🌀 nonlinear sciences

Block regularisation of the logarithm central problem

This paper proves that the logarithm central force problem in $\mathbb{R}^2$ , where the logarithm serves as the gravitational potential, is block regularizable, allowing its flow to be continuously extended over the singularity at the origin through appropriate re-parametrization.

Original authors: Archishman Saha, Cristina Stoica

Published 2026-09-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Archishman Saha, Cristina Stoica

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

In the vast theater of celestial mechanics, where planets trace their paths and stars dance in gravitational waltzes, there exists a persistent problem that has troubled mathematicians for over a century: the collision. When two objects in a gravitational system are predicted to crash into one another, the equations that describe their motion break down. The numbers involved shoot toward infinity, and the smooth flow of time seems to shatter. For decades, scientists have developed ways to smooth over these breaks, essentially rewriting the rules of the game just enough to let the story continue past the crash without the math collapsing. This work is not merely an academic exercise; it is essential for understanding the stability of star systems and for building accurate computer models of how galaxies evolve. While we have long known how to handle collisions in systems governed by the familiar inverse-square law of gravity, a different, equally important gravitational model has remained stubbornly resistant to these fixes. This model, which describes gravity in a flat, two-dimensional universe, relies on a logarithmic potential—a mathematical shape that behaves differently than the gravity we know in our three-dimensional world.

The researchers Archishman Saha and Cristina Stoica have now solved this specific puzzle. They have proven that the logarithm central force problem, a system where gravity pulls objects together according to a logarithmic rule, can be "block regularized." In plain terms, this means they found a way to transform the equations of motion so that the violent, undefined moment of a collision can be replaced with a continuous, predictable path. The team demonstrated that even though the original equations stop working when objects hit the center, the underlying physics does not actually stop. By changing the way time is measured and stretching the space around the collision point, they showed that a trajectory ending in a crash can be seamlessly connected to a trajectory emerging from that same point. The result is a complete, unbroken flow of motion that passes through the singularity without ever losing its mathematical integrity.

To understand why this is difficult, one must first grasp the nature of the logarithmic potential. In our three-dimensional universe, gravity weakens rapidly as you move away from a mass, following a specific rule where the force drops off with the square of the distance. In a two-dimensional world, however, the natural solution to the same physical laws produces a force that behaves like a logarithm. This force is weaker than standard gravity when objects are very close together, but it becomes stronger than standard gravity when they are far apart. Consequently, in this two-dimensional world, objects can never escape to infinity; they are always trapped in a bounded region, swinging back and forth. While this system is mathematically solvable in many ways, the moment of collision—where the distance between objects becomes zero—remains a wall that standard mathematical tools could not climb.

Saha and Stoica approached this wall by using a technique known as "blowing up" the singularity. Imagine the point of collision not as a single dot where everything breaks, but as a small, circular boundary that the objects approach. By expanding this point into a small ring or torus, the researchers created a new, larger space where the collision is no longer a sudden stop but a gradual approach to a specific boundary. They transformed the original equations, which were undefined at the center, into a new set of equations that are perfectly smooth and well-behaved all the way to this boundary. This process is similar to taking a map that has a torn spot in the middle and replacing that spot with a detailed, continuous landscape that connects the edges seamlessly.

The key to their success lay in two specific discoveries. First, they proved that a collision in this system can only happen if the objects have zero angular momentum, meaning they are moving in a perfectly straight line directly toward the center, with no sideways spin. If there is any spin at all, the objects will miss the center and swing around it safely. Second, they found that once the equations were transformed and the collision point was expanded into a ring, the motion on that ring became incredibly simple. On this expanded boundary, the flow of time and position follows a straightforward, predictable pattern. The objects move along the ring in a way that is easy to track, allowing the researchers to define exactly how an object enters the collision zone and how it must exit.

The authors constructed a specific "block," a defined region of space surrounding the expanded collision point, to test this behavior. They showed that any path entering this block from one side must exit from the other side in a continuous, predictable manner. Even for the rare cases where an object approaches the center and stops, the math shows that this path can be extended through the boundary and out the other side as if it had bounced off a mirror, but without any actual bounce or loss of energy. The researchers proved that this extension is not just a rough approximation but a continuous, smooth connection. They demonstrated that the map connecting the entry point to the exit point is a perfect, unbroken link, meaning the system is "block regularizable."

This finding is significant because it resolves a long-standing question about the behavior of two-dimensional gravitational systems. While previous work had successfully regularized collisions in three-dimensional gravity and other specific types of forces, the logarithmic case had remained an open problem. The researchers noted that their method relies on a specific type of mathematical transformation that is less smooth than those used in other successful cases, but they proved that this slight loss of smoothness does not prevent the system from being regularized. They also acknowledged that they were unable to find a specific type of geometric transformation, known as a conformal map, that would have simplified the problem further, leaving that particular avenue as an open question for future study.

Ultimately, the work of Saha and Stoica provides a complete mathematical description of how objects behave when they collide in a logarithmic gravitational field. By proving that the flow of motion can be extended continuously over the singularity, they have removed the mathematical barrier that previously made these collisions impossible to analyze. This allows scientists to treat the entire system, from the farthest reaches of the orbit to the very moment of impact, as a single, coherent whole. The result is a deeper understanding of the fundamental laws that govern motion in two-dimensional space, ensuring that even the most violent crashes in this theoretical universe can be understood as part of a continuous, unbroken story.

Technical Summary: Block Regularisation of the Logarithm Central Force Problem

Problem Statement
The paper addresses the singularity at the origin ( $r=0$ ) in the logarithm central force problem, where the potential is given by $V(r) = \ln r$ . This potential serves as the natural gravitational potential in two dimensions ( $\mathbb{R}^2$ ), analogous to the $-1/r$ potential in three dimensions. While the system is integrable and all trajectories are bounded, the equations of motion exhibit a singularity when a particle collides with the center. The primary challenge is that the flow is incomplete; solutions may reach the singularity in finite time, preventing the continuous extension of the trajectory through the collision point. The authors aim to prove that this system is "block regularizable," meaning the flow can be continuously extended over the singularity after a suitable re-parametrization and coordinate transformation.

Methodology
The authors employ the theory of "block regularisation" (or regularisation by surgery) originally developed by Conley and Easton, combined with McGehee's technique of "blowing up" the singularity. The methodology proceeds in the following steps:

Hamiltonian Formulation: The problem is formulated using a Hamiltonian $H = \frac{1}{2}p^2 + \ln|q|$ in polar coordinates. The system is reduced to one degree of freedom using the conservation of angular momentum ( $p_\theta = c$ ) and energy ( $H=h$ ).
Singularity Blow-up: To handle the singularity at $r=0$ , the authors introduce a coordinate transformation $(q, p) \to (s, u)$ and a subsequent change to variables $(w, \phi, \psi)$ . Crucially, unlike the homogeneous potentials treated by McGehee (which allow for analytic transformations), the logarithmic potential necessitates the use of $C^1$ transformations.
Time Re-parametrization: A new time variable $\tau$ is introduced via $d\tau = -\frac{1}{r^2} \exp(\frac{1}{r^2} - w) dt$ . This transformation extends the vector field smoothly to the set $r=0$ .
Collision Manifold: The singularity set $\{r=0\}$ is "blown up" into a compact invariant manifold $N$ , defined as a torus where $r=0$ and $e^{2w}=2$ . The flow on this manifold is fictitious but provides topological information about the real flow near the collision.
Isolating Blocks: The authors construct an isolating block $B$ around the collision manifold $N$ . They define the entry set ( $b_+$ ) and exit set ( $b_-$ ) of the block, as well as the subsets ( $a_+, a_-$ ) of trajectories that asymptotically approach the collision manifold (i.e., those with zero angular momentum).
Extension of the Map: The core of the proof involves analyzing the "map across the block" $\Psi$ , which maps entry points to exit points. The authors demonstrate that this map, initially defined on $b_+ \setminus a_+$ , can be continuously extended to the entire set $b_+$ , including the asymptotic trajectories.

Key Results

Block Regularizability: The main theorem establishes that the logarithm central force problem is block regularizable. The incomplete flow can be continuously extended over the singularity.
Structure of the Collision Manifold: The blown-up singularity forms an invariant torus $N$ . On this manifold, the flow is trivial: trajectories are given by $\psi(\tau) = \text{const.} = \psi_0$ , with the dynamics governed by $\dot{\phi} = 2\sin\phi$ and $\dot{\psi} = 0$ .
Asymptotic Behavior: The authors prove that trajectories approaching the singularity ( $r \to 0$ ) do so only if the angular momentum $c=0$ . These trajectories asymptotically approach specific equilibrium circles on the collision manifold ( $S_+$ and $S_-$ ).
Continuity of the Transition Map: The map $\Psi$ across the block extends to a homeomorphism from $b_+$ to $b_-$ . Specifically, for trajectories with zero angular momentum (which asymptotically tend to the collision manifold), the map is shown to be continuous by proving that the accumulated change in the angle $\psi$ vanishes as the initial angle approaches the equilibrium.
Smoothness Limitations: The authors note that while the regularisation is successful, the transformation is only $C^1$ , not analytic. Consequently, the extended map is a homeomorphism, not necessarily a diffeomorphism.

Significance and Claims
The paper claims to extend the theory of block regularisation to the logarithmic central force problem, a case distinct from the homogeneous potentials previously studied by McGehee. The significance lies in:

Theoretical Extension: Demonstrating that the "surgery" method works even when the potential does not permit the analytic conformal transformations (like $z \to z^n$ ) used for homogeneous potentials.
Handling Non-Analytic Potentials: Showing that the loss of smoothness (relying on $C^1$ rather than $C^\infty$ or analytic transformations) does not impede the proof of block regularizability.
Open Question: The authors modestly note that while they successfully regularized the system, they were unable to find a regularizing conformal transformation for the logarithm potential, leaving that specific aspect of the problem open.

The work does not propose new numerical applications or experimental proposals but contributes to the mathematical understanding of singularities in celestial mechanics and dynamical systems, specifically regarding the topological structure of flows near collision in two-dimensional logarithmic potentials.

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