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Mathematical Sciences Research Institute

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Program

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 (École Normale Supérieure de Lyon), Vadim Kaloshin (University of Maryland), Robert Littlejohn (University of California, Berkeley), Philip Morrison (University of Texas at Austin), Tere M. Seara (Universitat Politecnica de Catalunya), Serge Tabachnikov (Pennsylvania State University), Amie Wilkinson (University of Chicago)
Description

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. 

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Programmatic Workshops
August 16, 2018 - August 17, 2018 Connections for Women: Hamiltonian Systems, from topology to applications through analysis