On the existance of nonlinear stabilization of the circular
restricted problem of three bodies at mu = muc
Johannes Hagel
CERN - Genf, Switzerland
It has been known since long time that the linear stability of a
massless body close to the Langrangian points L4,5 in the
circular restricted problem of three bodies is fully determined by the
massparameter :
mu = m2/(m1+m2)
where m1 and m2 are the masses of the primaries. For 0 < mu
< muc = 0.03852 ... the linearized problem is bounded and the
appropriate solutions can be expressed in trigonometric functions. At
mu = muc the associated 4 x 4 matrix has degenerated
eigenvalues and the solution type turns to linear functions in time. For
mu > muc the motion becomes exponentially unbounded.
In contrary to this well known fact a numeric integration of the full
nonlinear equations shows clearly that orbits close to L4
survive for a time much longer than in the linear case. This suggests
that there exists the effect of nonlinear stabilisation in this
problem. In order to show this, we develop the full equation around
L4into a second order polynomial equation in the four coordinates
x, x dot, y, y dot and by diagonalizing the linear matrix contribution
we transform the system to action angle type variables. From these
equations we derive two integrals of motion (valid at mu = muc) and
by so reducing the problem to algebraic relations we see that infact
stability for small amplitude oscillations at mu = muc can be
established.