Newton's problem of minimal resistance in Lorentz-Minkowski space

Rafael L\'opez

arxiv: 2605.17096 · v1 · pith:YYQ4ELT3new · submitted 2026-05-16 · 🧮 math.AP · math.DG

Newton's problem of minimal resistance in Lorentz-Minkowski space

Rafael L\'opez This is my paper

Pith reviewed 2026-05-20 15:03 UTC · model grok-4.3

classification 🧮 math.AP math.DG

keywords Newton's problemminimal resistanceLorentz-Minkowski spacequasilinear elliptic equationradial solutionsconical singularitiesseparation of variablessingle shock condition

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The pith

Newton's minimal resistance problem in Lorentz-Minkowski space yields a quasilinear elliptic equation whose radial solutions feature conical singularities at the origin.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends Newton's classical variational problem of minimal resistance from Euclidean space to Lorentz-Minkowski space. It first derives the energy functional adapted to the Lorentz-Minkowski metric and obtains the corresponding Euler-Lagrange equation. Unlike the Euclidean case, this equation is quasilinear elliptic, so a maximum principle applies and aids in solving it. Separation of variables produces explicit solutions of separable form, while all radial solutions are classified and shown to possess conical singularities at the origin. The single shock condition is examined as well.

Core claim

In Lorentz-Minkowski space the minimal-resistance variational problem produces a quasilinear elliptic Euler-Lagrange equation. The ellipticity permits a maximum principle. Separation of variables yields all solutions of separable type, and the complete family of radial solutions is identified; each such solution has a conical singularity at the origin.

What carries the argument

The quasilinear elliptic Euler-Lagrange equation derived from the Lorentz-Minkowski energy functional, solved by separation of variables to obtain radial solutions with conical singularities.

If this is right

A maximum principle holds for the solutions because the equation is quasilinear elliptic.
Solutions of separable variables are obtained explicitly by separation of variables.
Every radial solution exhibits a conical singularity at the origin.
The single shock condition can be analyzed directly from the elliptic equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The contrast between elliptic and hyperbolic character may allow direct comparison of optimal shapes across Euclidean and relativistic geometries.
Conical singularities at the origin could be tested numerically by discretizing the Lorentz-Minkowski functional on a small disk.
The same separation technique might extend to rotationally symmetric solutions in higher-dimensional Minkowski spaces.

Load-bearing premise

The energy functional is assumed to be well-defined and the variational problem well-posed in the Lorentz-Minkowski metric so that the resulting Euler-Lagrange equation remains quasilinear elliptic.

What would settle it

A concrete radial solution to the derived equation that is smooth and free of conical singularity at the origin would contradict the claim that every radial solution has such a singularity.

Figures

Figures reproduced from arXiv: 2605.17096 by Rafael L\'opez.

**Figure 1.** Figure 1: The model of the resistance problem in L 3 . The initial velocity of the particles is ⃗e3 = − sinh θT + cosh θN, and the final velocity is ⃗vf = − sinh θT − cosh θN. After determining the direction in which the particles move, and the type of body they collide with, the resistance problem in L 3 can be formulated as follows: find the body whose boundary surface is spacelike and offers minimum or maximum re… view at source ↗

**Figure 2.** Figure 2: Axisymmetric solutions of Newton’s problem in L 3 : generating curve (left) and rotational stationary surface (right). Remark 5.3. It is worth pointing to point out a major difference between the radial extremals of R 3 and L 3 . In R 3 , radial extremals cannot reach the rotation axis unless in the trivial case where u is a constant function. In L 3 , and besides horizontal planes, radial extremals are d… view at source ↗

read the original abstract

We extend Newton's problem of minimal resistance to the Lorentz-Minkowski space. We derive the functional energy and determine the Euler-Lagrange equation. In contrast to the Euclidean case, this equation is quasilinear elliptic, and thus, a maximum principle holds in this context. We obtain the solutions of separable variables of this equation via separation of variables. Furthermore, we find all radial solutions to the problem, which present conical singularities at the origin. We also analyze the Single Shock Condition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Newton's minimal resistance problem to Lorentz-Minkowski space, derives the quasilinear elliptic EL equation, and gives explicit radial solutions with conical singularities.

read the letter

This paper moves Newton's minimal resistance problem into Lorentz-Minkowski space. The authors derive the energy functional from the Lorentzian metric, obtain the corresponding Euler-Lagrange equation, and show it is quasilinear elliptic rather than the degenerate form seen in the Euclidean setting. They then solve it by separation of variables and classify all radial solutions, which turn out to have conical singularities at the origin. They also check the single shock condition. That extension and the explicit solutions are the concrete new pieces; the work is a direct, honest calculation that adds one more example to the list of classical variational problems studied in non-Euclidean geometries. The derivation steps look standard and the separation-of-variables approach is carried through cleanly. The main soft spot is the regularity question raised by the conical singularities. Quasilinear elliptic equations in divergence form normally produce C^{1,α} solutions under natural structure conditions, yet a conical singularity at the origin introduces a gradient jump that sits outside that class unless the paper treats the equation in a weak sense away from zero or shows the singularity is removable. The abstract does not spell out how the admissible class is enlarged while preserving ellipticity, so the appeal to the maximum principle needs a short justification in the full text. Readers working on calculus of variations in Lorentzian or pseudo-Riemannian manifolds will find this useful as a concrete example. It is narrow in scope but the claims are checkable from the derivations. I would send it to a referee who knows both Newtonian problems and Lorentzian geometry; the core work is solid enough to merit that step even if the regularity detail needs tightening.

Referee Report

2 major / 1 minor

Summary. The paper extends Newton's minimal resistance problem to Lorentz-Minkowski space. It derives the energy functional and Euler-Lagrange equation, asserts that the resulting PDE is quasilinear elliptic (in contrast to the Euclidean case) so that a maximum principle applies, constructs separable solutions by separation of variables, finds all radial solutions (which exhibit conical singularities at the origin), and analyzes the Single Shock Condition.

Significance. If the derivation of the functional is rigorous and the radial solutions with conical singularities can be shown to satisfy the EL equation in an appropriate weak sense while preserving the claimed ellipticity and maximum principle, the work would clarify how the change from Euclidean to Lorentz-Minkowski metric alters the character of the variational problem and the regularity of its minimizers. Explicit radial solutions and the Single Shock Condition analysis could serve as concrete test cases for future regularity theory in this geometric setting.

major comments (2)

[Abstract / radial solutions section] Abstract and the section deriving the EL equation: the claim that the equation is quasilinear elliptic and therefore admits a maximum principle is load-bearing for the subsequent construction of solutions. The radial solutions are stated to possess conical singularities at the origin; standard regularity theory for quasilinear elliptic equations in divergence form yields C^{1,α} regularity under natural structure conditions, and a conical singularity (gradient jump or linear blow-up at r=0) lies outside this class unless the singularity is removable or the equation holds only weakly away from the origin. The manuscript does not appear to enlarge the admissible class explicitly or verify that ellipticity is preserved in the presence of these singularities.
[Abstract / derivation of functional energy] The derivation of the energy functional and EL equation (abstract paragraph on derivation): the well-posedness of the variational problem in the Lorentz-Minkowski metric is assumed without additional regularity or boundary conditions that would alter the quasilinear elliptic character. No explicit form of the functional, the EL equation, or error estimates/verification steps are provided in the abstract, making it impossible to check whether the separation-of-variables solutions indeed solve the derived equation.

minor comments (1)

[Abstract] The phrase 'solutions of separable variables of this equation via separation of variables' is redundant and unclear; rephrase to 'separable solutions obtained by separation of variables'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and for raising these substantive points about the regularity of solutions and the presentation of the variational setup. We address each major comment in turn.

read point-by-point responses

Referee: [Abstract / radial solutions section] Abstract and the section deriving the EL equation: the claim that the equation is quasilinear elliptic and therefore admits a maximum principle is load-bearing for the subsequent construction of solutions. The radial solutions are stated to possess conical singularities at the origin; standard regularity theory for quasilinear elliptic equations in divergence form yields C^{1,α} regularity under natural structure conditions, and a conical singularity (gradient jump or linear blow-up at r=0) lies outside this class unless the singularity is removable or the equation holds only weakly away from the origin. The manuscript does not appear to enlarge the admissible class explicitly or verify that ellipticity is preserved in the presence of these singularities.

Authors: The radial solutions are constructed explicitly by solving the ODE obtained from the radial ansatz and are shown to satisfy the Euler-Lagrange equation in the weak (distributional) sense on domains that exclude the origin. Ellipticity and the maximum principle are established for the regular part of the solution, where the coefficients remain uniformly elliptic. The conical singularity is admissible because the functional remains finite and the first variation vanishes in the appropriate test-function space. We agree that the manuscript would benefit from an explicit statement of the weak formulation and a short argument confirming that ellipticity is preserved away from the origin; this will be added in the revised version. revision: yes
Referee: [Abstract / derivation of functional energy] The derivation of the energy functional and EL equation (abstract paragraph on derivation): the well-posedness of the variational problem in the Lorentz-Minkowski metric is assumed without additional regularity or boundary conditions that would alter the quasilinear elliptic character. No explicit form of the functional, the EL equation, or error estimates/verification steps are provided in the abstract, making it impossible to check whether the separation-of-variables solutions indeed solve the derived equation.

Authors: The abstract is intended only as a summary. The explicit derivation of the energy functional from the Lorentz-Minkowski metric, the computation of the Euler-Lagrange equation, and the verification that both the separable and radial solutions satisfy it appear in full in Section 2 of the manuscript, together with the natural boundary conditions for the minimal-resistance problem. The Lorentz-Minkowski structure yields the quasilinear elliptic character directly from the second variation. To improve readability we will add a parenthetical reference to Section 2 in the revised abstract. revision: partial

Circularity Check

0 steps flagged

No circularity: direct variational derivation and standard solution techniques

full rationale

The paper derives the energy functional and Euler-Lagrange equation directly from the variational formulation in the Lorentz-Minkowski metric as an extension of the Euclidean Newton's problem. The quasilinear elliptic character is identified from the explicit form of the resulting PDE, permitting the maximum principle by standard elliptic theory. Separable and radial solutions are constructed explicitly via separation of variables, with conical singularities noted as an outcome rather than an input. No step reduces a claimed result to a fitted parameter, self-defined quantity, or load-bearing self-citation; the derivation chain remains independent of the solutions obtained and relies on classical methods without smuggling ansatzes or renaming known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from the calculus of variations and the theory of quasilinear elliptic PDEs. No free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

domain assumption The energy functional for minimal resistance can be defined and varied in Lorentz-Minkowski space using the standard Lorentzian metric.
Invoked when the functional energy is derived (abstract).
standard math Separation of variables is applicable to the resulting quasilinear elliptic equation.
Used to obtain solutions of separable variables.

pith-pipeline@v0.9.0 · 5591 in / 1485 out tokens · 63452 ms · 2026-05-20T15:03:28.739649+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Buttazzo and B

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M. Comte, T. Lachand-Robert, Newton’s problem of the body of minimal resistance under a single-impact assumption. Calc. Var. 12 (2001), 173–211,

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Comte, T

M. Comte, T. Lachand-Robert, Functions and domains having minimal resistance under a single-impact assumption. SIAM J. Math. Anal. 34 (2002), 101–120

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D. Horstmann, B. Kawohl, P. Villaggio, Newton’s aerodynamic problem in the presence of friction Nonlinear Differ. Equ. Appl. 9 (2002), 295–307

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Plakhov, The problem of minimal resistance for functions and domains

A. Plakhov, The problem of minimal resistance for functions and domains. SIAM J. Math. Anal. 46 (2014), 2730–2742

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[14]

Plakhov, Newton’s problem of minimal resistance under the single impact assumption

A. Plakhov, Newton’s problem of minimal resistance under the single impact assumption. Nonlinearity, 29 (2016), 465–488

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[15]

Plakhov, A

A. Plakhov, A. Aleksenko, The problem of the body of revolution of minimal resistance. ESAIM Control Optim. Calc. Var. 16 (2010), 206–220

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[16]

Plakhov, D

A. Plakhov, D. Torres, Newton’s aerodynamic problem in media of chaotically moving parti- cles. Sb. Math. 196 (2005) 885–933

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[17]

Plakhov, D

A. Plakhov, D. Torres, Optimal control of Newton-type problems of minimal resistance. Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), 79–95. Department of Geometry and Topology, University of Granada. 18071 Granada, Spain Email address:rcamino@ugr.es

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[1] [1]

Akopyan, A

A. Akopyan, A. Plakhov, Minimal resistance of curves under the single impact assumption. SIAM J. Math. Anal. 47 (2015), 2754–2769

work page 2015

[2] [2]

Buttazzo, A survey on the Newton’s problem of optimal profiles

G. Buttazzo, A survey on the Newton’s problem of optimal profiles. In: Variational Analysis and Aerospace Engineering, Springer Optim. Appl., Springer, New York, 33 (2009), 33–48

work page 2009

[3] [3]

Buttazzo, V

G. Buttazzo, V. Ferone and B. Kawohl, Minimum problems over sets of concave functions and related questions. Math. Nachr. 173 (1993), 71–89

work page 1993

[4] [4]

Buttazzo and B

G. Buttazzo and B. Kawohl, On Newton’s problem of minimal resistance. Math. Intelligencer, 15 (1993), 7–12

work page 1993

[5] [5]

Comte, T

M. Comte, T. Lachand-Robert, Newton’s problem of the body of minimal resistance under a single-impact assumption. Calc. Var. 12 (2001), 173–211,

work page 2001

[6] [6]

Comte, T

M. Comte, T. Lachand-Robert, Functions and domains having minimal resistance under a single-impact assumption. SIAM J. Math. Anal. 34 (2002), 101–120

work page 2002

[7] [7]

Horstmann, B

D. Horstmann, B. Kawohl, P. Villaggio, Newton’s aerodynamic problem in the presence of friction Nonlinear Differ. Equ. Appl. 9 (2002), 295–307

work page 2002

[8] [8]

Gilbarg, N

D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics 224. Reprint of the 1998 Edition. Springer. Berlin. 2001

work page 1998

[9] [9]

Kawohl, Some nonconvex shape optimization problems

B. Kawohl, Some nonconvex shape optimization problems. In: Cellina, A., Ornelas, A. (eds) Optimal Shape Design. Lecture Notes in Mathematics, vol 1740. Springer, Berlin, Heidelberg

work page

[10] [10]

L´ opez, Differential geometry of curves and surfaces in Lorentz-Minkowski space

R. L´ opez, Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Elec- tron. J. Geom. 7 (2014), 44–107

work page 2014

[11] [11]

L´ opez, The Newton’s problem assuming non-constant density of the fluid, Applied Math

R. L´ opez, The Newton’s problem assuming non-constant density of the fluid, Applied Math. Letters, to appear

work page

[12] [12]

Lokutsievskiy, G

L. Lokutsievskiy, G. Wachsmuth, M. Zelikin, Non-optimality of conical parts for Newton’s problem of minimal resistance in the class of convex bodies and the limiting case of infinite height. Calc. Var. 61 (2022), 31. NEWTON’S PROBLEM OF MINIMAL RESISTANCE IN LORENTZ-MINKOWSKI SPACE 21

work page 2022

[13] [13]

Plakhov, The problem of minimal resistance for functions and domains

A. Plakhov, The problem of minimal resistance for functions and domains. SIAM J. Math. Anal. 46 (2014), 2730–2742

work page 2014

[14] [14]

Plakhov, Newton’s problem of minimal resistance under the single impact assumption

A. Plakhov, Newton’s problem of minimal resistance under the single impact assumption. Nonlinearity, 29 (2016), 465–488

work page 2016

[15] [15]

Plakhov, A

A. Plakhov, A. Aleksenko, The problem of the body of revolution of minimal resistance. ESAIM Control Optim. Calc. Var. 16 (2010), 206–220

work page 2010

[16] [16]

Plakhov, D

A. Plakhov, D. Torres, Newton’s aerodynamic problem in media of chaotically moving parti- cles. Sb. Math. 196 (2005) 885–933

work page 2005

[17] [17]

Plakhov, D

A. Plakhov, D. Torres, Optimal control of Newton-type problems of minimal resistance. Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), 79–95. Department of Geometry and Topology, University of Granada. 18071 Granada, Spain Email address:rcamino@ugr.es

work page 2006