Cayley-Klein Poisson Homogeneous Spaces

Producción CientíficaThe nine two-dimensional Cayley–Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson–Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley–Klein geometries. The kinematical interpretation for the semiRiemannian and pseudo-Riemannian Cayley–Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.Ministerio de Ciencia, Innovación y Universidades (grant MTM2016-79639-P)European Cooperation in Science and Technology (COST Action MP1405 QSPACE

A new look at the Feynman ‘hodograph’ approach to the Kepler first law

Producción CientíficaHodographs for the Kepler problem are circles. This fact, known for almost two centuries, still provides the simplest path to derive the Kepler first law. Through Feynman's 'lost lecture', this derivation has now reached a wider audience. Here we look again at Feynman's approach to this problem, as well as the recently suggested modification by van Haandel and Heckman (vHH), with two aims in mind, both of which extend the scope of the approach. First we review the geometric constructions of the Feynman and vHH approaches (that prove the existence of elliptic orbits without making use of integral calculus or differential equations) and then extend the geometric approach to also cover the hyperbolic orbits (corresponding to $E\gt 0$). In the second part we analyse the properties of the director circles of the conics, which are used to simplify the approach, and we relate with the properties of the hodographs and Laplace–Runge–Lenz vector the constant of motion specific to the Kepler problem. Finally, we briefly discuss the generalisation of the geometric method to the Kepler problem in configuration spaces of constant curvature, i.e. in the sphere and the hyperbolic plane.Física Teórica, Atómica y ÓpticaMinisterio de Educación, Cultura y Deporte (project MTM-2012–33575)Gobierno de Aragón (project DGA E24/1)Ministerio de Economía, Industria y Competitividad (project MTM2014–57129