After Newton solved the problem of the orbit of a single planet around the Sun, the natural next challenge was to find the solution for two planets orbiting the Sun. Many of the best minds in mathematics and physics worked on this problem in the last century.

The current state of our model is given by five numbers: the position of Jupiter along it's orbit, and two velocities and positions for the Earth. One of these can be removed by moving to a coordinate system which rotates with Jupiter. Earth's angular momentum and energy are not conserved, because Jupiter provides an external periodic force. However, there is a conserved quantity

For our problem, I tried a lot of different maps. Taking a snapshot
once each Jovian year seemed natural, but leaves one with four coordinates
for the Earth: projecting the four dimensional space to two lost a lot
of information. The projection used in the program is where the Earth
lies directly between Jupiter and the Sun (the plane
**rE x rJ = 0** for **rE . rJ > 0**).
If you select ``Poincare (rotating reference frame)'' under View,
the program will plot the distance from the earth along the axis
to Jupiter along the horizontal axis, and the velocity component
towards Jupiter along the vertical axis (the parallel components
of the distance and the velocity). The perpendicular component of the
position is of course zero; the perpendicular component of the velocity
is not shown.

If you click on the graphics screen while viewing the Poincaré section, you'll start another trajectory with the appropriate parallel positions and velocities. The program selects the perpendicular velocity so that the ``Energy'' is conserved; if that's not possible, it writes a note to the original window and plots nothing. Thus, you can view a two-dimensional cross-section of the three-dimensional constant ``Energy'' surface.

- Who were the mathematicians and physicists who made serious contributions to the theory of the three-body problem? In particular, who proved that there were no further analytic integrals of the motion? Can you provide Mosaic links to bios already on the Net for these?
- What is the ``weird vector'' which provides the last integrals for the inverse-square two-body problem? Find out how it's related to the mapping from the inverse-square 3-D problem to the four-dimensional harmonic oscillator. How does it relate to the Laplace vector?
- Show that the ``Energy'' defined above is conserved. (What's its real name?) If we make the mass of the earth finite, the real energy and angular momentum are conserved. How do these turn into ``Energy'' in the limit where the earth's mass goes to zero? What happens to the other conserved quantities?
- Figure out what
*symplectic*means. Implement a symplectic algorithm, and check whether it runs faster than Runge Kutta. See,*e.g.*Olver, P. J., "App. of Lie Groups to Diff Eqn's" QA372.052, 1993 or Sanz-Serna, J. M., "Numerical Hamiltonian Problems" QA614.83.S26x.1994 in the Math library.

- Description of the Three-Body Problem.
- Jupiter, Earth, and Sun
- Why hasn't the Earth left the Solar System?
- Invariant Tori
- Chaos!
- Lagrange Points
- Poincaré Section for the L5 Point
- Questions for Further Research

Last modified: May 19, 1996

James P. Sethna, sethna@lassp.cornell.edu.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).