Continuation of relative equilibria in the n–body problem to spaces of constant curvature

2022

Preprint

Using the properties of the angular momentum, we develop a new geometrical technique to study relative equilibria for a system of 3bodies with positive masses, moving on the two sphere under the influence of an attractive potential depending only on the mutual distances among the bodies. With the above techniques we do an analysis of the relative equilibria for the case of three bodies when they are moving on the same geodesic (Euler configurations).

Section: Relative Equilibria On a Rotating Meridian For The Cotangent...supporting

confidence: 89%

Section: Relative Equilibria On a Rotating Meridian For The Cotangent...mentioning

confidence: 99%

See 1 more Smart Citation

Three-body relative equilibria on $\mathbb{S}^2$ I: Euler configurations

Fujiwara¹,

2022

Preprint

“…Therefore, the scalene relative equilibria cannot have continuation to the Euclidean plane. In [1] the authors proved that any relative equilibria on the plane can be extended to spaces of constant curvature κ when the parameter κ is small. The above result shows that the inverse is not true, the dynamics on the sphere is much richer that on Euclidean spaces.…”

Section: Scalene Relative Equilibriamentioning

confidence: 99%

Equal masses Eulerian relative equilibria on a rotating meridian of S^2

Fujiwara¹,

2022

Preprint

Relative equilibria on a rotating meridian on S 2 in equal-mass threebody problem under the cotangent potential are determined. We show the existence of scalene and isosceles relative equilibria.Almost all isosceles triangles, including equilateral, can form a relative equilibrium, except for the two equal arc angles θ = π/2. For θ ∈ (0, 2π/3) \ {π/2}, the mid mass must be on the rotation axis, in our case, at the north or south pole of S 2 .For θ ∈ (2π/3, π), the mid mass must be on the equator. For θ = 2π/3, we obtain the equilateral triangle, where the position of the masses is arbitrary.When the largest arc angle a is in a ∈ (π/2, a c ), with a c = 1.8124..., two scalene configurations exist for given a .

“…After that, some other authors have also studied Lagrangian configurations, but just for equilateral triangle shapes moving on a plane parallel to the equatorial plane in the context of the curved positive problem (that is using the cotangent potential), see for instance [4,5,10,11,12]. In a recent work, the authors were able to obtain a continuation of any relative equilibria of the Newtonian n-body problem to spaces of constant curvature, when the value of the curvature is small [1]. In particular they could extend Lagrangian RE with arbitrary masses to the sphere.…”

Section: Introductionmentioning

confidence: 99%

Three-body relative equilibria on $S^2$ II: Extended Lagrangian configurations

Fujiwara¹,

2022

Preprint

This is a natural continuation of our first paper [6], where we develop a new geometrical technique which allow us to study relative equilibria on the two sphere. We consider a system of three positive masses on S 2 moving under the influence of an generic attractive potential which only depends on the mutual distances among the masses. We reduce the problem of finding extended Lagrangian relative equilibria to the analysis of the inertia tensor, then we obtain a more manageable equivalent inertia tensor which allow us to find new families of Lagrangian configurations.