Poincaré Section for the L5 Point

Taka Koyama had to modify the Poincaré section used for the Earth-Jupiter-Sun system to make it useful here. In the old version, we took a snapshot of the Earth's position every time it crossed between Jupiter and the Sun. Of course, if the space-station, Moon, and Earth always form roughly an equilateral triangle, the space station will never pass between the Earth and Moon! Taka modified the program to take a snapshot whenever the light body (space station here) passed through an axis 60 degrees behind the line connecting the two heavy bodies (or more generally, an angle given by theta_Poincare, the angular lag for the Poincare section).

We've expanded the periodic orbit a lot, so instead of a dot (the intersection of a closed curve with the Poincare cross-section) it now looks like two small circles. (This happened because we input the initial conditions only to a few decimal places.) Remember how much we've expanded the view: the whole region around these orbits is extremely close to being periodic.

Click on areas near the left-hand Poincare section. (On the right, the trajectories will look different. Remember the energy surface has two sides, corresponding to the two solutions for v_perp for a given r_parallel and v_parallel: I think clicking on the right gives the "other side" of the energy surface.) Are there orbits starting near these circles which leave the vicinity? The region between the two circles is forbidden by energy conservation: the program finds a negative discriminant in the quadratic formula conserving energy for finding v_perp, and writes VERBOTEN. Near the forbidden region, it sure doesn't look stable!


  • Does "real" Lagrange point indeed lie between the two small circles? Verify this by increasing the precision of the initial positions and velocities for the space station. (In the configure menu, the space station is called the Earth, the Moon is called Jupiter, and the Earth is called Sun, because we're using a program which was intended to model the Earth-Jupiter-Sun system...) Taka found a book to give the equations for the coordinates: because the coordinates are centered at the center-of-mass, it isn't just the Earth-Moon distance at angle pi/3.
  • Find out why L5 is thought to be stable. (I believe it involved the eigenvalues of the linearization of the equations of motion about the fixed-point in the rotating reference frame...) Can you use this method to explain the quasiperiodic orbits (represented by the circles)? Should some be periodic (and thus chaotic)?
  • Increasing or decreasing the radial velocity (vertical coordinate on the display) seems to make for trajectories which leave the region. It appears that L5 is not stable, from our program. What does the linear stability analysis say about these trajectories?


    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).