Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups

E. A. Ladeishchikov; L. V. Lokutsievskiy; N. V. Prilepin

REVIEW 2 major objections 3 minor 18 references

Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read All sub-Lorentzian geodesics on 3D unimodular Lie groups have explicit formulas.

desk verdict A real advance in explicit timelike extremals for sub-Lorentzian problems, but the abstract overclaims: the anti-norm is not arbitrary and lightlike extremals are not explicitly integrated. read the letter →

arxiv 2507.00250 v1 pith:M5NAMCLX submitted 2025-06-30 math.OC

E. A. Ladeishchikov , L. V. Lokutsievskiy , N. V. Prilepin This is my paper

classification math.OC MSC 53C1749K15

keywords sub-Lorentziangeometrysub-Finslerconvexhyperbolicfunctionsanti-normPontryaginmaximumprincipleunimodularLiegroupsLobachevskyplaneHeisenberggroup

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that every Pontryagin extremal in left-invariant sub-Lorentzian problems on the five three-dimensional unimodular Lie groups, and in the Lorentzian geodesic problem on the Lobachevsky plane, can be written down explicitly once the unit ball $\Omega$ of the driving anti-norm satisfies two generic convexity-boundary assumptions. The formulas use newly introduced functions $\cosh_{\Omega}$ and $\sinh_{\Omega}$, which generalize ordinary hyperbolic functions to arbitrary unbounded convex sets. If correct, this integrates the vertical subsystem of Pontryagin's maximum principle for all these problems, reducing geodesic computation to evaluating the new functions and one quadrature. The same method also yields explicit geodesics in the Finsler problem on the Heisenberg group for unit balls admitting generalized spherical coordinates, including $\ell^p$ norms. A sympathetic reader would care because explicit geodesic formulas are the missing step for global reachability and optimal-synthesis questions in these model geometries.

What carries the argument

The load-bearing object is the anti-norm unit ball $\Omega$ together with its antipolar $\Omega^{\diamond}$, and the sector-area functions $\cosh_{\Omega}$ and $\sinh_{\Omega}$ built from them. For a boundary point $(\cosh_{\Omega}\theta, \sinh_{\Omega}\theta)$, the set ${}^{\diamond}\eta$ collects those angles $\eta$ whose antipolar covectors support $\Omega$ at that point; when the correspondence is single-valued the two functions differentiate exactly like hyperbolic sine and cosine. This machinery converts the Pontryagin maximum principle vertical subsystem into a two-dimensional ODE on the angle $\eta$ with a first integral $E$, and the explicit formulas in Theorems 7 and 8 are direct corollaries of those derivative rules.

What would settle it

Take $\Omega = \{(x,y)\mid |y|\leq x,\ x\geq 1\}$, a convex closed set satisfying the ray property but failing Assumption 2 because the boundary contains the ray $\{(x,x):x\geq 1\}$; solve the vertical PMP system (12.1) for the Heisenberg group with this anti-norm and check whether every timelike extremal can be written in the Theorem 8 form.

Watch

Extended reading notes

Core claim

The paper's central discovery is a duality-preserving extension of convex trigonometry to the hyperbolic regime. For an unbounded closed convex set $\Omega$ in the plane satisfying the ray property, the antipolar set $\Omega^{\diamond} = \{(p,q) \mid px - qy \geq 1 \text{ for all } (x,y)\in\Omega\}$ plays the role of the polar. Under Assumption 2, meaning no ray from the origin lies on $\partial\Omega$ and the boundary approaches the two boundary rays of its cone, every boundary point of $\Omega$ has a matching boundary covector, and the functions $\cosh_{\Omega}$ and $\sinh_{\Omega}$ defined by sector area satisfy the differentiation rules $d/d\theta\,\cosh_{\Omega}\theta = \sinh_{\Omega^{\diamond}}\eta$ and $d/d\theta\,\sinh_{\Omega}\theta = \cosh_{\Omega^{\diamond}}\eta$ when $\eta\in\theta^{\diamond}$, together with the inequality $\cosh_{\Omega}\theta\cosh_{\Omega^{\diamond}}\eta - \sinh_{\Omega}\theta\sinh_{\Omega^{\diamond}}\eta \geq 1$, with equality exactly for corresponding angles. These identities let the paper write every normal timelike extremal of a sub-Lorentzian problem on $SU(2)$, $SL(2)$, $SE(2)$, $SH(2)$, and $H_3$ in the form $h_1 = -\cosh_{\Omega^{\diamond}}\eta$, $h_2 = \sinh_{\Omega^{\diamond}}\eta$, $h_3 = \pm\sqrt{E - a(\sinh_{\Omega^{\diamond}}\eta)^2 + b(\cosh_{\Omega^{\diamond}}\eta)^2}$, $u_1 = \cosh_{\Omega}\theta$, $u_2 = \sinh_{\Omega}\theta$, with $\theta\in{}^{\diamond}\eta$ and $\eta$ given by a quadrature; lightlike extremals are exactly abnormal and run along the boundary rays of the cone. In the Lorentzian problem on the Lobachevsky plane, non-singular timelike extremals reduce to arcs of the reflected antipolar set, with the angle obeying the quadrature $t = \int d\eta / \cosh_{\Omega^{\diamond}}\eta$.

Load-bearing premise

The argument rests on the anti-norm's unit ball having two boundary properties: no ray from the origin lies on the boundary, and the boundary approaches the two side rays of its cone.

Editorial extensions

If this is right

On every 3D unimodular Lie group, timelike geodesics are parametrized by constants $E$, $\eta_0$ and a single quadrature, so no numerical integration of the covector ODE is needed.
Lightlike geodesics in these problems are precisely abnormal extremals running along the two boundary rays of the cone generated by $\Omega$, with at most $k+1$ switches when $h_3$ changes sign $k$ times.
On the Lobachevsky plane, non-singular timelike extremals are obtained by reflecting $\Omega^{\diamond}$ across the diagonal, scaling and shifting horizontally, while the singular vertical-like geodesics follow exponential motion.
For the Finsler problem on the Heisenberg group, unit balls admitting generalized spherical coordinates, including $\ell^p$ norms, yield explicit geodesic coordinates from convex sine and cosine rather than special functions or elliptic integrals.
The formulas extend the previously known quadratic Lorentzian case to arbitrary anti-norm unit balls satisfying Assumption 2, unifying the five unimodular groups in one scheme.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: because the angle correspondence $\theta\leftrightarrow\eta$ is the only place Assumption 2 enters, the same explicit scheme should extend to any anti-norm whose unit ball can be approximated from inside by sets satisfying both properties, at the price of a limiting argument.
Beyond the paper: for a wedge unit ball such as $\Omega=\{(x,y)\mid |y|\leq x,\ x\geq 1\}$, the boundary contains a ray from the origin and the correspondence fails, so no formula of this exact type should be expected; this gives a natural stress test of the hypothesis.
Beyond the paper: the quadrature $t = \pm\int d\eta / \sqrt{E - a(\sinh_{\Omega^{\diamond}}\eta)^2 + b(\cosh_{\Omega^{\diamond}}\eta)^2}$ suggests explicit periodicity and closed-geodesic criteria on $SL(2)$ and $SU(2)$ depending on the constants $a,b$ of the group, a question the paper does not pursue.
Beyond the paper: numerically integrating the vertical PMP system for a strictly convex $\alpha$-hyperbola with $\alpha>2$ would verify the finite-domain prediction for $\cosh_{\Omega_\alpha}$, since the defining sector area integral converges or diverges exactly according to the paper's threshold.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely new: it constructs cosh_Ω and sinh_Ω for unbounded convex sets, proves the expected duality and differentiability properties, and uses them to write explicit formulas for timelike extremals in sub-Lorentzian problems on all three-dimensional unimodular Lie groups, plus the Lorentzian Lobachevsky plane and a Finsler Heisenberg problem. The core calculations in Sections 7 and 12 are coherent, the first integrals check by differentiation, and reverse implications are supplied. The dependence on the authors' earlier convex trigonometry is real but not circular, since those results are published and derived independently. This is a serious contribution to geometric control.

The soft spots are mostly about framing, not math. The abstract and introduction promise "arbitrary anti-norm" and "explicit formulas for extremals." What the theorems actually require is Assumptions 1 and 2, and Example 2 shows that a perfectly natural anti-norm (the wedge) violates property (*). So the advertised coverage is broader than the hypotheses. The stress-test note is also correct: Theorem 8 does not explicitly integrate lightlike extremals. For λ0=0, the maximum condition is attained on an entire boundary ray, so the control contains an arbitrary measurable scalar function. The paper only says u(t)∈∂C a.e. and that h(t) stays on a fixed ray; it does not give u through cosh_Ω or sinh_Ω, and the quadrature for η only covers the timelike case where the first-integral inequality is strict. Saying "explicit formulas for extremals are obtained" is therefore an overclaim, even under Assumptions 1 and 2.

That said, these are issues of honest scope, not fatal flaws in the main construction. The timelike extremals are explicitly and correctly characterized, and the lightlike case is at least qualitatively classified. The geometric lemmas in Sections 9–10 are intricate but plausible, and I did not find a circular step or fitted parameter. The paper does not prove global optimality, but it does not claim it.

Who gets value: anyone working on sub-Lorentzian or sub-Finsler geometry, especially on Lie groups, will want to know these formulas. A careful referee should engage with it, but must push the authors to correct the abstract and intro so the claims match the assumptions: Assumption 2 throughout, and lightlike extremals described only qualitatively unless explicitly integrated. I would send it to peer review, expecting a revision on scope.

Referee Report

2 major / 3 minor

Summary. The paper introduces convex hyperbolic functions cosh_Ω and sinh_Ω for unbounded convex sets Ω⊂R² satisfying Assumptions 1 and 2, and applies them to write explicit formulas for Pontryagin extremals in three families of problems: left-invariant Finsler problems on the Heisenberg group (Theorem 2), the Lorentzian problem on the Lobachevsky plane modeled on Aff+(R) (Theorem 7), and sub-Lorentzian problems on the three-dimensional unimodular Lie groups SU(2), SL(2), SE(2), SH(2), and H3 (Theorem 8). The formulas integrate the vertical PMP subsystem in the normal timelike case using first integrals and quadratures, and the paper supplies reverse implications for the Heisenberg Finsler theorem. The abnormal lightlike case, however, is only classified qualitatively as u(t)∈∂C, and the introduction's claim of an arbitrary anti-norm is not supported by the hypotheses of the main theorems.

Significance. If the timelike formulas are correct, the paper provides a unified explicit integration of the vertical PMP subsystem for normal timelike extremals on all listed three-dimensional unimodular Lie groups and on the Lobachevsky plane, in terms of genuinely new functions that generalize the classical hyperbolic functions to non-quadratic anti-norms. The construction of the antipolar correspondence, the derivative formulas in Theorem 5, and the first-integral reduction in Theorem 8 are nontrivial and appear internally coherent. The paper also gives concrete examples (e.g., α-hyperbolic unit balls) and supplies reverse implications for the Heisenberg Finsler problem, which strengthens the reliability of the core calculations. The main weakness is that the advertised scope is broader than what is actually proved: abnormal lightlike extremals are not explicitly integrated, and the introductory claim of 'arbitrary anti-norm' is contradicted by Assumptions 1 and 2.

major comments (2)

[Section 12, Theorem 8] Theorem 8 explicitly integrates only the normal timelike case. For abnormal lightlike extremals it states only that u(t)∈∂C for almost every t and that h(t) stays on a ray of ∂C*, but the vertical subsystem (12.1) is not integrated. This is not a minor omission: in the H3 case (a=b=0) with h=(-1,1), h3=0, any u(t)=λ(t)(1,1) with bounded measurable λ≥0 satisfies the PMP maximum condition with H=0, and q(t) depends on the integral of λ(t). Thus there is an infinite-dimensional family of lightlike extremals that is not written through cosh_Ω, sinh_Ω and is not covered by the quadrature for η, which is derived under the strict inequality (12.2). Consequently, the abstract's claim that 'explicit formulas for extremals are obtained' is unsupported as stated. The claim should be narrowed to timelike extremals, or the lightlike abnormal case must be integrated separately.
[Introduction and Section 9.2] The introduction states that explicit formulas are obtained for left-invariant sub-Lorentzian problems with an arbitrary anti-norm on all three-dimensional unimodular Lie groups. However, the functions cosh_Ω and sinh_Ω are constructed only under Assumptions 1 and 2, and both Theorem 6 and Theorem 8 assume Assumption 2. Example 2 gives a natural anti-norm unit ball (the wedge Ω=R_1^+[P0;P1]) that satisfies Assumption 1 but violates property (∗) of Assumption 2, so such anti-norms are outside the scope of the main theorems. The word 'arbitrary' is therefore false as written and should be replaced by a precise statement of the assumptions, both in the introduction and in the abstract.

minor comments (3)

[Section 10, Theorem 6] In item 1 of Theorem 6, the formula for the covector is written as (h1,h2)=(−cosh_Ω η, sinh_Ω η); since h is an element of the dual space, the correct expression, as used in Theorems 7 and 8, should be (−cosh_{Ω⋄} η, sinh_{Ω⋄} η). The line also contains a garbled expression 'coshΩθΩθ' that should read (cosh_Ω θ, sinh_Ω θ).
[References] References [2] and [14] appear to be the same paper by L. V. Lokutsievskiy on explicit formulae for geodesics in left-invariant sub-Finsler problems on Heisenberg groups; they should be merged into a single reference to avoid confusion.
[Section 11, proof of Theorem 7] In the lightlike case, the proof discusses the subcase h3≡0 in detail and then states that h3 is either identically zero or sign-constant, concluding at most one switch. The reasoning would be clearer if it explicitly explained why a sign-constant nonzero h3 also forces the control to stay on a fixed boundary ray and why no switch can occur when h3 changes sign without h passing through zero.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the explicit extremal formulas are derived from the Pontryagin maximum principle and first integrals, not from fitted inputs or self-citation chains.

full rationale

The paper's central derivations are self-contained reductions of the PMP vertical subsystem, not re-statements of the assumptions. Theorem 6 and Proposition 10.2 show that normal timelike extremals force (-h1,h2) to lie on the boundary of the antipolar set Ω⋄, and then the formulas h1=-cosh_{Ω⋄}η, h2=sinh_{Ω⋄}η, u1=cosh_Ωθ, u2=sinh_Ωθ with θ∈⋄η are a boundary parametrization made possible by Proposition 9.1 and Assumptions 1-2. The nontrivial content of Theorem 8 is the derivation of the first integral and the quadrature for η from the dynamics ˙η=h3 and (12.1); this is a genuine integration step, not a definitional equivalence. The same holds for Theorem 7 on the Lobachevsky plane and Theorem 2 on the Heisenberg group, where the convex trigonometric parametrization is combined with explicit integrations of the PMP equations. The paper does rely on the authors' prior work [2], [6], and [14] for convex trigonometry and for the classification of 3D Lie algebra bases, but these are published, parameter-free results with stated assumptions that do not include the target sub-Lorentzian formulas; they are used as tools, not as the source of the conclusions. There are no fitted parameters, no quantity is tuned to data and then renamed a prediction, and no uniqueness theorem is imported from the authors to force the choice. Two limitations are present but they are not circularity: the formulas require Assumption 2, which excludes some natural anti-norms such as the wedge in Example 2, and Theorem 8 does not give explicit parametrizations for abnormal lightlike extremals beyond u(t)∈∂C a.e. These affect completeness and scope, not the logical dependence of the results on their inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central formulas rest on standard control theory, prior published convex trigonometry, anti-norm duality, and domain restrictions on Ω and U. No parameter is fitted to data, and no new physical entity is postulated; cosh_Ω and sinh_Ω are explicit mathematical definitions rather than invented entities. The main burden is Assumption 2, which is strictly narrower than the advertised 'arbitrary anti-norm'.

assumptions (6)

domain assumption The unit ball Ω of the anti-norm satisfies Assumption 1(i) and Assumption 2, including properties (*) and (**).
Invoked in Section 9 to define cosh_Ω and sinh_Ω and in Theorems 6, 7 and 8. Not every anti-norm unit ball satisfies it, so the introduction's 'arbitrary anti-norm' phrasing is overbroad.
domain assumption The Finsler unit ball U on H3 admits generalized spherical coordinates with concave profile f and convex base Ω, as in Section 3.1.
Required by Theorem 2 and Corollary 7.1; the paper itself notes that not every Finsler norm admits such coordinates, though ℓ_p norms do.
domain assumption The classification of possible planes Δ in 3D unimodular Lie algebras by brackets (3.1) is complete.
Section 3.3 uses this classification, imported from references [6] and [8], to reduce to SU(2), SL(2), SE(2), SH(2) and H3; completeness is not reproved in the paper.
standard math Pontryagin's maximum principle supplies valid necessary conditions for the length-extremal problems.
Used throughout Sections 4, 7, 10, 11 and 12; standard optimal-control background taken without proof.
standard math Anti-norm and unit-ball duality, including the bipolar theorem for antipolar sets.
Used in Sections 8 and 9, via Lemma 1 and Theorem 3, to justify Ω⋄⋄ = Ω and the ray property; the duality theory is cited to reference [15].
standard math The convex trigonometry of cos_Ω and sin_Ω from references [2] and [6] is correct.
Used in Section 7 for the Finsler Heisenberg problem; imported as an established tool rather than rederived.

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Cite this review

Pith. "Pith review of Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups." pith.science (2026). https://pith.science/paper/M5NAMCLX

@misc{pith2026250700250,
  author       = {Pith},
  title        = {Pith review of: Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5NAMCLX}},
  note         = {Machine review of arXiv:2507.00250}
}

abstract

In this paper, we consider the problem of finding geodesics in a series of left-invariant problems endowed with sub-Lorentzian and Finsler structures. Explicit formulas for extremals are obtained in terms of convex trigonometric functions. In the sub-Lorentzian setting, the new trigonometric functions $\cosh_\Omega$ and $\sinh_\Omega$, developed here, prove especially useful; they generalize the classical $\cosh$ and $\sinh$ to the case of an unbounded convex set $\Omega\subset\mathbb{R}^2$.

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Reference graph

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

The quasihyperbolic plane

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Lokutsievskiy, L.V. Explicit Formulae for Geodesics in Left–Invariant Sub–Finsler Problems on Heisenberg Groups via Convex Trigonometry. J Dyn Control Syst 27, 661–681 (2021). https://doi.org/10.1007/s10883-020-09516-z

work page doi:10.1007/s10883-020-09516-z 2021

[3] [3]

An Isoperimetric Problem on the Lobachevsky Plane with a Left- Invariant Finsler Structure

V. A. Myrikova, “An Isoperimetric Problem on the Lobachevsky Plane with a Left- Invariant Finsler Structure”, Optimal Control and Dynamical Systems, Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze, Trudy Mat. Inst. Steklova, 321, Steklov Math. Inst., Moscow, 2023, 223–236; Proc. Steklov Inst. Math., 32...

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Yu. L. Sachkov, “Lorentzian Geometry on the Lobachevsky Plane”, Mat. Zametki, 114:1 (2023), 154–157; Math. Notes, 114:1 (2023), 127–130

work page 2023

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Yu. L. Sachkov, E. F. Sachkova, “Sub-Lorentzian Problem on the Heisenberg Group”, Mat. Zametki, 113:1 (2023), 154–157; Math. Notes, 113:1 (2023), 159–162

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A.A.Ardentov, L.V.Lokutsievskiy, Yu.L.Sachkov, «Extremalsforaseriesofsub- Finsler problems with 2-dimensional control via convex trigonometry», ESAIM: COCV, 27 (2021), 32, 52 pp., arXiv: 2004.10194

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work page 2024

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Agrachev, D

A.A. Agrachev, D. Barilari, «Sub-Riemanian structures on 3D Lie groups», JDCS, 2012(18), p 21–44

work page 2012