Projective deduction of the non-trivial first integral to the Euler problem: an explicit computation
Gabriella Pinzari, Lei Zhao
TL;DR
The paper addresses the Euler two-fixed-center problem and its nontrivial first integral $E$ by constructing a projective dynamics framework that maps motion in $R^3$ to a tangential, conservative flow on a four-dimensional ellipsoid via a central projection with a $*$-norm. The main result shows that the ellipsoidal energy $G$ is conserved and satisfies $G = J + E/2 - \Theta^2/4$, providing an explicit, transparent derivation of $E$ and linking it to the energy of the projected system; the construction generalizes to ellipsoids $E_a$ and reveals limit cases corresponding to the Kepler problem. These findings connect the plane-projection viewpoint of Albouy and Zhao with a 4D ellipsoid projection, illuminate the super-integrability of the Kepler problem via two integrable mirror settings, and offer a concrete geometric mechanism for obtaining first integrals from projected dynamics.
Abstract
The validity of Kepler Laws for the {\it spherical Kepler problem} -- namely, the problem of the motion of a particle on the unit sphere {in $\mathbb R^3$} undergoing an attraction by another particle in the sphere, tangent to the geodesic line between the two and inversely proportional to its squared length -- prompted geometers to try to interpret such system as a '' projection'' of the familiar Kepler problem in the plane, with the hosting plane given by some affine plane in $\mathbb R^3$. At this respect, the most convenient mutual sphere-plane position has been object of a long debate, an account of which can be found in \cite{Albouy2013}. This fascinating topic, resumed %subject, firstly by A. Albouy in the aforementioned paper, has been expanded from the theoretical side in \cite{Albouy2015}. Further investigations recently appeared in \cite{AlbouyZhao2019, Zhao1, TakeuchiZhao1, TakeuchiZhao2}. As remarked in \cite{Albouy2013, Albouy2015}, extensions of the procedure to more dynamical systems would open to the possibility of finding first integrals to a given dynamical system simply looking at the energy of the mirror problem. In this note, we focus on the case of the problem of two fixed centers, already mentioned in \cite{Albouy2013}. We provide a{n explicit} geometrical construction allowing to interpret the first integral of the problem as the energy of its projection on an ellipsoid. Compared to previous papers on the same subject, ours -- besides being based on a somehow different construction -- includes complete explicit computations. {A byproduct of our construction is the existence of two integrable mirror problems (equivalently, three quadratic integrals, including the energy) for the Kepler problem, which is an aspect of its super-integrability.