Elementary Symplectic Topology and Mechanics

1. Front Matter

   Pages i-xvii

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2. Notes on Differential Geometry

   • Franco Cardin

   Pages 1-32

3. Symplectic Manifolds

   • Franco Cardin

   Pages 33-66

4. Poisson Brackets Environment

   • Franco Cardin

   Pages 67-72

5. Cauchy Problem for Hamilton-Jacobi Equations

   • Franco Cardin

   Pages 73-103

6. Calculus of Variations, Conjugate Points and Morse Index

   • Franco Cardin

   Pages 105-128

7. A Short Introduction to the Asymptotic Theory of Rapidly Oscillating Integrals

   • Franco Cardin

   Pages 129-135

8. Notes on Lusternik-Schnirelman and Morse Theories

   • Franco Cardin

   Pages 137-196

9. Finite Exact Reductions

   • Franco Cardin

   Pages 197-207

10. Other Instances: Generalized Elasticity

    • Franco Cardin

    Pages 209-215

11. Back Matter

    Pages 217-223

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This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincaré's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schrödinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects.

• 53D05,53D12,37J05,37J10,35F2,58E05,53Z05
• Hamilton-Jacobi equations
• Lusternik-Schnirelman theory
• Morse theory
• Symplectic geometry
• Viterbo symplectic topology

“This book brings the reader from the basics of symplectic topology through several concepts and issues relative to Hamiltonian and Lagrangian systems, Hamilton-Jacobi equations, the calculus of variations, and Morse and Lusternik-Schnirelman theories. … The book seems best suited to researchers and students from graduate level onwards.” (Maria Letizia Bertotti, Mathematical Reviews, August, 2015)

• Dipartimento di Matematica, Padova, Italy

  Franco Cardin

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