The N-body problem in d-dimension space has symmetry group SE(d).
Centre of mass reduction leads to a system with SO(d) symmetry acting
diagonally on positions and momenta. For N=3, d=4 reduction of the SO(4)
symmetry is complicated because the tensor of inertia is non-invertible.
The fully reduced system has 4 degrees of freedom and a Hamiltonian that
is not polynomial in the momenta. The most surprising property of the
reduced Hamiltonian is that it has equilibria that are minima.Updated on Oct 03, 2018 09:54 AM PDT
This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.
This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.Updated on Aug 17, 2018 03:29 PM PDT
In this course we will explore the connection between Hamilton-Jacobi PDE, Hamiltonian ODE and Symplectic Topology. Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. We regard a Hamiltonian system completely integrable if there exists a change of coordinates such that our Hamiltonian system in new coordinates is still Hamiltonian but now associated with a Hamiltonian function that is independent of position. For completely integrable systems the new momentum coordinates are conserved and the set of points at which the new momentum takes a fixed vector is invariant for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to Kolmogorov-Arnold-Moser (KAM) Theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. Aubry-Mather Theory construct a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. A. Fathi uses viscosity solutions of the associated Hamilton-Jacobi PDE to construct Aubry-Mather invariant measures. Recently there have been several interesting works to understand the connection between Aubry-Mather Theory and Symplectic Topology. The hope is to use tools from Symplectic Topology to construct interesting invariant sets/measures for Hamiltonian systems associated with non-convex Hamiltonian functions. In this course, we also explore the connection between Aubry-Mather Theory and the homogenization phenomena for Hamilton-Jacobi PDEs when the Hamiltonian function is selected randomly according to a translation invariant probability measure.Created on Aug 24, 2018 03:42 PM PDT
Updated on Oct 18, 2018 11:33 AM PDT
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