Projective dynamics and first integrals

We show the relation between Appell's remark on the central projection in the dynamic of a particle and T.-Y. Thomas and Nijenhuis studies of the polynomial first integrals of the geodesic flow on a space of constant curvature. Among the consequences: the space of leading terms of a quadratic first integral of an equation¨q = f (q), q ∈ IR n , is isomorphic to the space of quadrilin-ear forms on IR n+1 having the symmetries of the Riemannian curvature tensor. Such a leading term also appears as a quadratic form in what we call the pro-jective impulsion bivector. We characterize the cases where a quadratic first integral comes from a Lagrangian, thus giving a geometrical interpretation and a converse to some recent results by Lundmark.