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In this article, we define a circular restricted (N + 1) -body problem on the surfaces M κ 3 , with κ = ± 1 . The motion of the primaries corresponds to an elliptic relative equilibria studied in Diacu (Relative equilibria of the curved N-body problem. Atlantis Studies in Dynamical Systems, Atlantis Press, Paris, 2012), where N identical mass particles are rotating uniformly at the vertices of a regular polygon placed at a fixed parallel of a maximal sphere. By introducing rotating coordinates, this problem gives rise to a 3 d.o.f. Hamiltonian system. This problem has an equilibrium point placed at the poles of S 3 and the vertex of H 3 , for any value of the parameters. We give information about the linear and nonlinear stability of these equilibria. Finally, we carry out a study about the existence of periodic solutions and KAM tori. [ABSTRACT FROM AUTHOR]

Cornerstone models of physics, from the semi-classical mechanics in atomic and molecular physics to planetary systems, are represented by quasi-integrable Hamiltonian systems. Since Arnold's example, the long-term diffusion in Hamiltonian systems with more than two degrees of freedom has been represented as a slow diffusion within the 'Arnold web,' an intricate web formed by chaotic trajectories. With modern computers it became possible to perform numerical integrations which reveal this phenomenon for moderately small perturbations. Here we provide a semi-analytic model which predicts the extremely slow-time evolution of the action variables along the resonances of multiplicity one. We base our model on two concepts: (i) by considering a (quasi-)stationary-phase approach to the analysis of the Nekhoroshev normal form, we demonstrate that only a small fraction of the terms of the associated optimal remainder provide meaningful contributions to the evolution of the action variables. (ii) We provide rigorous analytical approximations to the Melnikov integrals of terms with stationary or quasi-stationary phase. Applying our model to an example of three degrees of freedom steep Hamiltonian provides the speed of Arnold diffusion, as well as a precise representation of the evolution of the action variables, in very good agreement (over several orders of magnitude) with the numerically computed one. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems H=h(y)+f(x,y,t),where y∈D⊆Rn with D being an open bounded domain, x∈Tn, f(x, y, t) is a real analytic almost periodic function in t with the frequency ω=(…,ωλ,…)λ∈Z∈RZ. As an application, we will prove the existence of almost periodic solutions and the boundedness of all solutions for the second-order differential equations with superquadratic potentials depending almost periodically on time. [ABSTRACT FROM AUTHOR]

This paper is a continuation to our work (Xu et al. in Ann Henri Poincaré 18(1):53-83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Rüssmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincaré 10(8):1419-1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincaré-Treshchev mechanism. [ABSTRACT FROM AUTHOR]

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $$(S^2)^n$$ . In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $$T^*\mathbf {R}^{2n}$$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity. [ABSTRACT FROM AUTHOR]