The N-body problem involves calculating the motion of multiple bodies under the influence of each other (gravity). This Python code implements a numerical solution for the N-body problem in 3D space using the Runge-Kutta of Order 4 (RK4) method, and then creates an animation to display it.

The N-body problem refers to the gravitational interaction between multiple bodies, and the specific Ordinary Differential Equation (ODE) being solved describes the gravitational interaction between these bodies. More precisely, the ODE being solved is a set of second-order ODE representing the gravitational interaction between all bodies. OBS: The gravitational force between two bodies is determined by Newton's law of gravitation, and the acceleration of each body is influenced by the gravitational forces exerted by all other bodies in the system.

n_body_solver_3d function:

• The code begins with parameter checks to ensure the coherence of input data.
• n_body_acceleration subfunction calculates the acceleration suffered by each body based on masses and current positions.
• For each time moment uses n_body_acceleration within the RK4 integration method, to numerically solve the system of ODEs and determine the motion suffered by each body (final position of each body).
• Returns a list with all bodies positions in each time moment (bodies_pos_list), the number of time moments calculated (num_frames), the number of bodies in the system (num_bodies).

animation_3d function:

• The code begins with parameter checks to ensure the coherence of input data.
• Creates a 3D graph with all bodies in their current positions. Do it to each time moment.
• Then synthesize all the graphics into an animation.

bodies_pos_list, num_frames, num_bodies = n_body_solver_3d(masses_list, initial_pos_list, initial_vel_list, G, h, t_max)
animation_3d(bodies_pos_list, num_frames, num_bodies, plot_scale)

n_body_solver_3d:

• masses_list: List of masses of the N bodies.
• initial_pos_list: List of initial positions of the N bodies in three dimensions.
• initial_vel_list: List of initial velocities of the N bodies in three dimensions.
• G: Gravitational constant.
• h: Time step for the RK4 integration method.
• t_max: Maximum simulation time.

animation_3d:

• bodies_pos_list: List of bodies positions, returned by n_body_solver.
• num_frames: Number of frames in animation.
• num_bodies: Number of bodies in the system.
• plot_scale: List (shape (3,2)) of plot scale limits in X,Y and Z .
• The other parameters are defined to customize the animation and they're very intuitive, however, I will leave some examples using all the available parameters.

• I made some usage examples calculating the motion of solar system, you can check both code and animations in the repository.
• Basically I took the current position/velocity/mass of each planet and used it as initial conditions, then I created 2 animations using different scales, one to better see the inner planets and another for the outer planets.
• I did this process to calculate the movement of solar system considering it alone in the universe (vacuum), and another considering the movement of the galaxy.