arXiv subjects

Gian Marco Canneori

Publications and source records attributed to Gian Marco Canneori.

8 recordsLinked to original sources

arXiv2026

Scattering Dynamics and Chaotic Motions in a Relativistic Two-Centre Problem

We study the planar relativistic two-centre problem at fixed energy, including a class of perturbations of the Keplerian potential. Up to a reparametrisation of time, the relativistic dynamics is equivalent to a classical two-centre system with critical strong-force singularities. This reduction allows us to apply variational methods based on the Maupertuis functional. We construct collision-free relativistic trajectories with prescribed symbolic itineraries and obtain a coding of the dynamics by admissible sequences. In particular, we prove the existence of bounded orbits, scattering solutions with prescribed asymptotic directions, and trapped trajectories that are asymptotically free in one time direction and exhibit prescribed symbolic behaviour in the other.

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arXiv2025

Symmetric solutions of the $n$-body problem: a numerical study of Floquet multipliers and Morse indices

In this paper, we consider periodic solutions of the $n$-body problem that satisfy symmetry constraints, expressed through invariance under finite group actions. We focus on their stability properties and present algorithms specifically designed for the computation of Floquet multipliers and Morse indices. Numerical results are provided to illustrate our methods in both two and three dimensional configuration spaces, and for different choices on the number of bodies.

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arXiv2025

Frozen planet orbits for the $n$-electron atom

We seek periodic trajectories of a system of multiple mutually repelling electrons on a half-line, with an attractive nucleus sitting at the origin. We adopt a variational viewpoint and study critical points of the associated Lagrange-action functional, by means of a modified Lusternik-Schnirelmann theory for manifolds with boundary. Additionally, when the charges of the electrons tend to zero, we show that frozen planet orbits converge to segments of a brake orbit for a Kepler-type problem, establishing a strong analogy with the Schubart orbits of the gravitational $n$-body problem.

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arXiv2024

Equivariant optimisation for the gravitational $n$-body problem: a computational factory of symmetric orbits

In this paper we present \texttt{SymOrb.jl}, a software which combines group representation theory and variational methods to provide numerical solutions of singular dynamical systems of paramount relevance in Celestial Mechanics and other interacting particles models. Among all, it prepares for large-scale search of symmetric periodic orbits for the classical $n$-body problem and their classification, paving the way towards a computational validation of Poincaré conjecture about the density of periodic orbits. Through the accessible language of Julia, \texttt{SymOrb.jl} offers a unified implementation of an earlier version. This paper provides theoretical and practical guidelines for the specific approach we adopt, complemented with examples.

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arXiv2024

Mountain pass frozen planet orbits in the helium atom model

We seek frozen planet orbits for the helium atom through an application of the Mountain Pass Lemma to the Lagrangian action functional. Our method applies to a wide class of gravitational-like interaction potentials thus generalising the results in [7] (Cieliebak, Frauenfelder and Volkov - 2023). We also let the charge of the two electrons tend to zero and perform the asymptotic analysis to prove convergence to a limit trajectory having a collision-reflection singularity between the electrons.

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arXiv2023

Chaotic phenomena for generalised N-centre problems

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

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arXiv2021

Symbolic dynamics for the anisotropic $N$-centre problem at negative energies

The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-α_j$: \[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional. Compared with the classical $N$-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.

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arXiv2020

Minimal collision arcs asymptotic to central configurations

We are concerned with the analysis of finite time collision trajectories for a class of singular anisotropic homogeneous potentials of degree $-α$, with $α\in(0,2)$ and their lower order perturbations. It is well known that, under reasonable generic assumptions, the asymptotic normalized configuration converges to a central configuration. Using McGehee coordinates, the flow can be extended to the collision manifold having central configurations as stationary points, endowed with their stable and unstable manifolds. We focus on the case when the asymptotic central configuration is a global minimizer of the potential on the sphere: our main goal is to show that, in a rather general setting, the local stable manifold coincides with that of the initial data of minimal collision arcs. This characterisation may be extremely useful in building complex trajectories with a broken geodesic method. The proof takes advantage of the generalised Sundman's monotonicity formula.

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