# Program

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)**

**Tags/Keywords**

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**

37Kxx - Infinite-dimensional Hamiltonian systems [See also 35Axx, 35Qxx]

70Fxx - Dynamics of a system of particles, including celestial mechanics

37J40 - Perturbations, normal forms, small divisors, KAM theory, Arnolʹd diffusion

37N05 - Dynamical systems in classical and celestial mechanics [See mainly 70Fxx, 70Hxx, 70Kxx]

**Secondary Mathematics Subject Classification**

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 |