José Lamas Rodríguez submitted his PhD thesis Oscillatory motions, parabolic orbits and collision orbits in the planar circular restricted three-body problem, supervised by Marcel Guàrdia Munárriz and Tere M-Seara Alonso at the Universitat Politècnica de Catalunya (April 2025).
Celestial mechanics is a classical meeting point between physics and mathematics. Its starting point is deceptively simple: write Newton’s laws for bodies that attract each other gravitationally and try to predict their motion. Yet, as soon as three bodies interact, long-term behavior becomes difficult to control, as already emphasized in Poincaré’s work [9]. Close encounters can produce large changes in a trajectory, and small uncertainties in the initial data may grow over time. For this reason, a modern approach does not aim primarily at explicit formulas for the motion, but at understanding which types of trajectories are possible, how they can transition from one type to another, and which geometric structures in phase space organize this complexity.
A standard model to study these questions is the planar circular restricted three-body problem (PCRTBP). In this setting, two primaries (the Sun and Jupiter) move on circular coplanar orbits, while a third body of negligible mass (for instance, an asteroid) evolves under their gravitational field. In rotating coordinates the primaries are fixed, the system is Hamiltonian, and trajectories are either defined for all time or end in collision with one of the primaries. For trajectories defined for all time, Chazy classified the possible asymptotic behaviors (final motions), which include unbounded motions (hyperbolic and parabolic), bounded motions, and oscillatory motions [1]. The main goal of this thesis is to incorporate the collision set into this picture and to understand how final motions and collision dynamics fit together in a single geometric framework.
The first part focuses on the dynamics close to the massive primary (the Sun), in a regime adapted to a small mass ratio. The main result is that every combination of past and future final motions can be realized by trajectories that pass arbitrarily close to collision with the Sun. Beyond this, the thesis constructs ejection-collision trajectories with the Sun that make arbitrarily large excursions before returning, and it builds hyperbolic invariant sets whose closure intersects both the collision set and infinity. These sets provide a precise form of chaotic dynamics: they contain in their closure ejection and collision orbits to the Sun, together with trajectories displaying prescribed asymptotic behavior, and their dynamics can be encoded by sequences of symbols [3, 10].
The techniques in this part are geometric and rely on regularization and invariant manifolds. Near the Sun, collision is regularized using McGehee coordinates: the singularity is replaced by invariant tori containing normally hyperbolic circles, each with stable and unstable manifolds. On the other end of phase space, «parabolic infinity» is compactified into an invariant object that also has stable and unstable manifolds. A central step is to bring these manifolds to a common transverse section and compare them by computing a distance function. Following the ideas in [5], this distance is analyzed through a Melnikov-type computation, which yields transverse intersections between the manifolds associated with collision and infinity. Combining these intersections with the classical description of the dynamics near infinity (in the sense of Moser [8]) produces the large ejection-collision excursions. A further local analysis close to the regularized collision set yields additional intersections near collision, and at a suitable energy level a more refined configuration leads to hyperbolic dynamics accumulating simultaneously on collision and on infinity.
The second part extends the analysis to the smaller primary (Jupiter) and to the interaction between both primaries. The conclusions mirror those near the Sun: all combinations of past and future final motions can be realized by trajectories that accumulate on collision with Jupiter; there exist ejection-collision orbits with Jupiter that perform arbitrarily large excursions; and there are hyperbolic invariant sets whose closure intersects Jupiter’s collision set and supports symbolic dynamics. Building on these constructions, we also prove the existence of ejection-collision orbits that connect the two primaries.
The methods differ because the local geometry near Jupiter is not the same as near the Sun when the mass ratio is small. To obtain a meaningful description as the mass parameter tends to zero, the analysis splits phase space into two regions: far from Jupiter, the motion is treated as a nearly integrable perturbation of the Kepler problem with the Sun; close to Jupiter, one zooms into a neighborhood whose size is determined by the mass ratio. In this near region, collision is regularized using Levi-Civita coordinates [4], which replace the singularity by a regular circle and allow ejection and collision trajectories to be treated as regular orbits passing through the transformed collision set. A local analysis then controls the passage near Jupiter (adapting ideas in the literature [6]), and the next step is to compare these near-Jupiter trajectories with the invariant manifolds associated with infinity by extending parameterizations to a transverse section near Jupiter. Since these manifolds are Lagrangian, the extension is carried out using Hamilton-Jacobi parameterizations together with complex-analytic estimates [2]. Finally, a Poincaré-Melnikov-like computation produces transverse intersections, from which the large-excursion ejection-collision orbits, the symbolic-dynamics structures, and the connecting trajectories between primaries are obtained.
References
[1] J. Chazy. Sur l’allure du mouvement dans le problème des trois corps quand le temps croît indéfiniment. Ann. Sci. École Norm. Sup. 39 (1922), 29–130.
[2] M. Guardià, P. Martín, T. M-Seara. Oscillatory motions for the restricted planar circular three body problem. Invent. Math. 203 (2016), 417–492.
[3] J. Lamas, M. Guardià, T. M. Seara. Oscillatory Motions, Parabolic Orbits and Collision Orbits in the Planar Circular Restricted Three-Body Problem. Commun. Math. Phys. 406 (2025), 106.
[4] T. Levi-Civita. Sur la régularisation du problème des trois corps. Acta Math. 42 (1920), 99–144.
[5] J. Llibre and C. Simó. Oscillatory solutions in the planar restricted threebody problem. Math. Ann. 248 (1980), 153–184.
[6] J.-P. Marco and L. Niederman. Sur la construction des solutions de seconde espèce dans le problème plan restreint des trois corps. Ann. Inst. H. Poincaré Phys. Théor. (1995), 211–249.
[7] R. McGehee. Triple collision in the collinear three-body problem. Invent. Math. 27 (1974), 191–227.
[8] J. Moser. Stable and random motions in dynamical systems: With special emphasis on celestial mechanics. Princeton University Press, 2001.
[9] H. Poincaré. Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13 (1890), 1–270.
[10] S. Smale. Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton University Press, 1965, 63–80.