Figure-eight choreographies with Lennard-Jones-type potentials

Hiroshi FUKUDA

(2016.07.12)

Preprint

animation by mp4 or gif

The following movies are the figure-eight solutions to the equation of motion

d²q_i/dt² = -∂U/∂q_i, i = 0,1,2

with the initial conditions

q₀ = (x₀, y₀), q₁ = (-2x₀, 0), q₂=(x₀, -y₀),

dq₀/dt = v(q₁-q₀)/|q₁-q₀|, dq₁/dt = (0, 2v(1+(3x₀/y₀)²)^-1/2), dq₂/dt = v(q₂-q₁)/|q₂-q₁|,

which are calculated by a set of parameters (x₀, y₀, v) given in each movie. Periods T's of the movies are relatively correct, and unit of T is 3/14.591=0.2056 [sec].

The Newtonian three-body problem in 2D

U = Σ_i_>_j log r_ij

(x₀, y₀, v) = (x₀, 0.407146x₀, 1.29879), T=9.76440x₀, E=3.75790+3logx₀

The Newtonian three-body problem

U = -Σ_i_>_j 1/r_ij

(x₀, y₀, v) = (x₀, 0.638775x₀, 0.824678x₀^-1/2), T=15.9191x₀^3/2, E=-0.695711x₀^-1

Associated periodic orbits H1-H3 found by Simó, T=1, E=-0.7055155727, S=2.116546718=-3ET:

A strong force potential

U = -Σ_i>j 1/r_ij²

(x₀, y₀, v) = (x₀, 0.778896x₀, 0.742100x₀^-1), T=18.1407x₀², E=0

A homogeneous potential

U = -Σ_i>j 1/r_ij⁶

(x₀, y₀, v) = (x₀, 0.985945x₀, 0.234675x₀^-3), T=61.2000x₀⁴, E=0.0467827x₀^-6

The Lennard-Jones-type potential

U = Σ_i_>_j( 1/r_ij¹² -1/r_ij⁶)

Series α

(x₀, y₀, v; E) = (0.682, 0.624904, 0.657065; 0.291225), T=14.5910

(x₀, y₀, v; E) = (1.5, 1.47862, 0.0694721; 0.00409703), T=310.066

(x₀, y₀, v; E) = (1.5, 0.502649, 0.181062; 0.00312146), T=234.081

Series β

(x₀, y₀, v; E) = (0.726, 0.766265, 0.302694; 0.0274632), T=26.4590

(x₀, y₀, v; E) = (1.0, 0.956733, 0.144241; 0.00263843), T=70.8963

(x₀, y₀, v; E) = (1.0, 1.241130, 0.0717890; 0.000697053), T=130.620

Series γ

(x₀, y₀, v; E) = (0.6007, 0.748371, 0.371779; 0.0622734), T=19.5424

(x₀, y₀, v; E) = (0.8, 1.08184, 0.126051; 0.00561328), T=68.2962

(x₀, y₀, v; E) = (0.8, 1.13674, 0.0749665; -0.00519619), T=77.2852

Series δ

(x₀, y₀, v; E) = (0.6501, 0.597985, 0.304229; -0.143858), T=24.4202

(x₀, y₀, v; E) = (0.84, 0.827038, 0.126408; -0.0330865), T=66.6927

(x₀, y₀, v; E) = (0.84, 0.848830, 0.0757119; -0.0387688), T=75.9582

Series ε

(x₀, y₀, v; E) = (0.7074, 0.579781, 0.204620; -0.211945), T=35.1563

(x₀, y₀, v; E) = (0.91, 0.803912, 0.0857343; -0.0497687), T=94.6090

(x₀, y₀, v; E) = (0.91, 0.811359, 0.0540501; -0.0521151), T=104.143