Description :
    This is a page for enjoying the motion of three bodies.
    The problem of three celestial bodies moving under their mutual gravitational attraction is known as the three-body problem. It is famous for being a difficult problem in which no general analytical solution exists. However, the motion can be simulated numerically by advancing time in small steps and solving the equations of motion.
    Here, we calculate and display the motion of three bodies of equal mass, all interacting through Newtonian gravity. For simplicity, the motion is restricted to a single plane, and the equations of motion are solved using the fourth-order Runge-Kutta method. Additionally, the system allows users to freely control the celestial bodies.
    Feel free to experiment and enjoy the complex and fascinating dynamics of the three-body system!
    While care has been taken to ensure high precision in the calculations, it is impossible to eliminate errors entirely due to limitations in numerical precision and computation time. For example, even in symmetric configurations such as equilateral triangle motion, slight numerical errors accumulate and eventually break the symmetry. Once symmetry is lost, the system can transition into unpredictable, chaotic motion—demonstrating how even tiny differences can lead to drastically different outcomes. Also, if two bodies come extremely close to each other, the numerical errors can become significant. In such cases, the simulation will stop, treating the event as a collision.

● You can start or stop the simulation using the [*Start/Stop:} button.
● You can change the speed of time progression using the [Fast/Slow] button.
● You can view the shape of the orbit using the [Strobo] button.
● You can return to the initial state using the [Reset] button.
● You can select a celestial body using the control checkboxes and drag near the velocity vector to control the velocity vector and try new scenarios. If the bodies collide, the simulation will stop.
● You can move the center of mass to the center and fix the center of mass frame using the [Fix] button.
● You can adjust the scale of the display using the slider at the bottom right of the screen.
● You can change the initial velocity of the three bodies using the Initial Velocity slider at the bottom.
● You can choose examples of the Euler solution, Lagrange solution, or the figure-eight solution discovered by Chenciner and Montgomery [1] using the radio buttons at the bottom right. By default, the Lagrange solution is selected.

[1] Sim´o C 2002 Dynamical properties of the figure eight solution of the three-body problem Celestial mechanics: Dedicated to Donald Saari for his 60th Birthday. Contemporary Mathematics 292 (Providence, R.I.: American Mathematical Society) pp 209–228

Copyright 2020 KATO, Noriyoshi