https://sites.google.com/view/gabrielebenedetti/teaching
All lectures are in room A133.
Hamiltonian dynamics describes the time evolution of physical systems, where the energy function uniquely determines the physical motions. Correspondingly, the existence of orbits with given boundary conditions is governed by an action functional: the physical motions are the minimizers or, more generally, the critical points of the action. Under global compactness assumptions, the evolution is typically recurrent, that is, orbits come back close to the initial starting point infinitely many times, given a mathematical counterpoint to Nietzsche's philosophy of "eternal return". In this course, we develop the calculus of variations of the action functional for detecting the easiest examples of recurrent motions: stationary points and periodic orbits. We study two flavors of the problem: orbits with fixed period and orbits with fixed energy. In the first part of the course, we consider convex Hamiltonians on configuration spaces. In this case, critical points of the Hamiltonian action functional can be found by minimizing the dual Lagrangian action functional. These methods can be refined to show the existence of Cantor-like invariant sets. They are the support of Mather's minimal measures or, equivalently, the uniqueness set of Fathi's viscosity solutions of the Hamilton--Jacobi equation. When the Hamiltonian is not convex, the above dual Lagrangian approach is not available and established mathematicians thought for years that the Hamiltonian action functional was ill-suited for doing calculus of variation since its critical points have infinite index and co-index. In the second part of course, we will discuss the breakthroughs of the 1980s that circumvented these difficulties and directly show the existence of critical points of the Hamiltonian action functional in great generality. In particular, we will compare the generating-function approach that works in euclidean space to the Floer-theory approach that is applicable much more broadly.
Prerequisites
We will only assume basic notions from Functional Analysis. No prior exposure to Hamiltonian systems is needed. The course is suitable also to interested Master students.
Literature
Our main sources for the course are
i) Hofer and Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser, 1994 (2011 reprint).
ii) Abbondandolo, Lectures on the free-period action functional, J. Fixed Point Theory Appl., 2013.
iii) Sorrentino, Action-minimizing methods in Hamiltonian dynamics. An introduction to Aubry-Mather theory, Princeton University Press, 2015.
iv) McDuff and Salamon, Introduction to Symplectic Topology, Third Edition, Oxford University Press, 2016.
v) Audin and Damian, Morse theory and Floer homology, Springer, 2014.
The exam will consist in a 50-minute seminar talk with questions. We will choose together the topic of the talk according to your interests and background. We expect that students attend the seminar talks of the other participants.