Projective dynamics and first integrals

TL;DR

This paper develops a tensor-based approach to polynomial first integrals in classical mechanics, linking special integrals to projectively flat manifolds and proposing a new theory for quadratic integrable systems.

Contribution

It introduces a tensor framework using Young tableau symmetry for polynomial integrals and connects these to projectively flat Riemannian manifolds, advancing the theory of quadratic integrable systems.

Findings

Relates special first integrals to Beltrami's theorem.

Develops a new theory for quadratic integrable systems.

Extends models to degenerate quadratic forms.

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.