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On the regularity of solutions to the Hamilton-Jacobi equations for the N-body problem

This paper establishes that suitably renormalized value functions for the $d$ -dimensional $N$ -body problem are viscosity solutions to the associated Hamilton-Jacobi equation, while characterizing their singularities and proving that the closure of the singular set is $\mathcal{H}^{d(N-1)-1}$ -rectifiable with a bounded Hausdorff dimension for regular conjugate points.

Original authors: Diego Berti, Davide Polimeni, Susanna Terracini

Published 2026-07-31

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CC BY 4.0

Original authors: Diego Berti, Davide Polimeni, Susanna Terracini

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

Imagine the universe as a giant, invisible dance floor where stars and planets are partners locked in a gravitational waltz. This is the realm of the N-body problem, a classic puzzle in physics that asks: if you have a bunch of masses pulling on each other, how will they move? Sometimes, they crash into each other; other times, they fly apart forever. Scientists use a special mathematical map called the Hamilton-Jacobi equation to predict these paths. Think of this equation as a "topography of time," where the height of the land tells you the cost (in energy and time) to get from one spot to another. The lowest points on this map represent the most efficient routes the universe takes.

However, this map isn't always smooth. Just like a mountain range has jagged peaks and sharp ridges where the ground suddenly changes, the solutions to these equations can have "singularities"—places where the path isn't unique or where the math gets messy. These are the spots where the universe seems to have a choice of directions, or where the rules of the dance get tricky. Understanding where these rough spots are, and how "big" they are, helps physicists understand the stability of our solar system and the fate of expanding galaxies. If the rough spots are rare and small, the universe is predictable; if they are everywhere, chaos reigns.

In this paper, the authors, Diego Berti, Davide Polimani, and Susanna Terracini, take a deep dive into these "rough spots" for a specific type of cosmic dance: one where the bodies are flying apart, expanding into the infinite void. They focus on three different ways this expansion can happen: bodies flying off at constant speeds (hyperbolic), bodies drifting apart slowly like a slow-motion explosion (parabolic), or a mix of both. Crucially, their analysis assumes that the center of mass of the entire system is fixed at the origin, effectively anchoring the dance floor so the whole group doesn't drift away while they spin.

The team proves that the mathematical map (the "value function") describing these expanding motions is indeed a valid solution to the Hamilton-Jacobi equation, even though it has some kinks. But the real magic is in measuring the kinks. They show that the set of "irregular" points—where the universe has more than one possible path to take—is surprisingly small. In fact, they prove that the closure of this set of confusing points is "rectifiable," which is a fancy way of saying it looks like a collection of smooth, lower-dimensional sheets (like a stack of paper or a line) rather than a messy, 3D blob filling up space.

Specifically, they calculate that the "size" of the set of points where the paths are not unique (the irregular set) is confined to a structure with a dimension of $d(N-1)-1$ . Even more interestingly, they look at "conjugate points"—a specific type of singularity where paths that started together might cross or fold back. They prove that the set of these conjugate points is even smaller, with a dimension no larger than $d(N-1)-2$ .

To put it in a playful analogy: Imagine the configuration space (all possible arrangements of the N bodies) as a giant, multi-dimensional room. The authors show that the "bad spots" where the rules get confusing are not scattered randomly throughout the room like dust motes. Instead, they are confined to very thin, flat structures, like a few sheets of paper floating in a vast hall. If you were to throw a dart at the room, the odds of hitting one of these confusing spots are incredibly low.

The paper also clarifies that while these solutions are smooth almost everywhere, they are not perfectly smooth at the very moment of a collision (where bodies hit each other). However, even at these collision points, the solutions remain "half-Hölder continuous," meaning they don't break apart completely; they just get a little bit rougher, like a fabric that is still connected but has a frayed edge.

In short, the authors have rigorously mapped the terrain of expanding cosmic systems. They've proven that while the universe's path can get a little bumpy, the bumps are confined to very specific, thin lines and surfaces. This gives us a clearer picture of how the N-body problem behaves over infinite time, confirming that the "chaos" of multiple paths is actually a very rare and structured phenomenon, not a chaotic mess.

Technical Summary: On the Regularity of Solutions to the Hamilton-Jacobi Equations for the N-Body Problem

Problem Statement
This work addresses the existence and regularity properties of viscosity solutions to the stationary Hamilton-Jacobi equation associated with the classical Newtonian $N$ -body problem in Euclidean space $\mathbb{R}^d$ ( $d \geq 2$ ). The study focuses on global solutions corresponding to expansive motions (where mutual distances diverge as $t \to +\infty$ ) at non-negative energy levels. Specifically, the authors investigate the regularity of "renormalized value functions" derived from a variational principle. These functions are defined on the configuration space $X$ (with zero center of mass) and are associated with different limiting shapes of expanding solutions: hyperbolic (asymptotic velocity $a \in \Omega$ ), parabolic (converging to a minimal central configuration $b_m$ ), and hyperbolic-parabolic (clusters of bodies with specific asymptotic behaviors).

The central challenge lies in characterizing the singular set of these value functions. In the context of the $N$ -body problem, the potential is singular at collision configurations, and the variational problem is defined over an unbounded time horizon. The paper seeks to define and estimate the size of the set of points where the minimizer of the associated variational problem is not unique (irregular points) and the set of conjugate points (where the linearized problem degenerates).

Methodology
The authors employ a unified variational framework based on the minimization of a suitably renormalized Lagrangian action functional. This approach builds upon recent results by Polimeni and Terracini [29], Maderna and Venturelli [22, 23], and extends the theory of viscosity solutions in Celestial Mechanics.

Renormalized Action Principle: To handle the non-integrability of the action on the half-line $[1, +\infty)$ for expansive motions, the authors define a renormalized action functional $A_x(\phi)$ on a Hilbert space $D$ of perturbations. The value function $v(x)$ is constructed as the minimum of this functional minus a linear correction term related to the asymptotic velocity or central configuration.
Finite Horizon Approximation: The stationary problem is approached via a sequence of finite-horizon problems. The authors prove that the finite-horizon value functions $v(T, x)$ converge uniformly on compact subsets to the global value function $v(x)$ as $T \to +\infty$ .
Viscosity Solution Theory: The paper utilizes the theory of viscosity solutions (Fréchet super- and sub-differentials) to handle the lack of classical differentiability. The authors establish that the value function $v$ is a viscosity solution to the Hamilton-Jacobi equation $H(x, \nabla v) = h$ .
Regularity Analysis: Drawing on foundational work by Cannarsa and Sinestrari [6] regarding the regularity of value functions in optimal control, the authors analyze the singular sets. They define:
- Irregular points ( $\Sigma$ ): Configurations where the minimizer of the action is not unique.
- Conjugate points ( $\Gamma$ ): Configurations where the minimizer is unique but the Hessian of the action (linearized operator) is not invertible.
Spectral and Geometric Tools: To estimate the dimension of the singular sets, the authors employ spectral theory along the flow, specifically analyzing the first eigenvalue of the Hessian operator (Sturm-Liouville type problems) and utilizing Sard-type theorems for Banach manifolds. They also utilize uniform coercivity estimates and Hardy-type inequalities to control the behavior of minimizers near collisions and at infinity.

Key Contributions and Results

Existence of Viscosity Solutions: The paper rigorously proves that for any initial configuration $x$ , and for any of the three types of expansive motions (Hyperbolic, Parabolic, Hyperbolic-Parabolic), there exists a viscosity solution to the Hamilton-Jacobi equation. These solutions are constructed as the renormalized value functions.
Regularity and Continuity: The value function $v$ is shown to be continuous on the entire configuration space $X$ , including collision configurations. While $v$ is not Lipschitz continuous at collisions, it is globally $1/2$ -Hölder continuous. On the collision-free set $\Omega$ , $v$ is locally Lipschitz and semiconcave with a linear modulus.
Structure of the Singular Set:
- The set of irregular points $\Sigma$ (where minimizers are non-unique) and the set of conjugate points $\Gamma$ (where the linearized operator is singular) are analyzed.
- The authors prove that the closure of the singular set $\overline{\Sigma}$ is countably $\mathcal{H}^{d(N-1)-1}$ -rectifiable.
- A significant result is the upper bound on the Hausdorff dimension of the set of regular conjugate points (points in $\Gamma$ but not in $\Sigma$ ). The paper establishes:
  $\dim_H(\Gamma \setminus \Sigma) \leq d(N-1) - 2$
- This implies that outside a set of codimension one (the irregular set), the value function is differentiable almost everywhere, and the set of points where it fails to be differentiable due to conjugacy is even smaller (codimension two).
Rectifiability: The set $\Gamma$ is proven to be countably $\mathcal{H}^{d(N-1)-1}$ -rectifiable. The proof involves decomposing $\Gamma$ into a part where the kernel of the Hessian has dimension 1 (handled via the Implicit Function Theorem and transversal sections) and a part where the kernel dimension is $\geq 2$ (handled via Sard-type arguments).

Significance and Claims
The paper claims to provide a rigorous characterization of the regularity of global viscosity solutions for the $N$ -body problem, extending previous results that were largely restricted to the hyperbolic case or specific low-dimensional scenarios. By establishing the rectifiability and dimension bounds of the singular sets, the work clarifies the geometric structure of the solution space for the Hamilton-Jacobi equation in this singular, non-compact setting.

The authors position their results as a necessary step in understanding the asymptotic behavior of the $N$ -body system through the lens of Hamilton-Jacobi theory. They note that their viscosity solutions extend continuously to collision configurations, providing a global definition of the value function that was previously lacking in full generality for mixed hyperbolic-parabolic motions. The work relies on and refines the variational framework established in [29], confirming that the renormalized action principle yields solutions that are fixed points of the Lax-Oleinik semigroup and satisfy the Hamilton-Jacobi equation in the viscosity sense.

The paper does not propose new experimental applications or future dynamical systems beyond the mathematical analysis of the regularity of these specific solutions. Its primary contribution is theoretical, solidifying the link between the variational properties of expansive motions and the regularity of the associated Hamilton-Jacobi solutions.

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