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Overview Classical dynamics and many-body systems

The motions of planets and other celestial bodies give the most convincing observational support for the laws of classical Newtonian mechanics. In this wonderful space laboratory all phenomena are observed in their purest form, without the complication of friction and air resistance that are inevitable in an ordinary earth laboratory.

The differential equations of motion for a body under the central inverse square gravitational force (for a planet orbiting a star or a satellite orbiting the planet) have exact analytic solutions (a single-body Kepler problem). The striking mathematical simplicity of trajectories is a distinctive feature of Keplerian motion. Any possible motion in the Newtonian inverse square gravitational field occurs along one of the conic sections - curves formed by the intersection of a circular cone by a plane. Exact analytic solutions exist also for the motions of two celestial bodies attracted by mutual gravitational forces - this two-body problem mathematically may be reduced to the case of a single body which moves in an effective stationary inverse-square gravitational field.

The most fascinating phenomena of celestial mechanics are revealed in the motions of three or more bodies attracted to one another by gravitational forces. If a third body is added to a system of two interacting bodies, the three-body problem generally becomes analytically unsolvable, that is, there exist no general formulas that describe the motion and permit the calculation of positions and velocities of the bodies from arbitrary initial conditions. The lack of analytic solutions is related to the extraordinary complexity of possible motions. Some examples included in the presented collection of Java applets allow us to observe fascinating trajectories of three-body motions that delight the eye and challenge our intuition. However, among the great variety of extremely complex motions there exist a finite subset of very simple regular motions. Some of these regular motions are also illustrated in this collection.

The simulations of this collection are implemented as Java applets. Java applets are run by web browsers (with Java plugin installed) under security restrictions to protect the user. In case Java applications are blocked by your security settings, as a workaround, you can use the Exception Site List feature of your operating system. Adding the URL of the blocked application (applet) to the Exception Site List allows the applet to run with some warnings. Unfortunately, this works only in Internet Explorer browser (IE).

Steps to Add URLs to the Exception Site list:

  • Go to the Java Control Panel (On Windows click Start, then Control Panel, and find Java there).
  • Click on the Security tab of the Java Control Panel.
  • Click on the Edit Site List button.
  • Click Add in the Exception Site List window.
  • Click in the empty field under the Location field to enter the URL.
  • Type in (or paste) the required URL that hosts the applet, namely
  • Click OK to save the URL that you entered.
  • Continue on the Security Warning dialog.
  • Reload the web page with the applet.

The following applets are included in the collection:

  1. Restricted three-body problem a satellite in the binary planet system
  2. Examples of the restricted three-body problem (in two frames of reference)
  3. Figure-eight periodic planar three-body motion
  4. Figure-eight three-body motion in two frames of reference
  5. Variations of figure-eight planar three-body motion
  6. Three-body motions in the equilateral configuration
  7. Regular planar motions of four bodies of equal masses

The applets of this collection are created with the help of Easy Java Simulations tool developed by Francisco Esquembre, professor at University of Murcia, Spain.

Many other examples of orbital motions can be found in the extensive package of educational software Planets and Satellites distributed by the American Institute of Physics. The package was developed by the author as a desk-top laboratory for individual highly interactive work to help students visualize and explore the laws of dynamics as they apply to both natural planetary systems and artificial satellites. The programs of this package illustrate Kepler's laws, trajectories in velocity space, various families of orbits, evolution of an orbit in the atmosphere, active manoeuvres in space and relative motions of orbiting bodies, precession of an orbit, motions of a binary star components, a planet in a double-star system, several planets orbiting a star, encountering planetary systems that exchange planets, and much more.

Besides working with the supplied preselected examples, with this package students have an opportunity to construct and investigate a model of the solar system, or to create an imaginary planetary system on their own, complete with the star, planets, moons, comets, asteroids, and satellites. Advanced students can use the package for mini-research physics projects designing active maneuvers for an interplanetary space flight and modelling it in the simulation experiment, or creating a planetary system and exploring its evolution, etc.

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Collection of remarkable three-body motions Overview