Bodies of Minimal Resistance

• Bodies of minimal resistance are defined as 3D sets that minimize momentum transfer in rarefied, non-interacting particle flows.
• They employ variational methods and geometric constructions, including billiard reflections and Kakeya-type arrangements, to optimize shape.
• These bodies showcase counterintuitive phenomena such as zero resistance and invisibility, impacting kinetic theory and aerodynamic design.

Bodies of minimal resistance are three-dimensional sets that, when placed in a rarefied flow of non-interacting particles, minimize the total resistance (momentum transfer) imparted by the particles on the body. This concept, with origins in Newton's Principia (1687), underlies classical and modern shape optimization in kinetic theory, ideal gas aerodynamics, and geometric scattering. The mathematical formalism rigorously defines resistance in terms of billiard reflections and admits a variety of admissible body classes, including convex, non-convex, and bodies permitting multiple reflections. Exploration of these classes has led to the discovery of paradoxical constructions (e.g., bodies of zero resistance and invisible bodies), deep connections with geometric measure theory, and sophisticated variational methods for minimization.

1. Mathematical Formulation of Resistance

Consider a bounded, connected body $B \subset \mathbb{R}^3$ with a piecewise-smooth boundary $\partial B$, placed in a parallel flow of non-interacting particles incident along a fixed direction $v_0 \in S^2$. Each particle starts from $x$ in the orthogonal plane $v_0^\perp$ and evolves with constant velocity $v(-\infty)=v_0$, reflecting specularly on $\partial B$ and exiting with velocity $v^+(x) \in S^2$ after a finite number of reflections. The total force (resistance) on $B$ in direction $v_0$ is given by

$$R_{v_0}(B) := \int_{x \in v_0^\perp} (v_0 - v^+(x))\,dx,$$

where the integral is over Lebesgue measure in $v_0^\perp$. The resistance is the component of $R_{v_0}(B)$ along $v_0$.

In convex or graphical settings, the body may be described as the epigraph of a function $u : \Omega \to [0, M]$, so

$$B = \{ (x, z) : x \in \Omega,\, 0 \leq z \leq u(x) \},$$

with $\Omega \subset \mathbb{R}^2$ a planar domain. For such bodies and unit-density flows, the classical Newton resistance functional becomes

$F(u) = \int_\Omega \frac{dx}{1 + |\nabla u(x)|^2}.$

The goal is to minimize $F$ over admissible profiles—typically concave, satisfying $u|_{\partial\Omega}=0$ and $0 \leq u(x) \leq M$.

2. Classes of Bodies and Impact Conditions

The analytic and geometric structure of the minimal resistance problem is highly sensitive to the class of admissible bodies and the imposed impact condition:

• Convex bodies: Restricting $B$ to convex forms ensures at most single specular reflection per particle.
• Single-Impact Condition (SIC): Graphical profiles $u$ must guarantee that no particle undergoes more than one reflection. Analytically, for every regular point $x$ and every $t > 0$ with $x - t \nabla u(x) \in \overline{\Omega}$,

$$\frac{u(x - t\nabla u(x)) - u(x)}{t} \le \frac{1 - |\nabla u(x)|^2}{2}.$$

This implies $|\nabla u(x)| < 1$; equivalently, the graph never exceeds 45° slope.

• Multiple reflections: Admitting bodies with multiple reflections leads to further class extensions and can collapse resistance to zero (0809.0108).

Variational minimization over these classes yields distinct optimal profiles, regularity properties, and in certain cases non-uniqueness or degeneracy of minimizers.

3. Existence and Qualitative Structure of Minimizers

In the convex class, the existence of minimizers is established by classical direct methods (sequential compactness in suitable Sobolev spaces, lower semicontinuity of $F$). For concave profiles over compact domains,

$$C_M = \{ u : \Omega \to [0, M]\,|\, u \text{ concave},\ u|_{\partial\Omega} = 0 \}$$

is compact in $W^{1, p}_{\mathrm{loc}}(\Omega)$ for any $p < \infty$, and $F$ is coercive (Buttazzo, 2 Nov 2025).

Qualitative analysis of minimizers reveals:

• Where $u$ is differentiable, either $\nabla u(x) = 0$ (flat region/stagnation zone) or $|\nabla u(x)| \geq 1$ (steep facet or developable surface).
• Global $C^1$ smoothness is impossible: minimizers have singularities (corners or discontinuous slope jumps) separating flat and steep regions. In portions where $u$ is $C^2$, second variation arguments force $\det D^2u = 0$, i.e., the surface is locally ruled or developable (Buttazzo, 2 Nov 2025, Plakhov, 2018).
• If all boundary points of $\Omega$ are regular and $f(p)$ satisfies a mild growth condition at infinity, any minimizer vanishes on $\partial\Omega$ (Plakhov, 2019).

4. Explicit Solutions and Main Theorems

4.1 Newton's Classical Problem

Newton’s minimal resistance problem, posed among bodies of revolution, produces a radially symmetric minimizer with a profile composed of segments joining a central flat cap to sloped sides, leading to a piecewise-linear or -parabolic solution. The resistance for such solutions is amenable to closed-form integration.

In analytical form, for a profile $u(r)$ over a disk of radius $R$,

$F(u) = 2\pi \int_0^R \frac{r\,dr}{1 + (u'(r))^2},$

and the optimal profile can be characterized piecewise (e.g., linear near the boundary, flat in the center) with parameters determined by the height constraint and regularity at the rim.

4.2 Non-Radial and Non-Convex Solutions

Recent work demonstrates that strictly convex $C^2$ regions cannot be minimal for the Newton resistance functional due to the existence of infinite-dimensional families of local perturbations that do not alter the resistance (Plakhov, 2018). Minimizers thus display regions of zero Gaussian curvature (flats or developables), and, crucially, minimizers on symmetric domains are non-radial; symmetry breaking occurs and optimal bodies need not be surfaces of revolution (Lokutsievskiy et al., 2020, Buttazzo, 2 Nov 2025).

4.3 Zero Resistance and Invisibility

Aleksenko and Plakhov constructed explicit three-dimensional bodies of zero resistance, no trace, and invisible bodies in one direction:

• Zero resistance: Let $B$ be such that for almost every incoming particle, after specular reflections, it emerges parallel to $v_0$, i.e., $v^+(x) = v_0$ a.e. This implies $R_{v_0}(B) = 0$.
• No trace: The mapping from initial to exit position $x \mapsto x^+(x)$ is measure-preserving in the orthogonal plane to $v_0$.
• Invisible bodies: Additionally, the mapping $x \mapsto x^+(x)$ satisfies $x^+(x) = x$ a.e.; outside a compact set, particles are unscattered.

The existence of such bodies is proved constructively: starting from a body $B_0$, formed as the surface of revolution of segments joining points in a plane arranged via equilateral triangles, each particle undergoes two (for $B_0$) or four (for a doubled body, $B_{\mathrm{inv}}$) reflections, always emerging with the original velocity. Invisibility is achieved by doubling the zero-resistance body about its mid-plane (0809.0108).

These constructions exploit precise geometric matching of deflection angles to achieve total cancellation of net momentum transfer. The construction violates classical single-impact or convexity hypotheses and is unattainable in Newton's classical setting.

5. Paradoxical Constructions and the Kakeya Analogy

The infimum of the resistance functional subject to the single-impact condition and without convexity or symmetry can be computed explicitly. Utilizing a Besicovitch-Kakeya-type construction, as in (Plakhov, 2013, Plakhov, 2014), profiles are engineered from arrangements of paraboloidal "mirrors" and small "valleys" such that almost all particles are reflected nearly tangentially (contributing minimal resistance), with the exceptional set (valleys) contributing resistance 1 but occupying vanishingly small area. As a result,

$\inf_{u \in S_{\Omega, M}} F(u) = \phi(\Omega, M),$

where $\phi$ can be driven down to $1/2$ in suitable circumstances. Allowing double-impacts, the infimum is zero: arbitrarily small resistance is achievable by constructing complex, slender paraboloidal arrangements enabling specular escape after two bounces (Plakhov, 2014, 0809.0108).

A summary of the implications is as follows:

Body Class	Minimal Resistance	Description
Convex, single impact	$F(u) > 1/2$	Unique minimizer, ruled/developable side
General (non-convex), single impact	$F(u) \to 1/2$	Besicovitch/Kakeya construction
Multiple impacts allowed (2+)	$F(u) \to 0$	Paraboloidal arrangements
Zero-resistance, invisible in a direction	$F(u) = 0$	Explicit billiard geometries

6. Applications, Open Problems, and Extensions

Bodies of minimal resistance are directly relevant to kinetic theory, rarefied gas dynamics, and foundational problems in calculus of variations and geometric measure theory. The theory is also intimately connected to extremal geometric optics (visibility, invisibility) and the design of retroreflectors.

Outstanding problems include:

• Characterizing minimizers in the convex, non-radial case: numerical and analytic methods suggest singularities are vital, but a full classification remains open (Buttazzo, 2 Nov 2025, Lokutsievskiy et al., 2020).
• Regularity at the free boundary: the structure and regularity at the transition between flat and steep regions is not fully understood.
• Extensions to viscous, compressible or thermally agitated media: models incorporating more realistic physics introduce additional constraints and functionals (Buttazzo, 2 Nov 2025).
• Constructibility of minimizers: Theoretical minimizers may possess microstructures that preclude physical manufacture; regularized versions incorporating penalization for curvature or higher regularity remain an active area.
• Visibility and scattering: The same billiard techniques underlie constructions of bodies invisible in a direction or with tailored scattering properties (0809.0108).

7. Historical Context and Significance

Newton's original formulation of the minimal resistance problem marked the first systematic paper of variational optimization of shape under particle collision models. Progress up to the mid-20th century was slow, limited by analytical challenges and the geometric complexity of admissible classes. Fundamental advances—such as the expansion to non-convex admissible sets, the exploitation of microstructures (Kakeya/Besicovitch), and explicit construction of zero-resistance and invisible bodies—have dramatically altered understanding. The modern theory elucidates how variational problems in geometric mechanics can admit "paradoxical" solutions, fundamentally altering the achievable minima compared to classical symmetry or regularity-restricted formulations (Plakhov, 2013, Plakhov, 2014, 0809.0108).

The field remains under active investigation, with ongoing work in the characterization of extremal sets, structure of minimizers under natural regularization, and connections to scattering theory, geometric optics, and dynamical billiards.