Projective dynamics and first integrals

Details Projective dynamics and first integrals Projective dynamics and first integrals

2006-12-11

arxiv.org/abs/math-ph/0612031v1

Physics

Mathematical Physics

28 pages, 1 figure

Scientific paper

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), where q is in R^n, is isomorphic to the space of quadrilinear forms on R^{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 projective 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.

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