Mikhail

The Three Body Problem

Abstract

The force exerted on the each of the bodies by each other can be described by the inverse square law:

where in this case, the constant "k" represents the gravitational constant multiplied by the masses of the two bodies. The potential energy of the non-fixed body can be described with:

Thus the equation of motion for a particle can be described in the following manner using Lagrange’s equations:

(first term is mass times acceleration, second term is a centripetal force, L is angular momentum, total force is negative derivative of potential)

which has a solution:

where theta_0 is a phase offset and e is eccentricity, which defines the shape of the orbit. When e = 0, the orbit is a circle, e < 1 is an ellipse, e = 1 is a parabola, and e > 1 is a hyperbola.

Thus, the two-body problem has a neat analytical solution. After Kepler was able to describe the orbits of two bodies in this way, Enlightenment-era scientists naturally wanted to take it a step further and add a third body, but they were not able to find general solutions, as the gravitational interactions between the bodies introduce chaos. However, in only a few highly specialized cases, an exact analytical solution can be found.

Theoretical Analysis

Specialized solutions to the Three Body Problem

The chaotic motion of three bodies makes developing a general analytical solution for the three-body problem incredibly difficult. However, there are a few specialized cases where analytical solutions can be found. In the 19th century, Euler found a solution for when a single particle orbits around two fixed bodies. The potential energy of the particle in such a system can be described as follows:

Since the two fixed bodies are collinear, they only have a difference in their y-position by distance a. mu_1 and mu_2 are proportionality constants that represent the mass of the fixed body multiplied by the gravitational constant and the mass of the particle.

We can convert the coordinates into elliptical coordinates in the following manner:

From these, we can create an elliptical graph where the two foci represent the fixed bodies:

With this coordinate system, the potential energy of the particle can be rewritten as:

And so can the kinetic energy:

Now, using the general solution of a Liouville dynamical system, these two equations can be used to solve for the parametric solution of the orbit:

We can then use this equation to describe the motion of the particle with regards to the two fixed bodies. Here are some example orbits of the particle:

Thus, in the case that a particle orbits two fixed bodies, Euler was able to develop an analytical solution. In the 18th century, Lagrange developed another specialized solution in which two particles orbit a fixed body and form an equilateral triangle. For many years, these exact solutions to the three-body problem were the only apparent ways of modeling the movement of orbits.

However, in 1908, a Finnish mathematician named Karl Sundman finally developed a general solution to the problem. The solution involved using a converging infinite series as the solution for the orbital equation. However, the extremely slow convergence of the series necessitated an incredible amount of terms, up to 10^8 million, in order to make an accurate prediction for a typical celestial mechanics problem. Thus, while a general analytical solution does exist, it is so verbose as to be useless. Luckily, we don’t need an exact analytical solution if we have computers, as modeling the movement of bodies is easier to approximate and simulate than to predict exactly.

Computational simulations

Using numerical integration, we can simulate the movement of however many particles we want under various conditions including different gravitational constants, masses, and initial velocities. Below are some comparisons between two-body and three-body simulations using code written by Dr. Philip Mocz at Princeton.

Two Bodies

As seen above, when two particles of equal mass are allowed to float in free space, they form periodic orbits with one another that can easily be described using analytical solutions. However, when a third body is added to the system, the neatness disappears:

Although the majority of three-body solutions fall into chaos like those above, in the late 20th century computational physicists found hundreds of configurations that exhibit periodic motion including the following figure-eight and ring shapes:

However, while one can derive analytical solutions from these configurations, they are unlikely to occur in nature, and so the significance of their patterns is relegated to their aesthetic value.

Conclusion

Bibliography

https://en.wikipedia.org/wiki/Kepler_problem

https://en.wikipedia.org/wiki/Euler%27s_three-body_problem

https://www.youtube.com/watch?v=et7XvBenEo8&ab_channel=PBSSpaceTime

https://medium.com/swlh/create-your-own-n-body-simulation-with-python-f417234885e9

https://www.youtube.com/watch?v=8_RRZcqBEAc&ab_channel=MatthewSheen

https://d10lpgp6xz60nq.cloudfront.net/physics_images/NVT_21_SCI_TECH_P1_X_C09_E01_004_S02.png

https://www.researchgate.net/publication/28574844_The_problem_of_two_fixed_centers_Bifurcations_actions_monodromy