Research Outputs
Periodic orbits and KAM tori of a particle around a homogeneous elongated body
We analyse the dynamics of an infinitesimal particle around an elongated body, which is modelled as a homogeneous fixed straight segment centred at the origin. We assume that the length of the segment is small compared with the distance to the particle. After a Lie–Deprit normalization, we end up with a Hamiltonian that has not only the mean anomaly but also the argument of the perigee relegated to terms or third order or higher. We employ invariant and reduction theories to reduce the artificial symmetries associated with the Kepler flow and the central action of the angular momentum. Analysing the relative equilibria in the first and second reduced spaces allows us to determine the existence of near-polar circular periodic orbits and KAM tori.
Geometric Numerical Test via Collective Integrators: A Tool for Orbital and Attitude Propagation
We propose a novel numerical test to evaluate the reliability of numerical propagations, leveraging the fiber bundle structure of phase space typically induced by Lie symmetries, though not exclusively. This geometric test simultaneously verifies two properties: (i) preservation of conservation principles, and (ii) faithfulness to the symmetry-induced fiber bundle structure. To generalize the approach to systems lacking inherent symmetries, we construct an associated collective system endowed with an artificial G-symmetry. The original system then emerges as the G-reduced version of this collective system. By integrating the collective system and monitoring G-fiber bundle conservation, our test quantifies numerical precision loss and detects geometric structure violations more effectively than classical integral-based checks. Numerical experiments demonstrate the superior performance of this method, particularly in long-term simulations of rigid body dynamics and perturbed Keplerian systems.