Families of periodic solutions of the 4- and 6-body problem using a gradient-free continuation method

This paper introduces a gradient-free stochastic black-box method to construct families of pseudo-periodic planar solutions for the 4- and 6-body problems, where bodies of equal masses move in symmetric opposite or triangular configurations subject to return conditions involving rotation and relabeling.

The Gist

Imagine the universe as a giant, cosmic dance floor. In this dance, stars and planets (which we'll call "bodies") are constantly pulling on each other with gravity, trying to figure out the perfect steps to stay in sync without crashing into one another. This is the famous N-Body Problem.

For a long time, mathematicians have been trying to find specific dance routines where the dancers return to their exact starting positions and repeat the same steps forever. These are called periodic solutions. While it's easy to predict the dance of two partners (like the Earth and the Sun), adding more dancers makes the choreography incredibly complex, like trying to solve a puzzle where every piece is moving and changing shape at the same time.

This paper by Oscar Perdomo is about finding new, beautiful dance routines for 4 dancers and 6 dancers.

The Challenge: A Puzzle Without a Map

Usually, to solve these complex equations, mathematicians use a "gradient" method. Think of this like hiking down a mountain in the fog. You feel the slope under your feet (the gradient) to know which way is down. You take a step, feel the slope again, and keep going until you reach the bottom (the solution).

However, in this specific cosmic dance, the "mountain" is so jagged and full of cliffs that the slope is impossible to calculate. The math breaks down if you try to measure the slope. It's like trying to hike down a mountain made of glass where you can't feel the ground, only see if you fall.

The Solution: The "Blindfolded Explorer"

Perdomo developed a new tool called a Gradient-Free Continuation Method. Instead of feeling the slope, imagine a blindfolded explorer trying to find the bottom of a valley.

1. The Guessing Game: The explorer stands at a spot and throws a handful of pebbles in random directions within a certain range.
2. Checking the Score: For each pebble, they check if it landed closer to the bottom (a better solution) or if it fell off a cliff (a failed calculation).
3. Learning from Success: If a pebble lands in a better spot, the explorer remembers how they threw it. They don't just throw randomly anymore; they start throwing more pebbles in that successful direction, but they also keep the range wide enough to explore new areas.
4. Adapting: If they keep failing, they shrink their throwing range to look more closely at the current spot. If they succeed, they expand their range to explore further.

This method is "stochastic" (random) and "black-box" (it doesn't need to know the internal rules of the math, it just needs to know if a guess is good or bad). It's a smart, adaptive way of guessing until the perfect dance routine is found.

The Dance Routines Found

Using this "blindfolded explorer" method, Perdomo discovered two families of dances:

1. The 4-Body Dance (The Double Pair)
Imagine two pairs of dancers.

• Pair A: Two heavy dancers (mass 1) holding hands and spinning around the center, always opposite each other.
• Pair B: Two lighter dancers (mass $m_2$) also holding hands and spinning opposite each other.
• The Twist: The two pairs are spinning at different speeds and angles. The paper finds the exact starting positions and speeds so that after a certain time, the whole group returns to the start, but the two pairs have swapped places (like a choreographed exchange).

2. The 6-Body Dance (The Double Triangle)
Imagine two triangles of dancers.

• Triangle A: Three heavy dancers forming an equilateral triangle, spinning around the center.
• Triangle B: Three lighter dancers forming their own equilateral triangle, also spinning.
• The Twist: The two triangles are rotating relative to each other. The method finds the precise starting conditions where, after a specific time, the entire formation looks exactly the same as the start, just rotated and with the dancers in the triangles swapped.

Why This Matters

The paper doesn't just find one solution; it finds a whole family of solutions. By changing a single angle (how much one pair is rotated relative to the other), the explorer finds a continuous stream of new, valid dances.

The author provides a "menu" of these dances (Tables 1 and 2 in the paper). If you plug these numbers into a computer, you can watch the 4 or 6 bodies dance in perfect, repeating loops. The paper even includes links to videos of these dances, showing that these aren't just abstract numbers, but real, beautiful motions that could theoretically exist in the universe.

In short: The author built a smart, random-guessing computer program to find the perfect choreography for 4 and 6 celestial bodies, discovering a whole new library of cosmic dances that were previously too hard to find.

Technical Summary

1. Problem Statement

The paper addresses the classical $n$-body problem in celestial mechanics, specifically seeking periodic solutions for $n=4$ and $n=6$ bodies under Newtonian gravity (with gravitational constant $G=1$).

The author investigates specific symmetric configurations:

• 4-Body Problem: Two pairs of bodies. Bodies 1 and 2 have mass $1$ and move on opposite sides of the origin (diametrically opposed). Bodies 3 and 4 have mass $m_2$ and also move diametrically opposed, but rotated $90^\circ$ relative to the first pair.
• 6-Body Problem: Two equilateral triangles centered at the origin. Bodies 1, 2, and 3 have mass $1$ forming one triangle. Bodies 4, 5, and 6 have mass $m_2$ forming a second triangle. The second triangle is rotated relative to the first.

The Challenge:
Finding periodic solutions requires satisfying "return conditions" where the system returns to its initial state (up to rotation and relabeling of bodies) after a period $T$.

• For the 4-body case, the system involves 6 variables ($r_1, r_2, \dot{\theta}, \dot{\beta}, m_2, T$) and 8 equations (return conditions for positions, velocities, and angular relationships).
• For the 6-body case, the system similarly involves 6 variables and 8 equations.
• Standard Newton-Raphson methods failed to converge for these specific return equations in the author's experiments, likely due to the sensitivity of the differential equations and the lack of good initial guesses.

2. Methodology

The core contribution of the paper is the development and application of a gradient-free, adaptive stochastic continuation method to solve the resulting system of non-linear equations.

A. Mathematical Formulation

The author derives the equations of motion for the specific ansatzes (symmetry assumptions) using polar coordinates.

• Differential Equations: The motion is reduced to a system of four second-order ODEs for the radial distances ($r_1, r_2$) and angular velocities ($\theta, \beta$).
• Return Conditions: To find a periodic solution, the author defines a vector function $h(Z) = 0$, where $Z$ contains the initial conditions and parameters. The components of $h(Z)$ represent the difference between the state at time $T$ and the required return conditions (e.g., $r_1(T) = r_1(0)$, $\theta(T) - \beta(T) = \pi$ for the 4-body case).
• Black-Box Nature: The evaluation of $h(Z)$ involves numerically integrating the ODEs. If the integration fails (e.g., singularity or non-termination), the point is considered outside the domain.

B. The Gradient-Free Solver (Algorithm 1)

Since gradients are unavailable or unreliable, the author employs a stochastic direct-search method with adaptive learning:

1. Initialization: Start with an initial guess $Z_0$ and a search box defined by radii $d$.
2. Sampling: At each iteration, generate $N$ random candidate points within an axis-aligned box centered at the current best solution ($Z_{best}$).
3. Evaluation: Compute the error functional $Err(Z) = \|h(Z)\|_\infty$ for all candidates. If evaluation fails, error is set to $+\infty$.
4. Selection: Identify the candidate with the minimum error ($Z^*$).
5. Adaptive Update:
   • If Improvement ($e_S < e_{best}$): Accept $Z^*$. Update the search box radii $d$ using the magnitude of the successful displacement ($\Delta_{last}$) to "learn" the local scale of the solution space.
   • If No Improvement: Shrink the search box using a factor $\rho$, but prevent it from collapsing below a minimum size $d_{min}$ by retaining information from the last successful move.
6. Termination: The process repeats until the error is below a target threshold ($10^{-7}$) or a maximum number of iterations is reached.

This method is distinct from standard direct-search algorithms because it uses a data-driven box adaptation based on successful displacements, allowing it to navigate complex, non-convex landscapes where traditional gradient methods fail.

3. Key Results

The author successfully computed families of periodic solutions parametrized by the final rotation angle $\theta(T) = \theta_1$.

• 4-Body Family:

  • Solutions were found for $\theta_1$ ranging from $30^\circ$ to $360^\circ$ in steps of $15^\circ$.
  • The method provided initial conditions (positions, velocities, mass ratio $m_2$, and period $T$) with errors $< 10^{-7}$.
  • Observation: The solutions exhibit a "pseudo-periodic" nature where the configuration returns to the initial state after a relabeling of bodies (e.g., bodies 3 and 4 are exchanged) and a rotation.

• 6-Body Family:

  • Solutions were found for $\theta_1$ ranging from $30^\circ$ to $180^\circ$.
  • Similar high-precision initial conditions were generated.
  • The trajectories show complex interactions where the two triangles rotate at different rates but maintain their equilateral shapes.

• Numerical Data: The paper provides Table 1 and Table 2, listing 12 decimal places of precision for the initial conditions ($r_1, r_2, \dot{\theta}, \dot{\beta}, m_2$) and the period $T$ for various rotation angles.

4. Significance and Contributions

1. Novel Algorithm: The paper introduces a robust, gradient-free continuation method tailored for celestial mechanics problems where standard Newton methods fail. The adaptive mechanism (using $\Delta_{last}$) effectively handles the anisotropy of the solution space.
2. New Families of Solutions: It discovers and numerically validates new families of periodic orbits for the 4- and 6-body problems that were previously difficult to compute.
3. Relaxation of Constraints: Unlike previous studies (e.g., García-Azpeitia and Ize) that imposed strict constraints like $\beta(t) = \theta(t) + c$ (leading to central configurations), this method allows the radii ratio $r_1(t)/r_2(t)$ to vary, exploring a broader class of solutions.
4. Reproducibility: The author provides highly precise initial conditions and links to video simulations, allowing other researchers to verify and build upon these results.

Conclusion

Oscar Perdomo demonstrates that a stochastic, black-box, gradient-free optimization strategy is highly effective for solving the complex, non-linear boundary value problems inherent in the $n$-body problem. By avoiding the need for analytical derivatives and adapting the search space based on successful moves, the method successfully uncovers families of periodic solutions that were previously inaccessible to standard numerical techniques.