The Restricted Three Body Problem

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.

Integrability

The first work went into finding an exact solution in analogy with the two-body problem. It was quickly recognized that the key was to find a sufficient number of conserved quantities. Energy, momentum, angular momentum, and a weird vector provide enough information to solve the two body problem. For problems where there are enough integrals, the motion is quasiperiodic: roughly speaking, there are several interdependent periodic motions, leading to a motion in phase space which lies on a multi-dimensional torus. After a while, it was proven that there were not enough conserved quantities: the three body problem is not ``integrable''.

Restrictions and ``Energy''

Gradually, the problem was simplified in order to explore the kernal of the difficulty. The original eighteen-dimensional problem becomes twelve when you move the center-of-mass coordinates. The planar three-body problem, simplified by restricting the planets to a plane, lies in eight dimensions. The restricted three-body problem sets one mass to zero: imagine Earth as a dust particle, wandering around Jupiter and the Sun which orbit one another. We study the circular, planar, restricted three-body problem: the eccentricity of Jupiter's orbit is set to zero.

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

which is output as the ``Energy'' in the program. Thus the motion of the dust particle representing the Earth lies on a three-dimensional surface of constant ``Energy'', in the rotating reference frame.

Algorithm

We use variable time-step Runge Kutta to integrate the equations of motion, using the Numerical Recipes routine odeint and rk4, suitably modified. Runge-Kutta is not the ideal approach to simulating Hamiltonian systems: even when it is quantitatively accurate, it is not qualitatively accurate. In particular, what small errors are left will not conserve energy (which will tend to drift up or down), and will violate Liouville's theorem. We should be using a symplectic algorithm, for which each time step is a canonical transformation. The classical Leapfrog or Verlet algorithm for molecular dynamics is an example of a symplectic algorithm: each time step is a canonical transformation, so it exactly conserves an approximate energy (rather than approximately conserving the real energy). Unfortunately, Verlet doesn't work with our periodically forced systems, even in the rotating reference frame.

Poincaré Sections

Visualizing periodic trajectories is easy, but when they get quasiperiodic or chaotic it's easier to use a Poincaré section. That is, one defines a hyperplane in the phase space, and draws a dot whenever the trajectory passes through the plane. This also provides a mapping from the plane to itself (the Poincaré first-return map), explaining why people use maps to understand dynamical systems even when most practical systems are continuously evolving.

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.

Questions

  1. 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?
  2. 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?
  3. 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?
  4. 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.

Jupiter:

How to Get Jupiter

Jupiter is available for Windows 95, Windows NT, Macintosh, and several Unix platforms (the IBM RS6000, Sun Sparc, Dec Alpha (courtesy Kamal Bhattacharya), Linux, and the PowerPC running AIX4.1). The files are available without charge by anonymous FTP (ftp.lassp.cornell.edu) or via the World Wide Web.
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).