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Hamiltonian systems, from topology to applications through analysis August 13, 2018 to December 14, 2018
Organizers Rafael de la Llave (Georgia Institute of Technology), LEAD Albert Fathi (Georgia Institute of Technology; École Normale Supérieure de Lyon), vadim kaloshin (University of Maryland), Robert Littlejohn (University of California, Berkeley), Philip Morrison (University of Texas, Austin), Tere Seara (Polytechnical University of Cataluña (Barcelona)), Sergei Tabachnikov (Pennsylvania State University), Amie Wilkinson (University of Chicago)
Hamiltonian mechanics was born out of optics. Sir William Rowan Hamilton developed a theory for studying the propagation of the phase in optical systems guided by Fermat's principle for light rays (i.e. high frequency systems). Shortly afterwards, he realized that ,based on the similarity of Fermat's principle with the action principle, one could adapt the machinery to mechanics. Hamiltonian methods are now a central topic in dynamics and mechanics. Many interesting PDE's appear as a limit of mechanical systems of many small particles (e.g. water waves, fluid mechanics, the equations of plasma physics), and therefore the Hamiltonian setting is essential for studying these types of PDE’s. It is interesting to note that Maxwell spent some time developing mechanical models for his equations for the electromagnetic field. Practical scientists appreciate the magic cancellations in the Hamiltonian setting that lead to efficient calculations. The interdisciplinary nature of Hamiltonian systems is deeply ingrained in its history. It is remarkable that the discovery in the 1980’s of the celebrated Aubry-Mather theory (one of the most important developments in decades) was accomplished simultaneously by a Physicist Serge Aubry and a Mathematician John Mather. Many of the people working in this area can talk to both mathematicians and physicists. This program is designed to mix the pure mathematical viewpoint with applications in physics, space mechanics, and theoretical chemistry. The two communities will be completely integrated for synergy. Workshops are designed with the priority of fomenting interactions.  We envision that during the whole semester the visitors will present tutorials aimed also to the people from different scientific backgrounds. The selection of the majority of visitors will be based on potential interactions. Mathematical topics include: 1) Arnold diffusion (using both the geometric and variational methods, including examples of diffusion in celestial mechanics). 2) Celestial mechanics (with a particular emphasis on minimizing orbits, and other surprising trajectories). 3) Connections between the weak (viscosity) solutions of the Hamilton-Jacobi equation and the Aubry-Mather theory of Lagrangian systems (Weak KAM theory). 4) PDE’s that can be thought of as infinite dimensional Hamiltonian Systems, to which the KAM methods can be applied. Applications include: 1) Astrodynamics and motions of satellites. 2) Plasma Physics and accelerator Physics confinement. 3) Theoretical Chemistry and atomic Physics.  Bibliography
Keywords and Mathematics Subject Classification (MSC)
  • Hamiltonian systems

  • Celestial Mechanics

  • Arnold diffusion

  • Hamilton-Jacobi

  • Aubry-Mather

  • KAM methods

  • Lagrangian systems

  • astrodynamics

  • plasma physics

  • accelerator physics

  • theoretical chemistry

  • atomic physics

  • weak KAM

  • viscosity solutions

  • symplectic

  • Poisson brackets

  • barriers and transport

  • perturbation theory

  • billiards

  • invariant manifolds

  • spectral rigidity

  • Hamiltonian invariants

  • twist maps

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
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Programmatic Workshops
August 16, 2018 - August 17, 2018 Connections for Women: Hamiltonian Systems, from topology to applications through analysis
August 20, 2018 - August 24, 2018 Introductory Workshop: Hamiltonian systems, from topology to applications through analysis
October 08, 2018 - October 12, 2018 Hamiltonian systems, from topology to applications through analysis I
November 26, 2018 - November 30, 2018 Hamiltonian systems, from topology to applications through analysis II