Affine transformations in Euclidean space generate a correspondence between
integrable systems on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in
$R^n$. Using this correspondence and the suitable coupling constant transformations, we can get
real integrals of motion in the hyperboloid case starting with real integrals of motion in the
sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic
and sextic polynomials in momenta.
Keywords:
completely integrable systems, Dirac brackets
Citation:
Tsiganov A. V., Integrable Systems on a Sphere, an Ellipsoid and a Hyperboloid, Regular and Chaotic Dynamics,
2023, Volume 28, Number 6,
pp. 805-821