2.1 Semianalytical theories

The secular evolution of the main belt asteroids, whose orbits are not planet crossing (so that the Hamiltonian has no singularity), have been studied by applying the averaging principle [Arnold 1976]: when there is no mean motion resonance between the fast angle variables, the long term evolution of the asteroids can be computed to first order in $\mu$ by averaging the right-hand side of Hamilton equations over the fast angles (in this case the mean anomaly $\ell$ and the phase $\ell'$ of the circular orbit of the planet). So the equations to be solved are:

$$\cases{ \dot{\overline g} = -\overline{\partial R \over \part... ...\overline Z} = \overline{\partial R \over \partial z} = 0 \cr}$$ (2)

where the overline bar means an integral average as in equation (1). $Z$, the component of the angular momentum orthogonal to the plane of the planetary orbits, and $L$, hence the mean semimajor axis $a$, are integrals for the averaged system because $\ell$ and $z$ (which coincides with $\Omega$, the longitude of the node) are cyclic variables.

This principle has been used to compute proper elements even beyond the circular planetary orbits approximation, by setting up a suitable perturbation theory [Williams 1969]; in more recent theories, even a significant part of the effects of second order in $\mu$ have been taken into account [Lemaitre and Morbidelli 1994]. However, this has not been possible so far for planet crossing orbits.