Periodic solutions of the N -body problem with Lennard-Jones-type potentials

We report on the Morse index and periodic solutions bifurcating from the figure-eight choreography for the equal mass three-body problem under homogeneous potential −1/r a for a ≥ 0, and under Lennard-Jones (LJ) type potential 1/r 12 − 1/r 6 , where r is a distance between bodies. It is shown that the Morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the Morse index. Inversely we observed bifurcation occurs at every point where the Morse index changes for the figure-eight choreography under −1/r a , and for α solution under LJ type potential, where α solution is a figure-eight choreography tending to that under −1/r 6 for infinitely large period. Thus, to our numerical studies, change of the Morse index is not only necessary but also sufficient condition for bifurcation for these choreographies. Further we observed that the change of the Morse index is equal to the number of bifurcated solutions regarding solutions with congruent orbits as the same solution.Submitted to: J. Phys. A: Math. Theor.

Section: Introductionmentioning

confidence: 99%

Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem

Fukuda

Fujiwara

Ozaki

2019

“…a model potential between atoms called Lennard-Jones-type (hereafter LJ) potential. Sbano and Southall [4] proved that there exist at least two N -body choreographic solutions for sufficiently large period T , and there exists no solution for small period T . Then we confirmed their theorem numerically and unexpectedly found a multitude of figure-eight choreographic solutions under LJ potential (1) [5].…”

Section: Introductionmentioning

confidence: 99%

Morse index for figure-eight choreographies of the planar equal mass three-body problem

Fukuda

Fujiwara

Ozaki

2018

We report on numerical calculations of Morse index for figure-eight choreographic solutions to a system of three identical bodies in a plane interacting through homogeneous potential, −1/r a , or through Lennard-Jones-type (LJ) potential, 1/r 12 − 1/r 6 , where r is a distance between the bodies. The Morse index is a number of independent variational functions giving negative second variation S (2) of action functional S. We calculated three kinds of Morse indices, N , N c and N e , in the domain of the periodic, the choreographic and the figure-eight choreographic function, respectively. For homogeneous system, we obtain N = 4 for 0 ≤ a < a 0 , N = 2 for a 0 < a < a 1 , N = 0 for a 1 < a, and N c = N e = 0 for 0 ≤ a, where a 0 = 0.9966 and a 1 = 1.3424. For a = 1, we show a strong relationship between the figureeight choreography and the periodic solution found by Simó through the S (2) . For LJ system, we calculated the index for the solution tending to the figure-eight solution of a = 6 homogeneous system for the period T → ∞. We obtain N , N c and N e as monotonically increasing functions of the gradual change in T from T → ∞, which start with N = N c = N e = 0, jump at the smallest T by 1, and reach N = 12, N c = 4, and N e = 1 for T → ∞ in the other branch.Submitted to: J. Phys. A: Math. Gen.

“…Sbano [3], and Sbano and Southall [4], after that, studied mathematically N -body choreographic solutions under an inhomogeneous potential u(r) = 1 r 12 − 1…”

Section: Introductionmentioning

confidence: 99%

Figure-eight choreographies of the equal mass three-body problem with Lennard-Jones-type potentials

Fukuda¹,

Fujiwara²,

Ozaki³

2017

Abstract. We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones-type, 1/r 12 − 1/r 6 where r is a distance between the particles. By numerical search, we found there are a multitude of such solutions. A series of them are close to a figure-eight solutions to a homogeneous system with no 1/r 12 term in the potential. The rest are very different from them and have several points with large curvatures in their figure-eight orbits, at which particles are repelled. Here figure-eight choreographies are the periodic motion whose shape is symmetric in both horizontal and vertical axis, starting with an isosceles triangle configuration and going back to an isosceles triangle configuration with opposite direction through Euler configuration. Thus the lobe of this figure-eight may be complex shape and needs not to be convex.