Workshop

Consider the Newtonian 4-body problem in 3-space with zero angular momentum and any mass ratios. Theorem: every bounded solution defined on an infinite time interval suffers infinitely many coplanar instants: instants where all four bodies lie in the same plane. This generalizes a theorem asserting that for the zero angular momentum 3-body problem in the plane, every bounded solution suffers infinitely many collinear instants, or ``syzygies''. The obvious generalization holds for any dimension $d$, that is, for the zero angular momentum $d+1$-body problem in $d$-space. After stating the results, I give a heuristic physical ``proof'' of this theorem inspired by Mark Levi. I then proceed to sketch the mathematical proof. Among its main ingredients are that (i) the translation-reduced problem has for its configuration space the space of $d \times d$ matrices (ii) the signed distance from the degeneration locus satisfies a Hamilton-Jacobi equation, (iii) the oriented shape space is homeomorphic to a Euclidean space (iv) the oriented shape space has non-negative sectional curvature where smooth. The theorem gives a ``hunting license'' to set up symbolic dynamics for the spatial 4-body problem, in analogy with the idea of syzygy sequences for the planar three-body problem.

Nontwist Hamiltonian systems have shearless invariant curves that act like barriers in phase space [1, 2]. Recently, secondary shearless curves have also been identified in the phase space of twist maps, in the neighbourhood of peculiar bifurcations of elliptic fixed points [3]. We use Slater’s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of area-preserving maps [4]. We also determine the breakup critical parameters, of the shearless curves, with a procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition [5]. Finally, we present evidences of transport barriers in plasmas confined in the tokamak TCABR [6] and in the Texas Helimak [7].

References 1- P. J. Morrison, Physics of Plasmas 7, 2279 [2000]. 2- D. Del-Castillo-Negrete, Physics of Plasmas 7, 1702 [2000]. 3- C. V. Abud, I. L. Caldas. Chaos 22, 033142 (2012). 4- C. V. Abud, I. L. Caldas. Physica D 308, 34 (2015) 5- M. S. Santos, M. Mugnaine, J. D. Szezech Jr, A. M. Batista, I. L. Caldas, M. S. Baptista, R. L. Viana. Chaos 28, 085717 (2018). 6-A. F. Marcus, I. L. Caldas, Z. O. Guimarães-Filho, P. J. Morrison, W. Horton, I. C. Nascimento, Yu. K. Kuznetsov. Phys. Plasmas 15, 112304 (2008). 7- D. L. Toufen, Z. O. Guimarães-Filho, I. L. Caldas, F. A. Marcus, K. W. Gentle. Phys. Plasmas (2012).

In this talk I give an overview on the existing results for two questions associated with a Hamiltonian function that is selected randomly according to a probability measure that is translation invariant (this includes Hamiltonian functions that are almost periodic in position or in both position and momentum) 1. Consider a Hamiltonian ODE with a Hamiltonian function that is periodic in time. As in Arnold's conjecture we may wonder whether or not such an ODE has periodic orbits. 2. Analogously, we may study Hamilton-Jacobi PDEs and examine the question of homogenization that is closely related to the long time behavior of the solutions to the corresponding Hamiltonian ODE.

Large particle accelerators and storage rings are represented by exceedingly complicated Hamiltonians. On the one hand the relativistic motion in electromagnetic fields leads to nonlinear connections between positions and momenta due to the presence of a square root. On the other hand, different from other examples including typical many body problems, an accelerator has thousands of separate sections with different localized fields. Finally it is often necessary to predict orbit stability for around billions of revolutions. All these make even attempts at symplectic tracking very difficult.

One method to describe the nonlinear motion is via perturbation theory describing the motion on a Poincare section for one revolution. Recent advances based on differential algebraic methods have allowed the generalization of such perturbation analysis from typically orders two and three to in principle arbitrary order, and in practice to orders around ten. Such representations of the Poincare map allow symplectic tracking in various ways.

More importantly, the  high order Poincare maps can be used to perform rigorous stability analysis. For this purpose, the differential algebraic approach allows performing automated high order normal form analysis of the motion, which upon adjusting system parameters of the accelerator leads to dynamics that is nearly integrable. Utilizing modern methods of rigorous global optimization based on high-order Taylor model methods, it is possible to perform rigorous Nekhoroshev-type stability estimates. Results are shown about such stability estimates, which can establish guaranteed stability within the regions typically occupied by the particle beam for operationally required times.  

A variety of interesting structures and dynamics stem from singularities of Poisson operators. As well known, the kernel of a finite-dimensional Poisson operator can be integrated “locally” to define Casimirs, and their level-sets foliate the phase space yielding symplectic submanifolds (Lie-Darboux theorem). However, the global structure can be more complex when the Poisson operator has “singularities” where its rank changes. Invoking some physical examples (both finite dimensional and infinite dimensional), we discuss how singularities cause seemingly non-Hamiltonian phenomena. Some theoretical methods for probing into singularities will be also introduced.

Taking as starting point several examples from Celestial mechanics where regularization techniques bring singularities in, we will introduce the geometry of contact structures where the regularity of the contact 1-form is relaxed. Contact structures also show up modelling problems in Fluid Dynamics and singularities also appear naturally in this context (ongoing joint work with Robert Cardona and Daniel Peralta-Salas).  

Two main geometrical problems will be addressed in this talk: The existence problem of contact structures with singularities on a given manifold and the study of its Reeb Dynamics, in particular, the existence of periodic orbits (Weinstein conjecture).

This is joint work with Cédric Oms.

In dynamical systems, one often encounters actions A ≡ RΩLx, v(x)%dx which
depend only on v, the velocity of the system and on % the distribution of the particles. In
this case, it is well–understood that convexity of L(x, ·) is the right notion to study variational
problems. In this talk, we consider a weaker notion of convexity which seems appropriate
when the action depends on other quantities such as electro–magnetic fields. Thanks to the
introduction of a gauge, we will argue why our problem reduces to understanding the relaxation
of a functional defined on the set of differential forms (Joint work with B. Dacorogna).

We will describe a  geometric method to prove instability in nearly integrable Hamiltonian systems of n-degrees of freedom. The approach is based on tracking the `outer dynamics’ along homoclinic orbits  to a normally hyperbolic invariant manifold (NHIM). Only little information is needed on the `inner dynamics' restricted to the NHIM, so this applies to rather general situations; for instance, the unperturbed Hamiltonian does not need to be convex.  The   conditions needed for this approach are transversality conditions and hence generic. Moreover, these conditions can be verified in concrete systems, such as  in celestial mechanics.

Consider a Markov chain on a one-dimensional torus $\mathbb T$, where a moving point jumps from $x$ to $x\pm \alpha$ with probabilities $p(x)$ and $1-p(x)$, respectively, for some fixed function $p\in C^{\infty}(\mathbb T, (0,1))$ and $\alpha\in\mathbb R\setminus \mathbb Q$. Such Markov chains are called random walks in a quasi-periodic environment. It was shown by Ya.~Sinai that for Diophantine $\alpha$ the corresponding random walk has an absolutely continuous invariant measure, and the distribution of any point after $n$ steps converges to this measure. Moreover, the Central Limit Theorem with linear drift and variance holds. In contrast to these results, we show that random walks with a Liouvillian frequency $\alpha$ generically exhibit an erratic statistical behaviour. In particular, for a generic $p$, the corresponding random walk does not have an absolutely continuous invariant measure, both drift and variance exhibit wild oscillations (being logarithmic at some times and almost linear at other times), Central Limit Theorem does not hold. These results are obtained in a joint work with Dmitry Dolgopyat and Bassam Fayad.