Fractional Variational Calculus

• Fractional variational calculus is an extension of classical variational methods that incorporates non-local operators like the Caputo derivative and Riemann–Liouville integral to capture memory and hereditary effects.
• It rigorously derives Euler–Lagrange and Legendre conditions in fractional settings, ensuring clear optimality criteria for systems with non-conservative dynamics.
• The framework is applied in physics, engineering, and applied mathematics, providing operator-theoretic insights and specialized integration-by-parts formulations.

Fractional variational calculus generalizes the classical calculus of variations by allowing the Lagrangian to depend on non-local fractional operators, most notably the Caputo fractional derivative and the Riemann–Liouville fractional integral. This extension accommodates systems with memory, hereditary behavior, and non-conservative effects, providing a rigorous framework for both theoretical developments and applications in physics, engineering, and applied mathematics. The field is characterized by its operator-theoretic richness, the necessity of non-local integration-by-parts formulations, and the subtleties of necessary (and in some cases, sufficient) optimality conditions for fractional variational problems.

1. Fractional Operators and Function Spaces

Let $[a,b]\subset\mathbb{R}$ and $X=\mathbb{R}^n$.

Riemann–Liouville Fractional Integrals:

For $\varphi\in L^1([a,b],X)$ and $\beta>0$, define

$$(I_{a+}^\beta \varphi)(t) = \frac{1}{\Gamma(\beta)} \int_a^t (t-\tau)^{\beta-1}\varphi(\tau)\,d\tau, \quad (I_{b-}^\beta \varphi)(t) = \frac{1}{\Gamma(\beta)} \int_t^b (\tau-t)^{\beta-1}\varphi(\tau)\,d\tau$$

These operators satisfy the semigroup property $$I_{a+}^\alpha \circ I_{a+}^\beta = I_{a+}^{\alpha+\beta}$$.

Caputo Fractional Derivative:

For $x \in C^1([a,b],X)$ and $0<\alpha \le 1$,

$${}^c D_{a+}^\alpha x(t) = \frac{1}{\Gamma(1-\alpha)} \int_a^t (t-\tau)^{-\alpha} x'(\tau)\, d\tau$$

$${}^c D_{b-}^\alpha x(t) = -\frac{1}{\Gamma(1-\alpha)} \int_t^b (\tau-t)^{-\alpha} x'(\tau)\, d\tau$$

For these operators, $$({}^c D_{a+}^\alpha \circ I_{a+}^\alpha)x = x(t) - x(a)$$ and $$I_{a+}^\alpha({}^c D_{a+}^\alpha x)(t) = x(t) - x(a)$$ (Yusubov et al., 7 Jun 2025).

Function spaces such as $C([a,b],X)$, $C^\alpha([a,b],X)$, and fractional Sobolev-type spaces provide the analytic setting for admissible trajectories.

2. Fractional Du Bois–Reymond Lemma

A fundamental lemma for weak variations is established for the Caputo derivative and Riemann–Liouville integral weighting: $$\int_a^b (b-t)^{\beta-1} f(t)\, {}^c D_{a+}^\alpha h(t)\, dt = 0 \quad \forall\, h\in C_0^\alpha([a,b])$$ gives:

• If $\beta > \alpha$: $(b-t)^{\beta-\alpha} f(t) \equiv 0$ $\implies$ $f(t)\equiv0$.
• If $0<\beta\leq\alpha\leq1$: $$f(t) = \frac{k}{\Gamma(\alpha)}(b-t)^{\alpha-\beta}$$ for some constant $k \in \mathbb{R}^n$.

The proof constructs a special fractional variation and reduces the vanishing of the integral to positivity of a quadratic form, enforcing strong constraints on $f$ (Yusubov et al., 7 Jun 2025).

This lemma underpins existence and uniqueness of extremals and is essential in deriving Euler–Lagrange equations in the fractional context.

3. Euler–Lagrange Equations in Fractional Variational Calculus

Given a Lagrangian $L=L(t,x,y)$ with $x \in C^\alpha([a,b], \mathbb{R}^n)$ and $y = {}^c D_{a+}^{\alpha} x$, consider

$$J[x] = \int_a^b (b-t)^{\beta-1} L(t, x(t), {}^c D_{a+}^\alpha x(t))\,dt,$$

subject to $x(a)=x_a$, $x(b)=x_b$.

The first variation yields

$$\delta J = \int_a^b (b-t)^{\beta-1} \big\langle L_x, h \rangle + \langle L_y, {}^c D_{a+}^\alpha h \rangle\, dt,$$

for all $h\in C_0^\alpha$.

The Du Bois–Reymond lemma then leads to two structurally distinct Euler–Lagrange equations:

• Case (i) $\beta > \alpha$:

$$(b-t)^{1-\alpha} I_{b-}^\alpha \big[ (b-\cdot)^{\beta-1} L_x(\cdot) \big](t) + (b-t)^{\beta-\alpha} L_y\big( t, x(t), {}^c D_{a+}^\alpha x(t) \big) = 0,$$

• Case (ii) $0<\beta\leq\alpha\leq1$:

$$(b-t)^{1-\beta} I_{b-}^\alpha \big[ (b-\cdot)^{\beta-1} L_x(\cdot) \big](t) + L_y \big( t, x(t), {}^c D_{a+}^\alpha x(t) \big) = \frac{k}{\Gamma(\alpha)} (b-t)^{\alpha-\beta}.$$

Here $I_{b-}^\alpha$ is the right Riemann–Liouville fractional integral (Yusubov et al., 7 Jun 2025). The constant $k$ is determined by boundary/transversality conditions or the problem's structure.

For the special case of functionals depending solely on the Caputo derivative (i.e., $L = L(t,y,{}^c D_{a+}^\alpha y)$), an Euler–Lagrange equation involving only Caputo derivatives can be established through a fractional generalization of the Du Bois–Reymond lemma (Lazo et al., 2012): $$L_y(x, y, {}^c D_{a+}^\alpha y) + {}_x^C D_{b}^{\alpha} L_v(x, y, {}^c D_{a+}^\alpha y) = 0,$$ with possible boundary contributions if endpoints are free.

4. Second-Order Conditions: The Legendre Criterion

Beyond first-order optimality, the Legendre necessary condition is extended to the fractional setting.

For a weak minimizer $x^0$ and $L$ twice differentiable in $(x, y)$: $$\delta^2 J[x^0; h] = \int_a^b (b-t)^{\beta-1} \big\{ \langle L_{yy} (t) {}^c D_{a+}^\alpha h, {}^c D_{a+}^\alpha h \rangle + 2 \langle L_{xy} (t) h, {}^c D_{a+}^\alpha h \rangle + \langle L_{xx} (t) h, h \rangle \} dt \ge 0$$ for all admissible $h$.

Applying a fractional “needle variation,” one deduces: $$\langle L_{yy}\big( \tau, x^0(\tau), {}^c D_{a+}^\alpha x^0(\tau) \big) r, r \rangle \ge 0, \quad \forall r\in\mathbb{R}^n,\, \forall \tau \in (a,b),$$ i.e., $L_{yy}$ is positive semidefinite along extremals (Yusubov et al., 7 Jun 2025, Lazo et al., 2013).

This criterion recovers the classical Legendre condition in the limit $\alpha\to1$ and is crucial for discarding false minimizers.

5. Illustrative Examples

**Example 1