lagrangian Grassmannian and the Maslov cycle. Conley-Zehnder index. Elliptic height and Bott iteration formula.PAPERS
from: Gordon, Palais, Portaluri, Rabinowitz, Robbin, Salamon, Smale, Zehnder- Oggetto:
