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7. Regular planar motions of four bodies of equal masses

Four bodies of equal masses coupled by mutual gravitational forces can synchronously execute surprisingly simple (Keplerian) motions, preserving the initial square configuration. This section shows examples of possible regular motions of four bodies in such equilateral configuration. The motion can be displayed either in the inertial centre-of-mass frame of reference, or in the non-inertial frame associated with one of the bodies ("geocentric" frame).

1. Circular motion of four bodies in the case of equal masses. Four bodies of equal masses coupled by mutual gravitation can move along a common circle circumscribing the square formed by the bodies. In this case the square rotates uniformly about its centre as a rigid body.

2. Instability of the circular motion of four bodies. In this example the initial velocity of one of the bodies of equal masses slightly differs from the value that provides the circular motion of the system. This tiny violation of symmetry in the initial conditions leads to a progressively increasing distortion of the equilateral configuration. The motion remains reversible until the bodies collide. (Click also here to see the applet.)

3. Elliptical motion of four bodies in the case of equal masses. If the initial velocities of the bodies, being equal in magnitudes (and directed perpendicularly to radius vectors of the bodies), differ from the value that provides the circular motion, the bodies synchronously trace congruent ellipses whose axes make angles of 90 degrees with one another (in the inertial centre-of-mass frame). In this case the square rotates nonuniformly about its centre, and the length of its sides vary periodically. (Click also here to see the applet.)

4. Instability of the elliptical motion of four bodies in the equilateral configuration. In this example the initial velocity of one of the bodies slightly differs from the initial velocities of the other bodies. This tiny violation of symmetry in the initial conditions leads to a progressively increasing distortion of the equilateral configuration. However, the motion remains reversible until the bodies collide. (Click also here to see the applet.)

Explanation

If four bodies of equal masses are located at the vertices of a square, the net force exerted on each of the bodies by the gravitation of the other three bodies is directed towards the centre of this square. The magnitude of this force is inversely proportional to the square of the distance of the body from this centre. That is, each body can behave as if it were subjected to the gravitational force produced by a single immovable point source (with some effective gravitational mass) located at the centre of the square, rather than to the gravitational forces exerted by the other three moving bodies. The same is true for all the four bodies, provided the equilateral configuration is preserved during their motion. Therefore the bodies can synchronously execute Keplerian motions along congruent (equal) conic sections. In particular, they can move along a common circle circumscribing the square formed by the bodies. In this case the square rotates uniformly about its centre. Click here (and once more here to see the applet) to observe an example of such a motion.

The circular motion in the equilateral configuration occurs if all the four bodies (of equal masses) have initial velocities directed perpendicularly to their radius vectors. These velocities must have equal and quite definite magnitudes. However, if equal magnitudes of the initial velocities differ from this value, and/or the velocities are not perpendicular to the radius vectors (but directed for all the bodies at the same angle with the radius vector), the bodies will synchronously trace congruent ellipses (generally conic sections) whose axes make angles of 90 degrees with one another (in the inertial centre-of-mass frame). Click here (and once more here to see the applet) to observe an example of such a motion.

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