arXiv subjects

Cristina Stoica

Publications and source records attributed to Cristina Stoica.

16 recordsLinked to original sources

arXiv2026

Continuation of Hamiltonian dynamics from the plane to constant-curvature surfaces

We investigate the deformation of symmetry on cotangent bundles from the Euclidean plane to two-dimensional constant-curvature surfaces and the continuation of local dynamics aspects in Hamiltonian systems. For a fixed curvature sign $σ\in\{+1,-1\}$, the curved problem is set up either on the sphere $(σ=+1)$ or on the hyperbolic plane $(σ=-1)$, both with radius $R=1/\varepsilon$, recovering flat space in the limit $\varepsilon\to 0$. The symmetry of these spaces is taken into account by using the Inönü--Wigner contraction of Lie algebras from $\mathfrak{so}(3)$ or $\mathfrak{so}(2,1)$ to $\mathfrak{se}(2)$. We use Riemannian exponential coordinates centred at the North pole together with the pull-back the associated momentum map and the symplectic form. Within this geometric setting we use a local slice construction and prove the persistence from flat to curved spaces of non-degenerate relative equilibria and relative periodic orbits of general cotangent bundle Hamiltonian systems. We apply the resulting framework to the Newtonian $n$-body problem.

math-ph↗

arXiv2026

A Note on Singular Boundary Regularisation in Hamiltonian Systems

Singular changes of variables in Hamiltonian systems, such as McGehee coordinates in celestial mechanics or renormalised variables in dispersive PDE blow-up, are designed to extend the equations of motion to a singular boundary. In contrast, it may be that near the boundary, the the induced symplectic two-form may rescale distinct geometric directions by distinct powers of the singular scale, and if the leading weighted part is degenerate, no single conformal factor makes the form extend as a smooth non-degenerate two-form. This anisotropic obstruction is complementary to the isotropic singularities studied in bm-symplectic geometry. We record it in an elementary finite-dimensional criterion and illustrate it with McGehee regularisation of homogeneous central-force collisions and, formally, with the modulation geometry of focusing nonlinear Schrödinger equation blow-up.

math.DS↗

arXiv2024

Stability of the regular $n$-gon rotating equilibria with logarithm interaction

We study the linear stability of regular $n$-gon rotating equilibria in the $n$-body problem with logarithm interaction. In the presence of a central mass $M$, linear stability is insured if $M$ is bounded below and above by constants depending on the number and mass of the (equal) outer $n$ bodies. Moreover, we provide explicit equations of these bounds. In the absence of a central mass we find that the regular $n$-gon is linearly stable for $n =2,3,\ldots 6$ only.

math.DS↗

arXiv2023

Block regularisation of the logarithm central problem

The logarithm function is the gravitational potential in $\mathbb{R}^2$. We prove that the logarithm central force problem is block regularizable, that is, the (incomplete) flow may be continuously extended over the singularity at the origin after an appropriate re-parametrization.

math-ph↗

arXiv2019

On the n-body problem in $\mathbb{R}^4$

Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in $\mathbb{R}^4$ with a general pairwise potential. We investigate the central force problem, set up the n-body problem and discuss certain properties of relative equilibria. We describe regular n-gons in $\mathbb{R}^4$ and when the masses are equal, we determine the invariant manifold of motions with regular n-gon configurations. In the case n=3 we reduce the dynamics to a six degrees of freedom system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable.

math-ph↗

arXiv2019

On the Manev spatial isosceles three-body problem

We study the isosceles three-body problem with Manev interaction. Using a McGehee-type technique, we blow up the triple collision singularity into an invariant manifold, called the collision manifold, pasted into the phase space for all energy levels. We find that orbits tending to/ejecting from total collision are present for a large set of angular momenta. We also find that as the angular momentum is increased, the collision manifold changes its topology. We discuss the flow near-by the collision manifold, study equilibria and homographic motions, and prove some statements on the global flow.

math-ph↗

arXiv2019

A note on the geometric modeling of the full two body problem

The two full body problem concerns the dynamics of two spatially extended rigid bodies (e.g. rocky asteroids) subject to mutual gravitational interaction. In this note we deduce the Euler-Poincare and Hamiltonian equations of motion using the geometric mechanics formalism.

physics.class-ph↗

arXiv2018

Notes on relative equilibria of isosceles molecules in classical approximation

We study a classical model of isosceles triatomic "A-B-A" molecules. The atoms, considered mass points, interact mutually via a generic repulsive-attractive binary potential. First we show that the steady states, or relative equilibria (RE), corresponding to rotations about the molecule symmetry axis may be determined qualitatively assuming the knowledge of 1) the shape of the binary interaction potential, 2) the equilibrium diatomic distances (i.e., the equilibrium bond length) of the A-A and A-B molecules, and 3) the distance at which the RE of the diatomic A-A molecule ceases to exist. No analytic expression for the interaction potentials is needed. Second we determine the stability of the isosceles RE modulo rotations using geometric mechanics methods and using Lennard-Jones diatomic potentials. As a by-product, we verify the qualitative results on RE existence and bifurcation. For isosceles RE we employ the Reduced Energy-Momentum method presented in [J.E. Marsden, Lectures in Mechanics, Cambridge University Press, 1992], whereas for linear (trivial isosceles) RE we apply the Symplectic Slice method, a technique based on the findings in the paper [R.M. Roberts, T. Schmah and C. Stoica, Relative equilibria for systems with configurations space isotropy, J. Geom. Phys., 56, 762, (2006)].

math.DS↗

arXiv2017

On the n-body problem on surfaces of revolution

We explore the $n$-body problem, $n\geq 3,$ on a surface of revolution with a general interaction depending on the pairwise geodesic distance. Using the geometric methods of classical mechanics we determine a large set of properties. In particular, we show that Saari's conjecture fails on surfaces of revolution admitting a geodesic circle. We define homographic motions and, using the discrete symmetries, prove that when the masses are equal, they form an invariant manifold. On this manifold the dynamics are reducible to a one-degree of freedom system. We also find that for attractive interactions, regular $n$-gon shaped relative equilibria with trajectories located on geodesic circles typically experience a pitchfork bifurcation. Some applications are included.

nlin.SI↗

arXiv2016

Central configurations of the curved $N$-body problem

We consider the $N$-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that we can recover the classical definition in the Euclidean case. After proving some criteria for the existence of relative equilibria, we find a natural way to define the concept of central configuration in curved spaces using the moment of inertia, and show that our definition is formally similar to the one that governs the classical problem. The existence criteria we develop for central configurations help us provide several examples and prove that, for any given point masses on spheres and hyperbolic spheres, central configurations always exist. We end our paper with results concerning the number of central configurations that lie on the same geodesic, thus extending the celebrated theorem of Moulton to hyperbolic spheres and pointing out that it has no straightforward generalization to spheres, where the count gets complicated even in the case $N=2$.

math.DS↗

arXiv2014

Dynamics in the Schwarzschild isosceles three body problem

The Schwarzschild potential, defined as U(r)=-A/r-B/r^3, where r is the distance between two mass points and A,B>0, models astrophysical and stellar dynamics systems in a classical context. In this paper we present a qualitative study of a three mass point system with mutual Schwarzschild interaction where the motion is restricted to isosceles configurations at all times. We retrieve the relative equilibria and provide the energy-momentum diagram. We further employ appropriate regularization transformations to analyse the behaviour of the flow near triple collision. We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for non-zero total angular momenta (e.g., when two of the mass points spin infinitely many times around the centre of mass). This phenomenon is known in celestial mechanics as the "black-hole effect" and it is understood as an analogue in the classical context of the behaviour near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits which are neither homothetic, nor homographic.

math.DS↗

arXiv2013

Normal forms for Lie symmetric cotangent bundle systems with free and proper actions

We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space suitable for the study of dynamics near relative equilibria, in particular for the Birkhoff-Poincare normal form method. For a general symmetry group, we observe that for the calculation of the truncated normal forms, one does not need an explicit coordinate transformation but only its higher derivatives at the relative equilibrium. We outline an iterative scheme using these derivatives for the computation of truncated Birkhoff-Poincare normal forms.

math.DS↗

arXiv2009

Smoothed dynamics in the central field problem

Consider the motion of a material point of unit mass in a central field determined by a homogeneous potential of the form $(-1/r^α)$, $α>0,$ where $r$ being the distance to the centre of the field. Due to the singularity at $r=0,$ in computer-based simulations, usually, the potential is replaced by a similar potential that is smooth, or at least continuous. In this paper, we compare the global flows given by the smoothed and non-smoothed potentials. It is shown that the two flows are topologically equivalent for $α< 2,$ while for $α\geq 2,$ smoothing introduces fake orbits. Further, we argue that for $α\geq 2,$ smoothing should be applied to the amended potential $c/(2r^2)-1/r^α,$ where $c$ denotes the angular momentum constant.

math-ph↗

arXiv2009

Escape dynamics in collinear atomic-like three mass point systems

The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise-interaction. Specifically, we focus on systems with non-negative total energy. We show that on the zero energy level set, most of the orbits lead to binary escape configurations and the set of initial conditions leading to escape configurations where all three separations infinitely increase as $t \to \infty 1$ has zero Lebesgue measure. We also give numerical evidence of the existence of a periodic orbit for the case when the two outer masses are equal. For positive energies, we prove that the set of initial conditions leading to escape configurations where all three separations infinitely increase as $t \to \infty$ has positive Lebesgue measure. Keywords: linear three point

math-ph↗

arXiv2005

Saari's Conjecture is True for Generic Vector Fields

The simplest non-collision solutions of the N-body problem are the "relative equilibria", in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured that the converse is also true for the planar Newtonian N-body problem: relative equilibria are the only constant-inertia solutions. A computer-assisted proof for the 3-body case was recently given by R. Moeckel. We present a different kind of answer: proofs that several generalisations of Saari's conjecture are generically true. Our main tool is jet transversality, including a new version suitable for the study of generic potential functions.

math.DS↗

arXiv2003

Global Dynamics in the Singular Logarithmic Potential

We present an analytical description of the motion in the singular logarithmic potential. This potential plays an important role in the modeling of triaxial systems (like elliptical galaxies) or bars in the centers of galaxy disks. In order to obtain information about the motion near the singularity, we resort to McGehee -type transformations and regularize the vector field. In the axis-symmetric case (b=1), we offer a complete description of the global dynamics. In the non axis-symmetric case (b<1), we prove that all orbits, with the exception of a negligible set, are centrophobic and retrieve numerically partial aspects of the orbital structure.

astro-ph↗