  The problem of stability of the triangular libration points in the planar
circular restricted three-body problem is considered. A software package,
intended for normalization of autonomous Hamiltonian systems by means of
computer algebra, is designed so that normalization problems of high analytical
complexity could be solved. It is used to obtain the Birkhoff normal form of
the Hamiltonian in the given problem. The normalization is carried out up to
the 6th order of expansion of the Hamiltonian in the coordinates and momenta.
Analytical expressions for the coefficients of the normal form of the 6th order
are derived. Though intermediary expressions occupy gigabytes of the computer
memory, the obtained coefficients of the normal form are compact enough for
presentation in typographic format. The analogue of the Deprit formula for the
stability criterion is derived in the 6th order of normalization. The obtained
floating-point numerical values for the normal form coefficients and the
stability criterion confirm the results by Markeev (1969) and Coppola and Rand
(1989), while the obtained analytical and exact numeric expressions confirm the
results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational
problem is solved without constructing a specialized algebraic processor, i.e.,
the designed computer algebra package has a broad field of applicability.

Earth and Planetary Astrophysics