Regular and Chaotic Dynamics最新文献

In this paper we provide estimates for mutual distances of periodic solutions for the Newtonian (N)-body problem.Our estimates are based on masses, total variations of turning angles for relative positions, and predetermined upper bounds foraction values. Explicit formulae will be proved by iterative arguments.We demonstrate some applications to action-minimizing solutions for three- and four-body problems.

Chirikov’s celebrated criterion of resonance overlap has been widely used in celestial mechanics and Hamiltonian dynamics to detect global instability, but is rarely rigorous. We introduce two simple Hamiltonian systems, each depending on two parameters measuring, respectively, the distance to resonance overlap and nonintegrability. Within some thin region of the parameter plane, classical perturbation theory shows the existence of global instability and symbolic dynamics, thus illustrating Chirikov’s criterion.

We propose a new approach to the theory of normal forms for Hamiltonian systems near a nonresonant elliptic singular point. We consider the space of all Hamiltonian functions with such an equilibrium position at the origin and construct a differential equation in this space. Solutions of this equation move Hamiltonian functions towards their normal forms. Shifts along the flow of this equation correspond to canonical coordinate changes. So, we have a continuous normalization procedure. The formal aspect of the theory presents no difficulties.As usual, the analytic aspect and the problems of convergence of series are nontrivial.

It is well known that a planar central configuration of the (n)-body problem gives rise to a solution where eachparticle moves in a Keplerian orbit with a common eccentricity (mathfrak{e}in[0,1)). We callthis solution an ellipticrelative equilibrium (ERE for short). Since each particle of the ERE is always in the sameplane, it is natural to regardit as a planar (n)-body problem. But in practical applications, it is more meaningful toconsider the ERE as a spatial (n)-body problem (i. e., each particle belongs to (mathbb{R}^{3})).In this paper, as a spatial (n)-body problem, we first decompose the linear system of ERE intotwo parts, the planar and the spatial part.Following the Meyer – Schmidt coordinate [19], we give an expression for the spatial part andfurther obtain a rigorous analytical method to study the linear stability ofthe spatial part by the Maslov-type index theory. As an application, we obtain stability results for some classical ERE, including theelliptic Lagrangian solution, the Euler solution and the (1+n)-gon solution.

We present the almost global in time existence result in [13]of small amplitude space periodicsolutions of the 1D gravity-capillary water waves equations with constant vorticityand we describe the ideas of proof.This is based on a novel Hamiltonian paradifferentialBirkhoff normal form approach for quasi-linear PDEs.

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that in the planar four-body problem there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its neighborhood.

In this paper, we study the projected Aubry set of a lift of a TonelliLagrangian (L) defined on the tangent bundle of a compact manifold (M) to an infinite cyclic covering of (M). Most of weak KAM and Aubry – Mather theory can be done in this setting. We give a necessary and sufficient condition for the emptiness of the projected Aubry set of the lifted Lagrangian involving both Mather minimizing measures and Mather classes of (L). Finally, we give Mañè examples on the two-dimensional torus showing that our results do not necessarily hold when the cover is not infinite cyclic.

In this short note, we prove that singular Reeb vector fields associated with generic (b)-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) (2N) or an infinite number of escape orbits, where (N) denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of (b)-Beltrami vector fields that are not (b)-Reeb. The proof is based on a more detailed analysis of the main result in [19].

For total collision solutions of the (n)-body problem, Chazy showed that the overall size of the configuration converges to zero with asymptotic rate proportional to (|T-t|^{frac{2}{3}}) where (T) is thecollision time. He also showed that the shape of the configuration converges to the set ofcentral configurations. If the limiting central configuration is nondegenerate, the rate of convergence of the shape is of order (O(|T-t|^{p})) for some (p>0). Here we show by example that in the planar four-bodyproblem there exist total collision solutions whose shape converges to a degenerate central configuration at a rate which is slower that any power of (|T-t|).

A brake orbit for the N-body problem is a solution for which, at some instant,all velocities of all bodies are zero. We reprove two “lost theorems” regarding brake orbits and use them to establish some surprising properties of the completion of theJacobi – Maupertuis metric for the N-body problem at negative energies.