# Mathematics Colloquium

#### Hamiltonian Evolution Equations - Where They Come From, What They Are Good For

**Speaker:**
Juerg Froehlich, ETH Zurich (& IAS)

**Location:**
Warren Weaver Hall 1302

**Date:**
Monday, December 2, 2013, 3:45 p.m.

**Synopsis:**

I start with a brief survey of examples of (non-linear) Hamiltonian evolution equations describing the dynamics of a variety of physical systems with infinitely many degrees of freedom. Among examples are the Vlasov-,Hartree- and Hartree-Fock equations.

I then focus on the example of equations describing the dynamics of a heavy pointparticle coupled to a wave medium. If the speed of the particle exceeds the speed of wave propagation in the medium then the particle tends to emit Cherenkov radiation. This results in a deceleration of the particle until its speed has dropped to a value smaller than or equal to the speed of wave propagation in the medium, where after its motion is ballistic. This provides an example of "Hamiltonian Friction". (A special case of this phenomenon concerns a charged particle moving through an optically dense medium at a speed exceeding the speed of light in the medium. The radiation it emits is the Cherenkov radiation observed in nuclear reactors. I will consider the example of a particle moving through a Bose-Einstein condensate.)