Three Body Problem

ThreeBody

System parameters:

• m₁: mass of body 1
• m₂: mass of body 2
• m₃: mass of body 3
• G: gravitational constant

The default initial condition is the figure-eight choreography (figure_eight, aliased as initial_condition) and the default window is one of its periods. The 4096-member initial_conditions grid of [6] is also provided, but every one of its members ends in a collision — see sympnets_initial_condition — so those are usable only on a window that stops short of it.

One period of the default figure_eight initial condition.

Four hundred steps per period of the default figure_eight initial condition. Over one period ImplicitMidpoint then conserves energy to $4 \times 10^{-9}$ and closes the orbit to $1.3 \times 10^{-3}$; Gauss(2) reaches $3 \times 10^{-15}$ and $1.3 \times 10^{-7}$.

Neither this nor any other step size makes sympnets_initial_condition integrable past its collision, and neither does exchanging the nonlinear solver. Over $t \in [0, 1]$ at $\Delta{}t = 0.5, 0.05, 0.01, 0.005, 0.001$ the default Newton method with a backtracking line search commits an energy error of 19, 15, 36, 399 and 249, and the trust-region DogLeg solver one of 9, 45, 12, 1615 and $2.2 \times 10^6$ — with the two ending up 0.6, 3.8, 5.0, 50 and 1209 apart in $q$. Neither is quieter than the other about it — they emit 5, 9, 4, 4, 4 and 6, 1, 1, 5, 6 solver messages respectively — though the count is not the thing to compare on: repeated reports are capped by maxlog and a stagnating solve stops after max_stalls, so it measures how a failure is reported rather than how badly it fails. The energy error is, and on it DogLeg is not an improvement but a regression: it is worse at the two finest step sizes by factors of 4 and $9 \times 10^3$, and better nowhere by more than a factor of 3. Since neither solver can integrate the collision and the trust region is the worse of the two where it differs, the default Newton solver is kept; on problems that are solvable the two agree bit for bit.

Constant taken from [6].

The default initial condition: the figure-eight choreography of [5], in which three equal masses chase one another along a single figure-eight curve.

It is the default because, unlike every member of initial_conditions (see sympnets_initial_condition), it has no close encounters — the smallest mutual distance over a period is 0.69 — so it can be integrated over many periods. Over one period DEFAULT_TIMESTEP conserves energy to $4 \times 10^{-9}$ and closes the orbit to $1.3 \times 10^{-3}$ with ImplicitMidpoint.

The momenta equal the velocities because the masses are one: $v_3 = (-0.93240737, -0.86473146)$ and $v_1 = v_2 = -v_3 / 2$.

Period of the figure_eight choreography for $G = m_1 = m_2 = m_3 = 1$.

Alias for figure_eight, the default initial condition of the problem constructors.

Here we get 4096 trajectories that should be similar to the data used in [6].

Constant taken from [6].

Constant taken from [6].

Constant taken from [6].

The first member of initial_conditions. It is not the default initial condition, because it ends in a collision: the bodies collide at $t \approx 0.08867$, where the solution ceases to exist. Successive RK4 refinements resolve a smallest mutual distance of $1.95 \times 10^{-3}$, $5.39 \times 10^{-4}$, $4.78 \times 10^{-5}$ and $2.93 \times 10^{-6}$ for $\Delta{}t = 10^{-4} \ldots 10^{-7}$ — the distance keeps collapsing at a fixed time rather than bottoming out — and two RK4 references at $\Delta{}t = 10^{-6}$ and $5 \times 10^{-7}$ disagree by 36 in position, so there is no computable trajectory to compare against either.

Every member of initial_conditions behaves this way: all 4096 come within 0.04 of a collision within $t \in [0, 5]$. Integrating any of them past its collision is not a matter of the step size or of the nonlinear solver (see DEFAULT_TIMESTEP); it needs regularization (Kustaanheimo–Stiefel, Levi-Civita) or an adaptive time transformation, neither of which this package provides. Use it only on a window that ends before the collision.

Turn array into a vector.

Default parameters taken from [6].

Hamiltonian ensemble for the three-body problem (varying initial conditions and/or parameters).

hodeproblem(q₀, p₀; timespan, timestep, parameters)

Hamiltonian version of the three-body problem

Constructor with default arguments:

hodeproblem(
    q₀ = [0.97000436, -0.24308753, -0.97000436, 0.24308753, 0.0, 0.0],
    p₀ = [0.46620368, 0.43236573, 0.46620368, 0.43236573, -0.93240737, -0.86473146];
    timespan = (0.0, 6.32591398),
    timestep = 0.01581478495,
    parameters = (m₁ = 1.0, m₂ = 1.0, m₃ = 1.0, G = 1.0)
)

Lagrangian ensemble for the three-body problem (varying initial conditions and/or parameters).

lodeproblem(q₀, p₀; timespan, timestep, parameters)

Lagrangian version of the three-body problem

Constructor with default arguments:

lodeproblem(
    q₀ = [0.97000436, -0.24308753, -0.97000436, 0.24308753, 0.0, 0.0],
    p₀ = [0.46620368, 0.43236573, 0.46620368, 0.43236573, -0.93240737, -0.86473146];
    timespan = (0.0, 6.32591398),
    timestep = 0.01581478495,
    parameters = (m₁ = 1.0, m₂ = 1.0, m₃ = 1.0, G = 1.0)
)