  Consider a swiveling arm on an oriented complete riemannian surface composed
of three geodesic intervals, attached one to another in a chain. Each interval
of the arm rotates with constant angular velocity around its extremity
contributing to a common motion of the arm. Does the extremity of such a chain
have an asymptotic velocity ? This question for the motion in the euclidian
plane, formulated by J.-L. Lagrange, was solved by P. Hartman, E. R. Van
Kampen, A. Wintner. We generalize their result to motions on any complete
orientable surface of non-zero (and even non-constant) curvature. In
particular, we give the answer to Lagrange's question for the movement of a
swiveling arm on the hyperbolic plane. The question we study here can be seen
as a dream about celestial mechanics on any riemannian surface : how many turns
around the Sun a satellite of a planet in the heliocentric epicycle model would
make in one billion years ?

Dynamical Systems