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This section illustrates the formalism we introduced in this tutorial
and it provides an example with long history dating back to
Poincaré.
Figure 3:
Restricted threebody problem

In restricted threebody problem one of the three masses is negligible
as compared to the other two masses. In the circular version we assume
that the remaining two masses move along circular orbits about their
center of mass (see Figure 3).
It is natural to introduce a coordinate system in which the two heavy
masses are at rest, say at points A and B and that the center of
mass is at 0. The potential energy
in this coordinate system is

U(Q) =  
 (51)

where and are suitable constants. We assume that the
light mass is 1 .
Thus the equations of motion are

=

P  x Q
 

=

 U(Q)  x P
 (52) 
where
P,Q . Of course, we may perform the
differentiations and obtain a completely explicit system of ODE
with 4 free parameters. Indeed, we may assume that
A and B are on the x axis and A = (a,0,0) , B = ( b,0,0) .
We must have
a = b which is the only relation between the
four parameters. We may also assume that
= (0,0,c) and
that c is another parameter.
Next: About this document ...
Up: Lagrangian and Hamiltonian mechanics
Previous: 5 Motion of a
Marek Rychlik
9/2/1997