Aug 20, 2018
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Introduction to weak KAM theory I
Marie-Claude Arnaud (Université d'Avignon)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Using a very simple example, we will introduce Lagrangian Dynamics and make the link with Hamiltonian Dynamics. We will explain the connection between Euler-Lagrange equations and the Lagrangian variational approach. This will be the way to introduce famous weak KAM theory. We will finish by the relation with Hamilton-Jacobi equation and the viscosity solutions.
- Supplements
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Notes
1.36 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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Hamilton description of plasmas and other models of matter: structure and applications I
Philip Morrison (University of Texas, Austin)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
1.09 MB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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02:00 PM - 03:00 PM
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Introduction to KAM theory in the planetary system I
Jacques Fejoz (Université de Paris IX (Paris-Dauphine))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
1.08 MB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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03:30 PM - 04:30 PM
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Introduction to Periodic Orbits and Invariant Manifolds in Celestial Mechanics with Applications to Space Missions I
Rodney Anderson (NASA Jet Propulsion Laboratory (JPL))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Periodic orbits, quasiperiodic orbits, and invariant manifolds are key to obtaining an overall understanding of motion and transport within celestial mechanics models. These topics will be introduced primarily in the circular restricted three-body problem with other examples in more complex models including the real-world ephemeris. Design consideration for trajectories using these components and heteroclinic connections for astrodynamics applications will also be introduced. Particular applications to the design of trajectories traveling to the moon and Jovian tours will be presented as well.
- Supplements
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Notes
1.14 MB application/pdf
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Aug 21, 2018
Tuesday
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09:30 AM - 10:30 AM
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Hamilton description of plasmas and other models of matter: structure and applications I
Philip Morrison (University of Texas, Austin)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
1.09 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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Introduction to weak KAM theory II
Marie-Claude Arnaud (Université d'Avignon)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Using a very simple example, we will introduce Lagrangian Dynamics and make the link with Hamiltonian Dynamics. We will explain the connection between Euler-Lagrange equations and the Lagrangian variational approach. This will be the way to introduce famous weak KAM theory. We will finish by the relation with Hamilton-Jacobi equation and the viscosity solutions.
- Supplements
-
Notes
1.36 MB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Introduction to Periodic Orbits and Invariant Manifolds in Celestial Mechanics with Applications to Space Missions II
Rodney Anderson (NASA Jet Propulsion Laboratory (JPL))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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--
- Abstract
- --
- Supplements
-
Notes
1.14 MB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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03:30 PM - 04:30 PM
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Introduction to KAM theory in the planetary system II
Jacques Fejoz (Université de Paris IX (Paris-Dauphine))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
1.08 MB application/pdf
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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Aug 22, 2018
Wednesday
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09:30 AM - 10:30 AM
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Introduction to applications of KAM theory to PED's I
Clarence Wayne (Boston University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We discuss, in the context of energy flow in high-dimensional systems the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We also demonstrate in a special model that the very slow dissipation is due to the slow drift along a family of approximate breather solutions.
- Supplements
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Notes
1.44 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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Introduction to Arnold's diffusion problems I
Tere Seara (Polytechnical University of Cataluña (Barcelona))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
1.45 MB application/pdf
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Aug 23, 2018
Thursday
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09:30 AM - 10:30 AM
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Introduction to Arnold's diffusion problems II
Ke Zhang (University of Toronto)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
1.45 MB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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11:00 AM - 12:00 PM
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An introduction to applications of KAM theory to PDE's II
Clarence Wayne (Boston University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Notes
1.45 MB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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02:00 PM - 03:00 PM
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Normal forms and KAM theory in Celestial Mechanics: from space debris to the rotation of the Moon
Alessandra Celletti (Seconda Università di Roma "Tor Vergata'')
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Since centuries, Celestial Mechanics is a test-bench for many theories of Dynamical Systems, among which perturbation theory and KAM theory. Realistic results in astronomical applications can be obtained through an accurate modeling and an appropriate study of the dynamics, which often requires a heavy computational effort.
After an overview on normal forms and (conservative and dissipative) KAM theory, I will consider some examples in Celestial Mechanics, where such theories give successful results.
In particular, I will review some results about the dynamics of space debris, which can be studied through averaging theory and normal forms computations. The stability of the rotation of the Moon, as well as the
orbital motion of asteroids in the framework of a particular 3-body problem, can be investigated through computer-assisted implementations of KAM theory.
- Supplements
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Notes
698 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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03:30 PM - 04:30 PM
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Introduction to Numerical Methods for Hamiltonian Systems
Michael Kraus (Max-Planck-Institut für Plasmaphysik (EURATOM))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Hamiltonian systems possess several important structures and conservation laws: most notably a symplectic or Poisson structure and conservation of energy, momentum maps and Casimir invariants. Numerical algorithms which preserve these structures usually show greatly reduced errors compared to algorithms that do not preserve these structures, as well as much better long-time stability.
In this lecture, important mathematical structures of Hamiltonian systems will be reviewed and consequences of their non-preservation in numerical simulations higlighted. Some basic structure-preserving algorithms for canonical Hamiltonian systems will be introduced and compared with their non-structure-preserving counterparts. Finally, and outlook will be given on the structure-preserving discretisation of noncanonical Hamiltonian systems like those found in fluid dynamics and plasma physics. For such systems, there are no standard methods available like the many that are known for canonical Hamiltonian systems and the development of new methods is much more challenging.
- Supplements
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Notes
788 KB application/pdf
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Aug 24, 2018
Friday
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09:00 AM - 09:30 AM
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The periodic orbits problem in Tonelli Hamiltonian dynamics
Marco Mazzucchelli (École Normale Supérieure de Lyon)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
My talk will be an introductory lecture on the periodic orbits problem for Tonelli Hamiltonian systems. In a first part of the talk, I will recall the variational formulation of the problem, and its connections to Aubry-Mather theory and Weak KAM theory. I will then review some of the main known results on the multiplicity of periodic orbits with prescribed energy, and some of the celebrated open questions in the field, including the closed geodesics problem in Riemannian and Finsler manifolds.
- Supplements
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Notes
691 KB application/pdf
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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Vizualizing the Dynamics of Symplectic Maps
James Meiss (University of Colorado at Boulder)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
A nearly-integrable dynamical system has a natural formulation in terms of actions that are nearly constant and angles that nearly rigidly rotate. Such maps arise naturally in the Hamiltonian, or symplectic case where the variables appear in canonical pairs. We study angle-action maps that are close to symplectic and have a twist that is positive-definite. When the map is symplectic, Nekhoroshev's theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volume-preserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rank-r resonances. A comparison with computations for a generalized Froeschlé map in four-dimensions, shows that this theory gives accurate results for the rank-one case.
- Supplements
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Notes
635 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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02:00 PM - 03:00 PM
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Dynamical and spectral properties of mathematical billiards
Alfonso Sorrentino (Seconda Università di Roma "Tor Vergata'')
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local, and there is a tight intertwinement between its dynamics and the shape of the domain. All of this translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures.
- Supplements
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Notes
717 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
- --
- Supplements
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03:30 PM - 04:30 PM
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Interaction between two charges in a uniform magnetic field
Robert MacKay (University of Warwick)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In joint work with Diogo Pinheiro, we reduced the dynamics of two charges in a uniform magnetic field to 3 degrees of freedom (DoF). If they have the same ratio of charge to mass we reduced to 2 DoF, by a symmetry we call locomotive coupling rod symmetry. For opposite signs of charge and charge to mass ratios not summing to zero, we proved every high enough energy level possesses chaotic coplanar motion, analogous to Poincaré’s second species orbits in celestial mechanics. The global motion can be considered as making transitions between ionic and free states. For same signs of charge, the state space is separated by the planar motions into pass-through and bounce-back motions. The work is the first step in an analysis of cross-field transport in quasi-symmetric magnetic fields.
- Supplements
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Notes
663 KB application/pdf
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