Let us consider, for simplicity, a restricted 3-body problem:
Sun-Earth-asteroid, in which the Earth is on a circular orbit
with radius
and the asteroid is on a Keplerian
elliptic orbit with semimajor axis
.
The Hamiltonian has the form
where
is constant
and
is the perturbing function
If there is an intersection between the two orbits, the perturbing
function
has a first order polar singularity, so we can compute
its average over the fast variables
because it is
an improper convergent integral:
On the other hand, if we try to study the qualitative behavior of the
solutions of Hamilton's equations by the classical averaging
principle, we have to average the right hand side of the equations of
motion, that is the derivatives of ,
which have second order polar
singularities; so these integrals are absolutely divergent.
We have previously proved (see Gronchi and Milani 1998)
that is possible to give a generalized definition of averaged
solutions for planet-crossing orbits by showing that Hamilton's
equations built with the averaged perturbing function
,
that is
Let us describe briefly this situation in the
plane: we have the conservation of the
component of the
angular momentum, that is of the action variable
;
so, for a given
value of
we have a maximum value of
(say
,
corresponding to inclination zero.
The Kozai domain, where the averaged solutions can evolve, is
represented by
in this plane; in
Figure 1 we have plotted also the ascending node crossing
line and the descending one (
and
respectively). If we call
the interior of the connected
components in which
is divided by the node crossing lines we
already know, as classical results, that
is continuous in
and differentiable in each
.
Using a Wetherill approximation (see Wetherill 1967) and Kantorovich's
method of singularity extraction we have shown that
is
differentiable in the closure of each
in
and, more
important, that we can define piecewise smooth solutions of Hamilton's
equations built with
simply by extending to open regions
including the node crossing lines the expressions of the derivatives
in the right-hand sides of the equations in a smooth, but two-valued,
way (see Gronchi and Milani 1998).
We can see in Figure 2 the level lines of the averaged
Hamiltonian of a Jupiter crosser asteroid (or ``Centaur'') with
semimajor axis
= 7 AU; these lines have a corner when they go
through a node crossing line and the presence of these corners has
been detected analytically, by isolating, from the averaged
Hamiltonian, the term that is responsible.
This term turns out to be the distance between the two straight lines that, according to Wetherill's approach, approximate the orbit of the Earth and of the asteroid in a neighborhood of the intersection points; thus it behaves like an absolute value function.