Workshop

Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. We show that studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provide an additional rich class of non-smooth systems that can be studied by perturbation methods. Moreover, the analysis can be extended to systems with soft steep potentials that limit to the impact systems. For example, for some of these systems, we show that KAM theory may be applied, proving that for a large portion of phase space the perturbed motion is conjugate to rotations on a torus [1]. On the other hand, other simple impact systems have inherently non-rotational motion – we show cases in which the motion is conjugate to geodesic flow on a flat torus with several handles [2]. [1] M. Pnueli & V. Rom-Kedar “On near integrability of some impact systems”, SIAM-DS, 2018 to appear. [2] L. Becker, S. Elliott, B. Firester, S. Gonen Cohen, M. Pnueli & V. Rom-Kedar, in preparation.

Celestial Mechanics systems have two fundamental conservation principles: conservation of momentum and conservation of (mechanical) energy. Of the two, conservation of momentum provides the most constraints on a general system, with three translational symmetries (which can be trivially removed) and three rotational symmetries. If no external force acts on the system, these quantities are always conserved independent of the internal interactions of the system. In contrast, conservation of energy involves assumptions on both the lack of exogenous forces and on the nature of internal interactions within the system. For this reason energy is often not conserved for “real” systems that involve internal interactions, such as tidal deformations or impacts, even though such systems conserve total momentum. Thus, mechanical energy generally decays through dissipation until the system has found a local or global minimum energy configuration that corresponds to its constant level of angular momentum. This observation motivates a fundamental question for celestial mechanics: What is the minimum energy configuration of a N-body system with a fixed level of angular momentum? It can be shown that this is an ill-defined question and has no answer for traditional point-mass celestial mechanics systems. If, instead, the system and problem are formulated accounting for finite density distributions this question becomes well posed and we can prove the existence of minimum energy configurations for all suitably formulated celestial mechanics systems (Scheeres, CMDA 113(3): 291-320, 2012). This small change also leads to fundamental changes in the nature and stability properties of relative equilibria and, ultimately, the dynamics of these systems. Finally, we can also show that this naturally leads to a “granular mechanics” extension of celestial mechanics, with fundamental links between this topic and the science of small solar system bodies.

We consider a general form of electromagnetic gyrokinetic Vlasov-Maxwell theory in which the gyrocenter symplectic structure contains electric and magnetic perturbations that are necessary to cause the first-order gyrocenter polarization displacement to vanish. The gyrocenter Hamilton equations, which are expressed in terms of a gyrocenter Poisson bracket that contains electromagnetic perturbations and a gyrocenter Hamiltonian, satisfy the Liouville property exactly with a time-dependent gyrocenter Jacobian. The gyrokinetic Vlasov-Maxwell equations are derived from a variational principle, which also yields exact conservation laws through the Noether method. We show that the new symplectic gyrokinetic Vlasov-Maxwell equations retain all first-order polarization and magnetization effects without the need to consider second-order contributions in the gyrocenter Hamiltonian.

We consider a Hamiltonian system slowly depending on time: $$ H=H(x,\tau),\qquad \dot\tau=\epsilon\ll 1. $$ For small $\epsilon$ the energy $H$ changes slowly: $\dot H=O(\epsilon)$. If the frozen autonomous system with Hamiltonian $H(\cdot,\tau)$ has one degree of freedom and energy levels are closed curves, there is an adiabatic invariant $I$ which changes much slower. Then the energy $H=h(\tau,I)$ changes gradually. However, the adiabatic invariant is destroyed for trajectories passing near equilibria. Neishtadt showed that if the frozen system has a hyperbolic equilibrium with a figure eight separatrix, then generically the energy will have quasi-random jumps of order $\epsilon$ with frequency of order $1/|\ln\epsilon|$. We partly extend Neishtadt's result to multidimensional systems such that for each $\tau$ the frozen system has a hyperbolic equilibrium possessing several transverse homoclinics. The trajectories we construct have quasirandom jumps of energy of order $\epsilon$ with frequency $1/|\ln\epsilon|$ while staying distance of order $\epsilon$ away from the homoclinic set. Gelfreigh and Turayev showed that if the frozen system has compact uniformly hyperbolic chaotic invariant sets on energy levels, then generically there exist trajectories with energy having quasirandom jumps of order $\epsilon$ with frequency of order 1. Thus the energy may grow with rate of order $\epsilon$. However, this result does not work near a homoclinic set, where dynamics of the frozen system is slow, so there is no uniform hyperbolicity.

In this talk we discuss recent work on a geometric approach to certain optimal control problems and the relationship of the solutions of these problems to some classical integrable dynamical systems. These systems include the rigid body equations, geodesic flows on the ellipsoid and the Toda lattice flows. We discuss the Hamiltonian structure of these systems and relate our work to some classical work of Moser. We also discuss the link to discrete dynamics and symplectic integration. The work is joint with Francois Gay-Balmaz and Tudor Ratiu.

I shall present recent results about the complex dynamics of the water waves equations of a bi-dimensional fluid under the action of gravity and eventually capillary forces, with space periodic boundary conditions. This is an infinite dimensional Hamiltonian system. We shall discuss both long time existence results as well as bifurcation of small amplitude time quasi-periodic solutions. Major difficulties are the quasi-linear nature of the water waves equations and complex resonance phenomena.

How the fundamental mathematical tools such as Noether's Theorem can be implemented for practical purposes, such as control of quality of prediction of fusion plasmas discharges? In this talk I will try to respond on this intricate question. Below are some motivations. In fusion plasmas the strong magnetic field allows the fast gyro motion to be systematically removed from the description of the dynamics, resulting in a considerable model simplification and gain of computational time. The dynamically reduced kinetic model obtained in the given setup is called gyrokinetic model. Performing dynamical reduction in the framework of Lagrangian formalism allows one to control the exactness of reduction procedure and derive robust models for numerical implementations. In their turn, specific results of gyrokinetic (GK) simulations performed prior to consideralby costly fusion plasma experiments allows to predict and optimise experimental setup. In particular, numerical simulations of fusion plasma discharges can impact our predictions of the plasma behavior in ITER and future fusion reactors. Confidence in these predictions requires a rigorous and systematic verification of the underlying model, which should be regarded as an indispensable step before any validation of the numerical results against experiments can be considered meaningful. A new and generic theoretical framework and specific numerical applications to test the validity and the domain of applicability of existing GK codes will be presented. In particular the role of the energy invariants issued from the Noether theorem will be explicitly highlighted throughout the analysis of energy balance and generic plasma instabilities mechanisms providing the ultimate connection between the fundamental mathematical tools and practical implementations.

We study the asymptotic behavior of the solutions to a family of discounted Hamilton--Jacobi equations when the discount factor goes to zero.

The new point is that we tackle the problem in a noncompact setting. We prove that a distinguished critical solution of the equation with vanishing  discount is selected at the limit. Our approach is based on  some duality results between suitable cones of Lagrangian functions and families of probability measures defined on the tangent bundle of the ambient space.

Many physical problems are described by conformally symplectic systems. We study the existence of whiskered tori in a family $f_\mu$ of conformally symplectic maps depending on parameters $\mu$. Whiskered tori are tori on which the motion is a rotation but having as many contracting/expanding directions as allowed by the preservation of the geometric structure. Our main result is formulated in an a-posteriori format. Given an approximately invariant embedding of the torus for a parameter value $\mu_0$ with an approximately invariant splitting of the tangent space at the range of the embedding into stable/unstable/center bundles, there is an invariant embedding and invariant splittings for new parameters. Using the results of formal expansions as the starting point for the a-posteriori method, we study the domains of analiticity of parameterizations of whiskered tori in perturbations of Hamiltonian Systems with dissipation. The proofs of the results lead to efficient algorithms that are quite practical to implement. Joint work with A. Celletti and R. de la Llave.

Many physical situations involve the interplay of different phenomena at different scales. The corresponding description requires the use of multi-physics models, whose mathematical formulation poses several challenges. Examples are found in the classical-quantum coupling in molecular dynamics or in the coupling between mean flow and fluctuation kinetics in turbulence. In plasma physics, the interaction of energetic particles (obeying kinetic theory) with a fluid bulk (obeying magnetohydrodynamics) requires formulating hybrid kinetic-fluid models, which are often obtained by making assumptions that destroy the correct energy balance. This talk shows how momentum-map techniques in geometric mechanics provide a powerful unifying framework for both kinetic-fluid and classical-quantum coupling, thereby leading to new hybrid models in different contexts.

Some Hamiltonian PDEs which are invariant under spatial translations possess traveling wave solutions which form finite dimensional invariant manifolds parametrized by their spatial locations. Extensive studies have been carried out for their stability analysis. In this talks we shall focus on local dynamics and invariant manifolds of the traveling wave manifolds for the Gross-Pitaevskii equation in $R^3$ and the gKdV equation as our main PDE models, while our approach works for a general class of problems. Noting that the symplectic operators of some of these models happen to be unbounded in the energy space, violating a commonly assumed assumption for the study of the linearized systems at these traveling waves, we could carry out linearized analysis in a general framework we developed recently. Nonlinearly our main results are the existence of local invariant manifolds of unstable traveling waving manifolds and the implications on the local dynamics. In addition to applying certain space-time estimates, we use a bundle coordinate system to handle an issue of a seemingly regularity loss caused by the spatial translation parametrization.