We apply Newton's laws of motion to planetary motion and other systems of a few particles and explore some of the counter-intuitive consequences of Newton's laws.

The following EJS models are described in Chapter 5 of the EJS adaptation of An Introduction to Computer Simulation Methods.

• Kepler Law: Reads the period T and the semimajor axis a of the planets and displays a plot of these data. The unit of length is the astronomical unit (AU) and the unit of time is one (Earth) year. This model is used to verify Kepler's third law and to show how text data is imported and processed in an EJS model.
   
• Newton Planet: Solves the dynamical equations of motion of a mass attracted to a massive object by an inverse square law force. This model is used as a starting point for various exercises including a test of Kepler's laws. This model also demonstrates how different Runge-Kutta-Fehlberg differential equation solvers adjust their internal step size to achieve a desired accuracy (solution tolerance).
   
• Two Planets: Solves the time evolution of a two-dimensional solar system with two planets in orbit about a fixed massive sun. The presence of a interacting planets implies that the total force on a given planet is not a central force. This new model has twice as many dynamical variables and twice as many differential equations as the Newtonian Planet model.
   
• Central Force Scattering: Solves the dynamical equations of motion of Rutherford scattering from a beam of non-interacting particles interacting with a 1/r² repulsive central force. The differential cross section is approximated after every time step by binning the trajectories according to the particle scattering angle. This teaching model is used in various problems and exercises to show how the true differential cross section emerges when the particles pass the scattering center.
   
• Three Body Models: Computes periodic trajectories of three particles of equal mass moving in a plane and interacting under the influence of gravity. Although Poincar\'{e}\index{Poincar\'{e}} showed that it is impossible to obtain an analytical solution for the unrestricted motion of these particles, solutions are known for a few special cases and it is instructive to study the properties of these solutions.
  • Euler: In 1767 Euler discovered an analytical solution for co-linear periodic orbits. The first mass is placed at the center and the other masses are placed on opposite sides with velocities that are equal but opposite. The center of mass is located at the fixed mass and moving masses are always attracted toward this point.
  • Lagrange: A second analytic solution to the unrestricted three-body problem was found by Lagrange in 1772. This solution starts with three masses at the corners of an equilateral triangle. Each mass moves in an ellipse in such a way that the triangle formed by the masses remains equilateral while periodically shrinking and expanding.
  • Moore/Montgomery: A spectacular new figure-eight solution was first discovered numerically by Chris Moore in 1993 and proven to be stable by Alain Chenciner and Richard Montgomery in 2000.

Download

• Chapter 5 XML source code in a zip archive.  Unzip this model in the EJS workspace.
• Chapter 5 ready to run examples in a jar file.
• Chapter 5 text in PDF format