We consider the classical three-dimensional motion in a potential which is the sum of $n$ attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the $n$ centres, we find a universal behaviour for all energies $E$ above a positive threshold.
Whereas for $n=1$ there are no bounded orbits, and for $n=2$ there is just one closed orbit, for $n\geq 3$ the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set.
Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates.
The theory includes the $n$--centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted $n$-{\em body} problems.
To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres.
Finally we show that different, non-universal, phenomena occur for collinear configurations.
In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis') are collision-free. This methods applies to $3$-dimensional problems of celestial mechanics exhibiting a particular cylindrical symmetry, as well as to planar problems of $N$-centre type, where it gives optimal results.
In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincaré.
In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of $N$-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types o…
We construct a normal form suited to {\it fast driven systems}. We call so systems including actions ${\rm I}$, angles {$ψ$}, and one fast coordinate $y$, moving under the action of a vector--field $N$ depending only on ${\rm I}$ and $y$ and with vanishing ${\rm I}$--components. {In absence of the coordinate $y$, such systems have been extensively investigated and it is known that, after a small perturbing term is switched on, the normalised actions ${\rm I}$ turn to have exponentially small variations compared to the size of the perturbation. We obtain the same result of the classical situati…
Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proven for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here we show for a class of pair potentials including the homogeneous ones of degree -a for 0<a<2, that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface.
Our idea is to imitate Smale's list of problems, in a restricted domain of mathematical aspects of Celestial Mechanics. All the problems are on the n-body problem, some with different homogeneity of the potential, addressing many aspects such as central configurations, stability of relative equilibrium, singularities, integral manifolds, etc. Following Steve Smale in his list, the criteria for our selection are: (1) Simple statement. Also preferably mathematically precise, and best even with a yes or no answer. (2) Personal acquaintance with the problem, having found it not easy. (3) A belief …
We consider the 3-body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces, and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of new classes of orbits. In particular, we find some families of isosceles triangles, which occur in elliptic space.
In this article, we give two refinements of Franks' theorem: For orientation and area preserving homeomorphisms of the closed or open annulus, the existence of $k$-periodic orbits ($(k,n_0)=1$) forces the existence of infinitely many periodic orbits with periods prime to $n_0$. Moreover, if $f$ is reversible, the periodic orbits above could be symmetric. Our improvements of Franks' theorem can be applied to Reeb dynamics and celestial mechanics, for example, we give some precise information about the symmetries of periodic orbits found in Hofer, Wysocki and Zehnder's dichotomy theorem when the…
Given a volume preserving dynamical system with non-compact phase space, one is sometimes interested in special subsets of its wandering set. One example from celestial mechanics is the set of initial values leading to collision. Another one is the set of initial values of semi-orbits, whose asymptotic velocity does not exist as a limit. We introduce techniques that can be helpful in showing that these sets are of measure zero: by defining a sequence of hypersurfaces, that are eventually hit by each of those semi-orbits and whose total surface area decreases to zero.
We prove that if for relative equilibrium solutions of a generalisation of the $n$-body problem of celestial mechanics the masses and rotation are given, then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero. We furthermore prove that the set of such relative equilibria is compact.
We prove that if the masses of the $n$-body problem of celestial mechanics are given, the number of classes of relative equilibria that solve the $n$-body problem is finite.
We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.
We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark-Zeeman systems and study their behaviour.
