This paper considers the planar N-body problem with a quasi-homogeneous potential given by

$$ \begin{align*}W = \sum_{1\leq k<j\leq N} \left[ \frac{ m_k m_j}{\|\boldsymbol{r}_k-\boldsymbol{r}_j\|} +\frac{m_k m_j C_{jk}}{\|\boldsymbol{r}_k-\boldsymbol{r}_j\|^p} \right], \end{align*} $$

where $ m_k>0 $ are the masses and $C_{jk}= C_{kj}$ are nonzero real constants, and the exponent g being $ p> $1. Generalizing techniques of the classical N-body problem, we first characterize the periodic solutions that form a regular polygon (relative equilibria) with equal masses ($m_k= m$, $k=1, \ldots , N$) and equal constants $C_{jk}= C$, for all $j, k=1, \ldots , N$ (for short, N-gon solutions). Indeed, for $C>0$ we prove that there exists a unique regular N-gon solution for each fixed positive mass m. In contrast, for the case $C <0$, we demonstrate that there can be a maximum of two distinct regular N-gon solutions for a fixed positive mass m. More precisely, there is a range of values for the mass parameter m for which no solutions of the form of an N-gon exist. Furthermore, we examine the linear stability of these solutions, with a particular focus on the special case $ N=3 $, which is fully characterized.

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

In this paper, we identify the five dimensional analogue of the finite energy foliations introduced by Hofer–Wysocki–Zehnder for the study of three dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudo-holomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. We discuss examples, further geometric structures, and speculate on possible applications.

In this paper we study a Fermi–Ulam model where a pingpong ball bounces elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion $f(t)$ is 1-periodic and piecewise $C^3$ with a singularity, $\dot {f}(0+)\ne \dot {f}(1-)$ . If the second derivative $\ddot {f}(t)$ of the platform motion is either always positive or always less than $-g$ , where g is the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits co-exist with bounded orbits at arbitrarily high energy levels.

We present a generalization of Moser’s theorem on the regularization of Keplerian systems that include positive and negative energies. Our approach does not consider the geodesics of the hyperboloid embedded in a Lorentz space for the unbounded orbits, as it is previously done in the literature. Instead, we connect the Keplerian positive and negative energy orbits with the harmonic oscillator with negative and positive frequencies. The connection is established through the canonical extension of the stereographic projection, as it is done in Moser’s original paper. How we base our study reveals that Kustaanheimo–Stiefel map KS and Moser regularizations are alternative ways of showing the spatial Kepler system as a subdynamics of the 4D harmonic oscillator.

We study a second-order ordinary differential equation coming from the Kepler problem on ${{\mathbb{S}}^{2}}$. The forcing term under consideration is a piecewise constant with singular nonlinearity that changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

We give a uniform description, in terms of Coxeter diagrams, of the Voronoi domains of the root and weight lattices of any semisimple Lie algebra. This description provides a classification not only of all the facets of these Voronoi domains but simultaneously a classification of their dual or Delaunay cells and their facets. It is based on a much more general theory that we develop here providing the same sort of information in the setting of chamber geometries defined by arbitrary reflection groups. These generalized kaleidoscopes include the classical spherical, Euclidean, and hyperbolic kaleidoscopes as special cases. We prove that under certain conditions the Delaunay cells are Voronoi cells for the vertices of the Voronoi complex. This leads to the description in terms of Wythoff polytopes of the Voronoi cells of the weight lattices.