arXiv subjects

Roberto Ciccarelli

Publications and source records attributed to Roberto Ciccarelli.

4 recordsLinked to original sources

arXiv2026

A Lagrangian framework for canonical analysis for the Holst model with $β= 0$

We perform a canonical analysis of the Holst model for General Relativity, within the framework laid out in arXiv:2401.07307 and arXiv:2010.07725, distinguishing our approach by setting the Barbero parameter to $β=0$ and leaving the lapse and shift functions unconstrained. The $β= 0$ choice is of particular interest because it is viable across all dimensions, providing a necessary foundation for extending the Loop Quantum Gravity formalism beyond $3+1$ dimensions. Through field decomposition and the projection of the field equations, we derive a system of 37 equations (10 differential constraints, 21 algebraic constraints, and 6 evolution equations) exactly matching the 37 field components to be determined. Moreover, leaving the gauge unfixed reveals that three equations, which are typically identically satisfied under normal evolution, are actually differential constraints whose triviality depends on specific gauge choices. The resulting framework remains fully consistent with the standard $3+1$ decomposition of the Einstein equations without requiring any constraints on the lapse and shift functions.

gr-qc↗

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.

math.DS↗

arXiv2025

New solutions for the symmetrical n-body problem through variational approach and optimisation techniques

Advances in the variational approach to the $n$-body problem have led to significant progress in celestial mechanics, uncovering new types of possible orbits. In this paper, critical points of the Lagrangian action associated with the $n$-body problem are analysed using evolutionary algorithms to identify periodic and symmetrical solutions of the discretised system. A key objective is to locate minimum points of the action functional, as these correspond to feasible periodic solutions that satisfy the system's differential equations. By employing both stochastic and deterministic algorithms, we explore the solution space and obtain numerical representations of these orbits. Next, we examine the stability of these orbits by treating them as critical points. One approach is to compute their discrete Morse index to distinguish between minimum points and saddle points. Another is to classify them based on their action levels. Finally, analysing the boundaries of their attraction basins allows us to identify non-minimal critical points via the Ambrosetti-Rabinowitz Mountain Pass Theorem. This leads to an updated version of the algorithm that provides a constructive proof of the theorem, yielding new orbits in specific cases. This paper builds upon and extends the results presented in \cite{nostro}, providing a more detailed theoretical framework and deeper insights into the formulation. Additionally, we present new numerical results and an extended analysis of the critical points found, further enhancing the findings of the previous study.

math.DS↗

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.

math.DS↗