**Next:** Acknowledgement
**Up:** Non-integrable
cases in satellite **Previous:** The
truncated zonal satellite

The *Theorem 1* by Yoshida presented in the previous section is
useful to establish the non-integrability of Hamiltonian systems with homogeneous
potential provided it has only two degrees of freedom. If it is not the
case, other alternative procedures have to be used. In this line, Yoshida
([12]) obtained a criterion, also based
on Ziglin's theorem ([14]), for the non-existence
of an additional meromorphic integral in Hamiltonian systems with *n*
degrees of freedom. Such a criterion can be stated as follows:

*Theorem 2*: Consider a Hamiltonian system with *n* degrees
of freedom of the form

(11) |

where
is assumed to be a homogeneous function of integer degree *k* ( ).
Since the equations of motion are scale-invariant, one can compute the
Kowalevski exponents (KE), ,
,
,
(see [8]). The KE characterize the branching
of the solution in the complex *t*-plane as follows,

(12) |

Here, *g*=2/(*k*-2),
is a constant vector, *I _{j}* are integration constants, and
the term
denotes a convergent Taylor series. For (11),
the KE have a pairing property,
.
Because of this pairing property,
and
can be rational numbers if and only if the difference
is a rational number. Further, one can always assume that
is rational, .

Then, if the numbers are rationally independent, then the Hamiltonian system (11) has no additional global analytic integral besides the Hamiltonian itself.

In general, it is quite obvious that the greater the number of degrees
of freedom is, the more complicated the problem becomes. Moreover, let
us point out that the analysis carried out through the above theorem is
much more involved than that performed in the previous section, in which
we only had to compute the integrability coefficient, while in the n-degrees
of freedom criterion the knowledge of the eigenvalues of an *n*-dimensional
matrix is required to determine the rational independence of the KE.

Next, we will consider the case for the satellite problem in which the
only acting perturbation is that due to the term corresponding to the harmonic
(2,2) of the development in spherical harmonics of the potential of the
planet (i.e. the *J*_{22}-problem).

By using Cartesian coordinates in the rotating system *Oxyz*, attached
to the rigid body, and which turns with uniform angular velocity
with respect to the inertial axis, the Hamiltonian of the problem is

(13) |

where

and the term is due to the rotation of the reference frame.

Since the available non-integrability results are not directly applicable
to the original problem (13), we first consider
an auxiliary problem obtained by the reduction of the system deduced from
(13) to a similarity invariant one. This
system will be proved not to have additional meromorphic integrals by using
the above quoted *Theorem 2*.

By performing the change of scale
defined by ,
,
and taking limits when ,
the Hamiltonian (13) becomes

(14) |

which is taken as Hamiltonian of an auxiliary problem.

Taking into account that (14) is of the
type (11), with homogeneity degree *k*=-3,
we can apply the *Theorem 2*. Since the algebraic equation
admits as solutions
and ,
we can compute the Hessian
and its eigenvalues to find that, in this case, ,
and .
Therefore, as the imaginary numbers ,
are rationally independent, the auxiliary Hamiltonian *K _{s}*
(14) does not have any global meromorphic
integral independent of the Hamiltonian.

Now, starting from the non-integrability of the auxiliary problem, we will try to get as much information as possible about the non-integrability of the original Hamiltonian (13). Let us remember that the forementioned theorem by Yoshida [12] is proved by using a particular straight-line solution of the form , with satisfying . Afterwards, the linear variational equations around that solution are set up. According to Ziglin's result [14], the problem is not integrable if two different non-resonant monodromy matrices (corresponding to suitable loops on a certain Riemann surface) not commuting can be found. In the case of homogeneous potential, the normal variational equations can be transformed into a Gauss hypergeometric equation and explicit expressions for the monodromy matrices are known (see [12]).

However, in our problem, the presence of the gyroscopic term in makes the aforementioned analysis more involved. In fact, it is not trivial to find a particular solution which, in the limit , tends to the straight-line solution used by Yoshida, which is decisive for calculating the monodromy matrices explicitly - apart from possible inconvenient variations of the Riemann surface and the necessary loops as varies. So, alternative procedures should be used to ensure that (13) is not analytically integrable.

Nevertheless, what we can easily conclude is the non-existence of additional
rational integrals in the original problem (13).
Let us first note that if
is a rational first integral, by performing the forementioned change of
variables, it becomes

(15) |

where *f _{m}* and

The expression (15) is a rational first integral of (13). Then, multiplying by an adequate power of and taking limits when , we obtain a rational integral of the auxiliary problem (14).

Since (14) does not have any additional
meromorphic integral, it cannot admit any additional rational integral
and thus, neither does (13). Notice that
the same assertion still holds if
gathers not only the Cartesian coordinates but also the radius *r*
(or any other homogeneous function in the coordinates).

Another result, not formulated before, is the non-existence of any additional
meromorphic integral without poles in *z*=0 and
since, otherwise, the planar *J*_{22}-problem would have a
meromorphic integral which is analytical in .
But this is impossible and can be easily deduced from Yoshida's results
([11], theorem 4.1). Moreover, we have the
feeling that it could be quite difficult to have a spatial *J*_{22}-problem
without the restriction of non-singularity in *z*=0 and .

Let us remark that the case under study in this section is an example
of sectorial truncation of the satellite problem (i.e. the problem of the
satellite only perturbed by the main tesseral harmonic *J*_{22}).
In this respect, new results concerning the non-integrability of sectorial
and tesseral truncations of the satellite problem, when more suitable coordinates
than the Cartesian ones are used, have been recently obtained by the authors
and will be published in a forthcoming paper ([4]).