• collisions
• Kepler problem
• Levi-Civita
• regularization
• Three-body

In the three-body problem, three point masses move in space under their mutual
gravitational interactions as described by Newton’s theory of gravity. In celestial
mechanics, the study of the dynamics intends to predict the motions of gravitationally
interacting astronomical bodies. As typical examples we may consider the Sun-Earth-
Moon or Sun-planet-asteroid systems.
In accordance with the properties of the Newtonian gravitational force field, the
forces acting between particles approach infinity when the mutual distances approach
zero. Therefore, at collision the equations of motion show singularities. The analytic
study of the behavior of the system in the immediate vicinity of a collision was initiated
by Painlevé in 1897 [14]. He rigorously demonstrated what seems obvious, namely that
if the motion does not remain regular as 𝑡 approaches 𝑡∗, this can only occur because
one of the mutual distances or all three approach zero as 𝑡 → 𝑡∗ (1).
Encouraged by Painlevé’s results, many mathematicians devoted themselves in those
same years to study the dynamics of collisions. Sundman [19] was the first to study, in
1907, the dynamics of a collision of the three bodies, showing that this can only occur if
the resulting momentum of the three bodies vanishes.
He was the first to completely regularize the three-body problem in his Mémoire sur
le problème des trois corps, dated 1912 [18]. However, the method that he proposed is
indirect: it requires the introduction of several auxiliary variables and gives rise to a
system that is no longer within the framework of the equations of dynamics.
Levi-Civita devoted a great deal of time and effort to the problem of regularizations,
and proposed methods for regularizing the restricted, the planar and the spatial problem
in several publications. The Levi-Civita regularizations transform the singular motion
into a smooth one where all the variables involved remain finite, and which continues
after the collision, with the great advantage of maintaining the canonical form of
the equations. Levi-Civita obtained in 1904 a first regularizing transformation of the
planar restricted three-body problem based on the map C → C, 𝑧 → √𝑧. This was a
well-known transformation that sends the Kepler ellipses into Hooke’s circles. He made also use of the so-called Darboux inversion [5].
For the spatial problem the idea of Levi-Civita was to first regularize a parabolic
binary collision. Then, he introduced an auxiliary reference system to increase the
number of variables involved in the transformation. These techniques surprisingly
provide contact transformations that make the Hamiltonian of the three-body problem
perfectly regular.
The main focus of this thesis is investigating the regularizations proposed by Levi-
Civita, and relate them to modern regularizations of the Kepler problem. The space of
the Keplerian orbits with fixed negative energy is a product of two spheres. This remark
comes from Pauli, who in 1926 made use of this formalism in an attempt to quantize the
hydrogen atom [15]. Following Pauli’s work, Fock, a Soviet physicist expert on quantum
mechanics and quantum electrodynamics, compared the stereographic projection of
the great circle of the sphere with Keplerian orbits [6]. This construction gave rise to a
regularization, often attributed to Moser [13], which has a profound bound with the
one of Levi-Civita. It thus appear that Levi-Civita, a master of celestial mechanics, and
Fock, a master of quantum mechanics, independently discovered the same construction,
which is thus designated as unique and fundamental.
In 2019, Knauf [7] proposed a geometric regularization of the Kepler problem using
modern developments in differential geometry. Symplectic geometry now provides a
suitable language for the description of the Hamiltonian model of classical mechanics,
and Knauf regularization fits perfectly into this framework. The great advantage of
using a rather complex formalism for the description of the Kepler problem in the
immediate vicinity of collisions lies in the fact that all energy surfaces can be regularized
simultaneously, a long sought but never achieved result even by Levi-Civita himself.
It remains subject of study and research how such results can be used for numerical
implementation and practical applications.