  Among the ODEs peculiarities -- specially those of Mechanics -- besides the
problem of leading them to quadratures and to solve them either in series or in
closed form, one is faced with the inversion. E.g. when one wishes to pass from
time as function of lagrangian coordinates to these last as functions of time.
This paper solves in almost closed form the system of non linear ODEs of the
2D-motion (say, co-ordinates $\theta$ and $\psi$) of a gravity-free double
pendulum (GFDP) not subjected to any force. In such a way its movement is ruled
by initial conditions only. The relevant strongly non linear ODEs, have been
put back to hyper-elliptic quadratures which, through the Integral
Representation Theorem (hereinafter IRT) have been driven to the Lauricella
hypergeometric functions $F_D^{(j)}, j=3, 4, 5, 6 $. The IRT has been applied
after a change of variable which improves their use and accelerates the series
convergence. The $\psi$ is given in terms of $F_D^{(4)}$ -- which is inverted
by means of the Fourier Series tool and put as an argument inside the
$F_D^{(5)}$ -- in such a way allowing the $\theta$ computations. We succeed in
a insight knowledge of time laws and trajectories of both bobs forming the
GFDP, which -- after the inversion -- is therefore completely solved in
explicit closed form. Suitable sample problems of the three possible cases of
motion are carried out and their analysis closes the work. The Lauricella
functions employed here to solve the differential equations -- in lack of
specific SW packages -- have been implemented thanks to some reduction theorems
which will form the object of a next paper. To our best knowledge, this work
adds a new contribution as it concerns detection and inversion of solutions of
nonlinear hamiltonian systems.
