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Two-boost problem for the rotating Kepler problem

This paper resolves the two-boost problem for the rotating Kepler problem by extending Lagrangian Rabinowitz Floer homology to a broader class of settings and successfully computing the corresponding homology despite the challenges posed by the non-compactness of the energy hypersurface near collision and infinity.

Original authors: Jagna Wiśniewska

Published 2026-07-07

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jagna Wiśniewska

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Question: Can We Get There with Just Two Pushes?

Imagine you are trying to fly a spaceship from Point A to Point B in space. In the real world, rockets usually burn fuel constantly to steer and adjust their path. But what if your rocket engine is broken and can only fire twice? Once at the very beginning to get moving, and once at the very end to stop or land?

This is the "Two-Boost Problem." It asks: Is it mathematically possible to connect any two points in space using only these two engine bursts?

For a long time, we knew the answer was "yes" if the universe was perfectly still and simple (the classic "Kepler problem"). But our universe isn't still; the Earth and other planets are spinning. This creates a "Coriolis force" (the same force that makes hurricanes spin), which makes the path of a spaceship much more complicated.

This paper, written by Jagna Wiśniewska, proves that yes, even in this spinning, complicated universe, you can still get from Point A to Point B with just two boosts.

The Mathematical Tools: A New Kind of Map

To prove this, the author uses a very advanced branch of math called Symplectic Geometry and a specific tool called Floer Homology.

The Analogy: Imagine you are trying to find a path through a dense, foggy forest. You can't see the whole forest at once. Instead of trying to walk the whole way, you look for "peaks" and "valleys" in the terrain.
The "Action Functional": In this math, the "terrain" is a landscape of energy. The spaceship's path is like a hiker trying to walk along a specific contour line (an energy level). The "Action Functional" is a map that tells us how much "effort" a path takes.
The Critical Points: The solutions to the problem (the perfect paths) are like the peaks and valleys on this map. If we can prove these peaks and valleys exist, we prove the paths exist.

The Problem: The Map Has Holes and Infinite Edges

The author faces a massive technical hurdle. The "terrain" she is mapping has two dangerous features:

The Singularity (The Black Hole): If the spaceship gets too close to the planet, it crashes. In the math, this is a hole where the map breaks down.
Infinity (The Edge of the World): The spaceship could fly off into deep space forever. The map doesn't stop; it goes on forever.

Standard math tools usually require the map to be a closed, finite shape (like a sphere). Because this map has holes and goes to infinity, the standard tools fail. The author had to invent a way to stretch these tools to cover this broken, infinite terrain.

The Solution: Smoothing the Rough Edges

To fix the "crash" problem, the author uses a technique called Levi-Civita Regularization.

The Analogy: Imagine a road that ends abruptly at a cliff. Instead of driving off, you fold the map over itself. Now, the cliff becomes a smooth tunnel. You can drive through it without falling off.
In the paper, this mathematical "folding" turns the dangerous crash point into a smooth, passable path. This allows the math to handle collisions (or near-collisions) without breaking.

The Main Result: The "Two-Boost" Guarantee

The paper combines several advanced steps to reach its conclusion:

Expanding the Toolbox: The author takes a known mathematical method (Lagrangian Rabinowitz Floer Homology) and upgrades it so it works on these broken, infinite maps.
Proving the Path Exists: She shows that on this upgraded map, there are always "peaks" (solutions). Specifically, she proves there are at least two distinct paths connecting any two points.
The "Double Cover" Trick: Because of the way the math folds the space (to handle the crash), one path in the "folded" world actually corresponds to two paths in the real world. This guarantees that you don't just have one lucky path; you have at least two options.

The Conclusion

The paper concludes with Theorem 1.1: If you have a spaceship in the gravitational field of a spinning planet, and you want to go from point A to point B, you can always find a trajectory that connects them using exactly two engine boosts, provided you have enough energy.

In short: Even with the confusing forces of a spinning planet and the risk of crashing, the universe is "nice" enough that a two-push journey is always possible. The author didn't just say "it works"; she built a new mathematical bridge to prove it, handling the tricky parts where the math usually falls apart.

Technical Summary: The Two-Boost Problem for the Rotating Kepler Problem

Problem Statement
The paper addresses the "two-boost problem" within the context of the rotating Kepler problem. This problem asks whether two arbitrary points in phase space, $q_0$ and $q_1$ , can be connected by a Hamiltonian orbit of a fixed energy level $c$ using only two impulsive maneuvers (boosts) at the start and end of the trajectory. While the classical Kepler problem (stationary) was solved by Hohmann using elliptical transfers, the rotating Kepler problem introduces a Coriolis force, complicating the dynamics. The Hamiltonian for this system, $K$ , is non-compact due to two distinct singularities: the collision singularity at the origin (representing a crash into the celestial body) and the behavior at infinity (representing escape to space). The primary technical challenge is establishing the existence of solutions despite the non-compactness of the energy hypersurface $K^{-1}(c)$ .

Methodology
The author employs a variational approach using the Rabinowitz action functional, whose critical points correspond to the desired Hamiltonian chords (orbits connecting the fibers over $q_0$ and $q_1$ ). To handle the non-compactness and singularities, the paper utilizes the following framework:

Levi-Civita Regularization: The singularity at the origin is removed via the Levi-Civita transformation. The original Hamiltonian $K$ is related to a regularized Hamiltonian $K_c$ defined on the entire cotangent bundle $T^*\mathbb{R}^2$ . Solutions to the original problem correspond to solutions of the regularized problem, with collision trajectories in the original system mapping to specific solutions in the regularized system that intersect the zero section.
Lagrangian Rabinowitz Floer Homology (LRFH): The paper generalizes the definition of LRFH, previously defined for compact hypersurfaces, to a strictly broader class of non-compact hypersurfaces. This involves constructing a chain complex generated by the critical points of the Rabinowitz action functional and defining a boundary operator via Floer trajectories (gradient flow lines).
Perturbation and Compactness: To ensure the well-definedness of the homology, the author introduces a specific class of perturbations $\mathcal{H}$ to the Hamiltonian. The core technical effort involves proving that for Hamiltonians of the form $H_a - h$ (where $H_a$ is a quadratic Hamiltonian and $h \in \mathcal{H}$ ), the critical sets are "continuously compact" and the space of Floer trajectories is uniformly bounded in the $L^\infty$ -norm. This relies on establishing specific bounds for the action, energy, and the Lagrange multiplier $\eta$ , utilizing a maximum principle and careful estimates of the Hamiltonian's growth properties.
Homology Calculation: The author computes the positive Lagrangian Rabinowitz Floer homology for the perturbed system. By showing this homology is non-vanishing (specifically isomorphic to $\mathbb{Z}_2$ in degree $1/2$ ), the existence of critical points (solutions) is guaranteed.
Relating Hamiltonians: A specific perturbation $h$ is constructed such that the dynamics of the regularized rotating Kepler Hamiltonian $K_c$ on a relevant compact subset coincide with the dynamics of the perturbed Hamiltonian $H_a - h$ . This allows the transfer of the existence results from the abstract homological setting back to the specific physical problem.

Key Contributions and Results

Generalization of LRFH: The paper extends the construction of Lagrangian Rabinowitz Floer homology to a class of non-compact hypersurfaces, overcoming the dual non-compactness issues (infinity and collision) inherent in the rotating Kepler problem.
Existence of Solutions (Theorem 1.1): The main result proves that for the rotating Kepler problem, given any two points $q_0, q_1 \in \mathbb{R}^2 \setminus \{0\}$ and an energy level $c \geq 2R^2$ (where $R = \max\{|q_0|, |q_1|\}$ ), there exist at least two solutions to the two-boost problem on the energy level $K^{-1}(c)$ . These solutions include both regular trajectories and collision trajectories.
Invariance and Non-vanishing (Theorems 1.2 & 1.3): The paper establishes that the LRFH is well-defined and invariant under the specified class of perturbations. Furthermore, it calculates the positive LRFH explicitly, showing it is non-trivial ( $\mathbb{Z}_2$ in degree $1/2$ ), which directly implies the non-emptiness of the critical set.
Technical Bounds: The work provides rigorous proofs for the boundedness of Floer trajectories and the Novikov finiteness condition in the non-compact setting, which are prerequisites for the homology to be well-defined.

Significance
The paper provides a positive answer to the two-boost problem for the rotating Kepler problem, a fundamental model in astrodynamics for spacecraft motion in rotating gravitational fields. By successfully adapting Floer homological methods to non-compact settings with singularities, the author demonstrates that topological invariants can be used to guarantee the existence of mission-critical trajectories (transfers between orbits) even in the presence of collision risks and unbounded energy levels. The result confirms that, similar to the classical Hohmann transfer in the stationary case, two boosts are sufficient to connect phase space points in the rotating frame, provided the energy is sufficiently high. The methodology offers a robust framework for analyzing similar problems in celestial mechanics where standard compactness assumptions fail.