The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations

Elimhan N. Mahmudov; Shakir Sh. Yusubov; Shikhi Sh. Yusubov

REVIEW 3 major objections 3 minor 35 references

The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that the classical proof of the Legendre necessary condition can be adapted to fractional calculus of variations with fixed endpoints, contrary to a claim in the existing literature.

desk verdict The new Du Bois-Reymond lemma and integral-form Euler-Lagrange equations are worth attention, but the advertised proof of the Legendre condition for fixed endpoints has a sign/direction error in its key estimate and is not valid as written. read the letter →

arxiv 2506.06736 v1 pith:VZFJCNTJ submitted 2025-06-07 math.OC

Shikhi Sh. Yusubov , Shakir Sh. Yusubov , Elimhan N. Mahmudov This is my paper

classification math.OC MSC 26A3349K9949K05

keywords fractionalcalculusofvariationsCaputoderivativeRiemann-LiouvilleintegralEuler-LagrangeequationLegendreconditionDuBois-Reymondlemmaweaklocalminimumnecessaryconditions

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

The paper studies minimization problems in which the cost is a Riemann-Liouville fractional integral of order $\beta>0$ of a Lagrangian that depends on a Caputo fractional derivative of order $0<\alpha\le 1$, under fixed, free, or mixed endpoint conditions. Its central claim is that, for the problems with a fixed final endpoint, a weak local minimum must satisfy the classical Legendre condition: the second derivative of the Lagrangian with respect to the derivative variable is positive semidefinite along the extremal, at every point where the Caputo derivative is continuous. This is proved by the standard classical method, using a fractional analogue of the Du Bois-Reymond lemma and a needle variation whose fractional derivative has support concentrated on a small interval. The result directly contradicts a well-known assertion in the literature that the classical fixed-endpoint proof cannot be adapted to fractional problems. If the claim is correct, fractional variational problems inherit the same second-order test as classical ones, and the relation between $\alpha$ and $\beta$ enters the Euler-Lagrange equations explicitly.

What carries the argument

The main tool is the generalized Du Bois-Reymond lemma (Lemma 3.1), which characterizes when $\int_{t_0}^{t_1}(t_1-t)^{\beta-1}f(t)(cD_{t_0+}^\alpha h)(t)\,dt=0$ for every zero-endpoint $h$: if $\beta>\alpha$ the function $(t_1-t)^{\beta-\alpha}f(t)$ must vanish, and if $0<\beta\le\alpha$ then $f$ must be a constant multiple of $(t_1-t)^{\alpha-\beta}$. This lemma converts the first variation into the Euler-Lagrange equations in integral form. The second mechanism is the needle variation (16), $h(t)=\frac{1}{\Gamma(\alpha)}\int_{t_0}^t (t-\tau)^{\alpha-1}g(\tau)\,d\tau$ with $g=(f(t)-k)r$ on the needle and zero elsewhere; because the Caputo derivative of $h$ equals $g$, the variation has a compactly supported fractional derivative, which is exactly the feature the classical proof needs and which the earlier literature said could not be arranged under final constraints.

What would settle it

For the needle variation in Theorem 4.2, take $f(t)=1+\delta\cos(\pi(t-\sigma)/\varepsilon)$ on $[\sigma-\varepsilon,\sigma+\varepsilon]$; then the integral of $(f(t)-f(c))^2$ over the needle is $O(\delta^2\varepsilon)$ rather than the $O(\varepsilon)$ needed for estimate (19), so for small $\delta$ the negative term no longer dominates and the contradiction argument fails, which would refute the paper's claim that the standard classical proof goes through without additional hypotheses.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.2: for the fixed-endpoint problem $(P)$, if $x_0$ is a weak local minimum and $L,L_x,L_y,L_{xx},L_{xy},L_{yy}$ are continuous near its graph, then $\langle P(t)r,r\rangle\ge 0$ for every $t$ at which the Caputo derivative of $x_0$ is continuous and every $r\in\mathbb{R}^n$, where $P(t)=L_{yy}(t,x_0(t),(cD_{t_0+}^\alpha x_0)(t))$. The proof constructs a variation $h$ whose Caputo derivative equals $(f(t)-k)r$ on a small interval $[\sigma-\varepsilon,\sigma+\varepsilon]$ and vanishes outside; the constant $k$ is fixed by $h(t_1)=0$. Substituting this $h$ into the second variation, the paper shows that a negative value of $\langle P(\sigma)r,r\rangle$ would make the leading negative term dominate the remaining terms for sufficiently small $\varepsilon$, contradicting minimality. The same argument is repeated for the free-initial and fixed-final problem in Theorem 4.4.

Load-bearing premise

The proof of the Legendre condition rests on the unstated assumption that the needle test function $f$ deviates from its midpoint value by at least a fixed fraction of its maximum amplitude over a non-negligible portion of the tiny interval, whereas the paper only assumes $f$ is continuous and non-constant.

Editorial extensions

If this is right

Fixed-endpoint fractional variational problems now have a second-order necessary condition of the same form as in the classical calculus of variations.
The Euler-Lagrange equations come in two regimes depending on the ordering of the two fractional orders, and ignoring this relation can produce systems with no admissible solution, as Example 4.1 shows.
For problems with a free right endpoint in the regime where the integral order exceeds the derivative order, a necessary condition for existence is that the partial derivative of the terminal cost with respect to the final value vanishes along the extremal.
The fractional Du Bois-Reymond lemma reduces to the classical lemma when both orders equal one and to an earlier fractional lemma when the two orders are equal, so the new proof contains those earlier results as special cases.
The paper's variation method is presented as a template that can be reused for other fractional variational problems under constraints.

Reading between the lines

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

If the Legendre condition does hold for all problems of this type, it yields a quick nonexistence check: any candidate extremal along which the second derivative of the Lagrangian in the derivative variable has a negative eigenvalue at a continuity point cannot be a weak local minimum, just as in the classical theory.
The power-law family appearing in the fractional Du Bois-Reymond lemma suggests that fractional first integrals may take a similar power-law form, which could be used to construct conserved quantities for fractional Euler-Lagrange equations.
Because the paper's needle variation has nonzero Caputo derivative only on a small interval, similar variations may be used to prove higher-order necessary conditions or to treat inequality constraints in fractional variational problems.

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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

Two things to know. First, the generalized Du Bois-Reymond lemma (Lemma 3.1) with the beta>alpha vs beta<=alpha split is genuinely new, and the integral-form Euler-Lagrange equations that display the alpha-beta coupling are not in [9] or [13]. Those parts are largely coherent, direct, and not circular. Second, the paper's headline claim — that the Legendre condition for fixed-final-constraint problems can be proved by the standard classical method — fails at the key estimate in Theorem 4.2. The stress-test note is right: the proof replaces (f(t)-f(c))^2 by its upper bound 4M^2 inside an integral with a negative coefficient, which gives a lower bound, not the claimed negative O(epsilon) upper bound. To make the argument work you need a positive lower bound on the integral of (f-f(c))^2 over the needle interval, and the paper neither states nor constructs an f with that property. For f(t)=t, the integral is O(epsilon^3), while the Q-term is O(epsilon^{1+alpha}); for 0

Credit where it is due. The proof of Lemma 3.1 is direct and the special variation with parameter k is a real idea. Theorem 4.4, for the free-initial/fixed-final problem, avoids the problematic f because the variation's fractional derivative is constant on the needle, so its first-term estimate has the correct sign and order. The examples illustrating the alpha-beta distinction are useful. The k!=0 restriction in Lemmas 3.1/3.3 and Theorem 4.1 is a smaller technical issue: for f=0 the equality holds but the stated condition requires k!=0, making the "if and only if" literally false. That is fixable by allowing k=0 or adding a separate zero case.

The reader's skeptic is mostly on target. I would add that Theorem 4.4's proof appears sound in the part that matters, which the reader's report does not emphasize. But the central assertion of the paper, Theorem 4.2, is not supported as written.

Who is this for? People working on fractional variational necessary conditions. The lemma and EL equations deserve a serious referee, and the Legendre question is important enough that a referee could push the authors to either fix the proof or substantially qualify the claim. I would not desk-reject; I would send it out with a clear request to repair the k issue and rewrite the Legendre proof or remove the unsupported claim.

Referee Report

3 major / 3 minor

Summary. The paper studies a fractional variational problem whose functional is a Riemann-Liouville fractional integral of order β of a Lagrangian depending on the Caputo fractional derivative of order α. The main results are a fractional Du Bois-Reymond lemma, Euler-Lagrange equations in integral form for four endpoint regimes, and — as the advertised novelty — a Legendre necessary condition for the fixed-endpoint and free-initial/fixed-final cases. The authors claim this Legendre proof adapts the classical needle variation and thereby overcomes the obstruction described in [9]. The Euler-Lagrange part is developed through a direct, non-circular proof of the Du Bois-Reymond lemma, and several illustrative examples are included.

Significance. If correct, the explicit dependence of the necessary conditions on the relation between α and β would be a useful contribution, and the Legendre condition for fixed final constraints would settle a question raised in [9]. The Du Bois-Reymond lemma and the Euler-Lagrange derivations are largely coherent and are supported by explicit constructions. However, the central Legendre proof contains a load-bearing estimate with the wrong direction, and the theorem is not established as written. The examples and the comparison with [9] are helpful, but they do not compensate for the gap in the main proof.

major comments (3)

[Section 4.1, proof of Theorem 4.2, inequality (19)] The estimate of the first term in the second variation is not an upper bound. From (15), that term equals ∫(t1−t)^{β−1}(f(t)−f(c))²⟨P(t)r,r⟩dt ≤ −γ∫(t1−t)^{β−1}(f(t)−f(c))²dt. Replacing (f(t)−f(c))² by its maximum 4M² gives a lower bound, not an upper bound: the integral is ≥ −4M²γ∫(t1−t)^{β−1}dt. To conclude that the first term is at most −M₁γε, the proof would need a lower bound of the form ∫_{σ−ε}^{σ+ε}(t1−t)^{β−1}(f(t)−f(c))²dt ≥ cε for some c>0. No such bound is stated or proved for the arbitrary continuous nonconstant f. For instance, f(t)=t gives an integral of order ε³, while the Q-term in (12) is O(ε^{1+α}); for 0<α<1 the latter dominates, so the claimed contradiction δ²J<0 does not follow. Thus Theorem 4.2 is unsupported as written.
[Lemmas 3.1 and 3.3; Theorems 4.1, 4.3, 4.5] In the case 0<β≤α≤1, the statements require a constant k≠0 (or k∈R^n\{0}) in conditions 2). This makes the stated 'necessary and sufficient' claims false: f≡0 satisfies identity (1), and the corresponding Euler-Lagrange condition (10) may hold with k=0, but these are excluded by the nonzero-constant requirement. The proof itself determines k from a boundary integral and never uses k≠0. The conditions should allow arbitrary k∈R (or k∈R^n), including zero.
[Section 4.1, setup before Theorem 4.2] The weak local minimum in problem (P) is defined in the space C^α([t0,t1],R^n), whose Caputo derivative is continuous. The Legendre proof, however, works with x₀∈PC^α and variations h∈PC^α₀, whose Caputo derivatives may be piecewise continuous with jump discontinuities. Then x₀+λh need not belong to the admissible class C^α, so the second-variation inequality (12) is not justified for the variations used in the proof. The admissible class or the notion of weak local minimum must be extended to the piecewise-smooth setting before Theorem 4.2 can be applied.

minor comments (3)

[Section 2] In the definition of PC^α₀([t0,t1],R^n), the right-hand side should refer to the set PC^α([t0,t1],R^n), not PC^α₀, otherwise the definition is circular.
[Remark 4.1] The notation cAC^{α,∞}_{0+} is invoked without definition or reference; please state or cite the space explicitly.
[Throughout] The manuscript contains several LaTeX artifacts and inconsistent notations, such as a manuscript header 'JOT A manuscript No.', unclear spacing in author names, and the repeated use of '0 ⁄= k' where the intended meaning is 'k ≠ 0'. A careful editorial pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Euler-Lagrange and Legendre derivations are self-contained, and cited prior work is not used as a load-bearing premise.

full rationale

The paper's central derivation chain is self-contained. Lemma 3.1, the fractional Du Bois-Reymond lemma, is proved directly by constructing a test function h through a fractional integral and choosing the constant k from the endpoint condition h(t1)=0; this is a standard proof technique, not a parameter fitted to the target conclusion. Lemma 3.3 and Theorem 4.1 then derive the Euler-Lagrange equations from first variations using that lemma, with no reliance on circular definitions. Theorem 4.2 derives the Legendre condition from the second variation inequality: the test function h is built from a prescribed g with compact support inside a needle interval, and the constant k is determined by h(t1)=0, not chosen to force the Legendre inequality. The paper's self-citations [32-35] are used only to reference known Legendre conditions in endpoint cases already treated in those works, not to justify the new cases proved here. The citation to [9] is explicitly a foil whose claimed obstruction is being overcome; it is not used as evidence for the new result. The proof may contain a nontrivial mathematical error — the estimate (19) appears to turn an upper bound on (f(t)-f(c))^2 into a negative O(epsilon) upper bound without a lower-bound argument, which is a correctness concern rather than circularity. No step reduces by construction to its own input, and no fitted quantity is renamed as a prediction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard fractional calculus background, the mean value theorem, Fermat's theorem, and one inequality taken from [9]. The only unstated load-bearing assumption is the shape condition on the test function f in Theorem 4.2, which the proof needs but does not supply. No new physical or mathematical entities are introduced, and no parameters are fitted to data.

assumptions (4)

domain assumption Fractional integrals and derivatives as defined in Definitions 2.1-2.3, and the composition identities in Propositions 2.1-2.2.
The paper uses the standard Riemann-Liouville and Caputo fractional calculus without proving these background facts, citing monographs [19,29].
standard math Fermat's theorem and the Taylor expansion of the functional at a local minimum.
Used to derive the first and second variation conditions in Section 4.
standard math Lemma 2.1 (from [9]) bounding (sigma2^alpha - sigma1^alpha)^2.
Used in the proof of Theorem 4.2 to estimate the R-term outside the support of the Caputo derivative of the variation; the lemma is quoted from existing literature with an independent citation.
ad hoc to paper Existence of a continuous non-constant f with integral of (f-f(c))^2 of order epsilon on the needle interval.
Assumed implicitly in the proof of Theorem 4.2 to make the negative P-term dominate; the paper neither states this condition nor provides a construction, and the printed estimate does not follow from the stated assumptions on f.

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Pith. "Pith review of The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations." pith.science (2026). https://pith.science/paper/VZFJCNTJ

@misc{pith2026250606736,
  author       = {Pith},
  title        = {Pith review of: The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VZFJCNTJ}},
  note         = {Machine review of arXiv:2506.06736}
}

abstract

In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< \alpha \leq 1$ and the Riemann- Liouville fractional integral of order $\beta >0$ under certain constraints. A fractional analogue of the Du Bois-Reymond lemma is proved. Using this lemma for various weak local minimum problems, the Euler-Lagrange equation is derived in integral form. Some serious works in the literature claim that the standard proof of the Legendre condition in the classical case $\alpha=1$ cannot be adapted to the fractional case $0<\alpha <1$ with final constraints. In spite of this, we prove the Legendre conditions using the standard classical method. The obtained necessary conditions are illustrated by appropriate examples.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [9]

Bourdin, L., Ferreira, R.A.C.: Legendre’s necessary con dition for fractional Bolza func- tionals with mixed initial/ﬁnal constraints. J.Opim. Theo ry Appl. 190, 672-708 (2021)

work page 2021

[2] [13]

Ferreira, R.A.C.: Fractional calculus of variations: a novel way to look at it. Fract. Calc. Appl. Anal. 22, 1133-1144 (2019)

work page 2019

[3] [1]

Acta Appl.Math.109, 973-1033 (2010)

Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on exis tence results for bound- ary value problems of nonlinear fractional diﬀerential equ ations and inclusions. Acta Appl.Math.109, 973-1033 (2010)

work page 2010

[4] [2]

Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational prob- lems. J. Math. Anal. Appl. 272, 368-379 (2002)

work page 2002

[5] [3]

Fract.Dynam.Contr

Almeida, R., Malinovska, A., Torres, D.: Fractional Eule r-Lagrange diﬀerential equation via Caputo derivatives. Fract.Dynam.Contr. II, 109-118 (2012)

work page 2012

[6] [4]

Atanackovic, T.M., Konjik, S., Pilipovic, S.: Variation al problems with fractional deriva- tives: Euler-Lagrange equations. J. Phys. A. 41, 095201 (2008)

work page 2008

[7] [5]

Nonlin.Dynam

Baleanu, D., Trujillo, J.: On exact solutions of a class of fractional Euler-Lagrange equations. Nonlin.Dynam. 52, 331-335 (2008)

work page 2008

[8] [6]

Discrete Contin

Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Neces sary optimality conditions for fractional diﬀerence problems of the calculus of variation s. Discrete Contin. Dynam. Syst. 29, 417-437 (2011)

work page 2011