**Next:** Non-integrability
of the J22-problem **Up:** Non-integrable
cases in satellite **Previous:** Introduction

Let us begin by recalling a non-integrability criterion for 2-degrees of freedom Hamiltonian systems, with homogeneous potential of integer degree, obtained by Yoshida in 1987 [10]. Such a result is expressed as follows:

*Theorem 1*: Let
be a homogeneous potential function of degree
and compute the quantity (integrability coefficient)
defined by

(1) |

where is the Hessian matrix of and is a solution of the algebraic equation .

If
is in the so-called non-integrability regions *S _{k}*, then
the 2-degrees of freedom Hamitonian system
is non-integrable, i.e. there cannot exist an additional integral
which is complex analytic in .

For our purposes, it suffices to consider
and the *S _{k}* given by

(2) |

Now, let us consider the Hamiltonian of the truncated zonal satellite
of order
in Cartesian canonical variables

(3) |

where

(4) |

is the disturbing potential, *P _{k}*(

If we carry out a change to cylindrical variables the
Hamiltonian (3) is transformed into

(5) |

where and the vertical component of the angular momentum, is a first integral since the coordinate is ignorable.

The Hamiltonian (5) has the integrals *H*=
const. and
const., which are independent and are in involution (i.e. the Poisson bracket
).
Let us suppose that there exists a third first integral *F*, independent
of the other two and in involution with them, that is, verifying
since
always holds provided *F* is a first integral.

As ,
it is obvious that *F* does not depend on .
If we perform the reduction of order of the Hamiltonian (5),
by considering as
a parameter, it is clear that if (5) is completely
integrable with integrals *H*,
and *F*, the reduction will also be so with integrals
and .
Likewise, if the integrals of (5) are meromorphic,
so are those of the reduction. Moreover, we can assume that *F* is
analytical in
since otherwise it suffices to multiply *F* by a suitable power of
,
as
is a first integral
because
and *F* are so. In consequence, if (3)
is Liouville integrable, so is

(6) |

which is obtained by making in the reduction of (5) to two degrees of freedom.

Notice that, as the potential of this 2-degrees of freedom Hamiltonian
consists of a finite number of homogeneous terms, whose degrees vary from
-1 (corresponding to the Keplerian term) to -*n*-1, we are in conditions
to apply the Yoshida theorem ([11], theorem
4.1), which allows us to establish the non-existence of an additional meromorphic
integral if the integrability coeficient
of either the lowest or highest order part is in their corresponding non-integrability
regions. To this end, proceeding as Yoshida [9],
by performing a suitable change of scale, the Hamiltonian (6)
becomes

(7) |

which is taken as Hamiltonian of an auxiliary problem (see [2] for details).

As the potential *V _{n}* is homogeneous of order

(8) |

and on the value of the integrability coefficient
defined by

(9) |

It is easily checked that the system (8)
admits a solution of the form ,
where *z*_{0} is a solution to the equation

(10) |

On the other hand, by using well known properties of Legengre polynomials,
straightforward calculations allow us to compute the trace
to find that the integrability coefficient
is

As the non integrability regions *S*_{-n-1} defined
in (2) contain the interval ,
we conclude that the auxiliary Hamiltonian (7)
is non-integrable.

Now, coming back to the Hamiltonian (6),
according to Yoshida ([11], theorem 4.1),
in our case, it holds for the lowest order -*n*-1 and hence (6)
is non-integrable. In consequence, as explained before, the original problem
(5) is not Liouville integrable through meromorphic
integrals.

Let us remark that the choice of the solution to (8)
carried out here has allowed us to prove the non-integrability of any truncation
of the zonal satellite problem irrespective of whether it ends in even
or in odd harmonics, while the solution chosen by Irigoyen and Simó
[3] to set up the non-integrability of the
*J*_{2}-problem (whose truncation ends in *J*_{2})
would only be useful to prove the non-integrability of truncations ending
in even harmonics.