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.

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. 

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.

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.

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.

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.

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.