Projective dynamics and first integrals

Abstract

We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.

39 pages, 2 figures

References in corpus (2)

• Projective dynamics and classical gravitation
• On bi-integrable natural Hamiltonian systems on the Riemannian manifolds

Cited by in corpus (4)

• Darboux inversions of the Kepler problem
• Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere
• Orbits in the problem of two fixed centers on the sphere
• Curvature as an integrable deformation