<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META content="text/html; charset=utf-8" http-equiv=Content-Type>
<META name=GENERATOR content="MSHTML 8.00.6001.23588">
<STYLE></STYLE>
</HEAD>
<BODY bgColor=#ffffff>
<DIV><FONT size=2 face=Arial>I might have an idea for you guys. About two
hundred years ago, Laplace and Lagrange worked out the gravitational
consequences of Newton's laws of gravitation. It's really interesting
stuff. That was all "simple" (Euclidean) calculus, and one of the
consequences that they figured out the Lagrange L4 and L5 points, both for
Jupiter and for the Earth. A company is even discussing building a HOTEL
at the Earth's L4 point.</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>In any case, I thinnk that if you duplicate the
math of Lagrange and LaPlace, your understanding of gravitation may
grow.</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>Further, there are some more advanced gravitational
effects at both L4 and at L5, where the objects there follow very peculiar
Lissajous orbits (in three dimensions). I believe that math requires
Riemannian Calculus to predict the orbits. The Jupiter Trojans also move
around in Lissajous orbits around each other.</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>In any case, I think these objects might provide
for you a more thorough math of the gravitation. Many years ago, I
duplicated much of Lagrange's Calculus (all simplly based on Newton's equations)
and I think I learned a lot from that "homework". As of yet, I have not
tried to do the math regarding the Lissajous orbits.</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>The Jupiter Trojans include thousands of asteroids
in both groups. Their Lissajous orbits are really complex, where they
rarely hit each other, even in such tight orbits. I don't think anybody
knows yet WHY they don't collide more. The Earth Trojans are pretty rare,
mostly in our L4 and I think that only a couple small objects have been found in
our L5 as of yet.</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>Carl Johnson</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV></BODY></HTML>