Second Order Conditions and Sufficient Conditions for Extremum

The Euler-Lagrange equation is only a necessary condition, analogous to the equality $f'(x) = 0$ in regular analysis. But $f'(x) = 0$ vanishes the same way at a minimum, at a maximum, and at a saddle point. To distinguish a true minimum, second order conditions are needed. In the calculus of variations, the situation is even subtler: not only can the second derivative change sign, but the very "directions" $\delta y$ are infinite-dimensional. Therefore, the theory has developed several levels of verification: from the simple Legendre condition to the deep Weierstrass theorem.

Second Variation

Analogous to the Taylor expansion $f(x + \varepsilon) \approx f(x) + \varepsilon f'(x) + \frac{\varepsilon^2}{2} f''(x)$, for a functional $J[y + \varepsilon\eta]$ one has the expansion:

$ J[y + \varepsilon\eta] = J[y] + \varepsilon \cdot \delta J[y, \eta] + \frac{\varepsilon^2}{2} \cdot \delta^2 J[y, \eta] + O(\varepsilon^3). $

When $\delta J = 0$ (EL is satisfied), the sign of $\delta^2 J$ determines whether it is a minimum.

Formula for the second variation: $\delta^2 J[y, \eta] = \int_a^b [P(x)\cdot {\eta'}^2 + Q(x)\cdot \eta^2],dx$, where $P = \frac{\partial^2 F}{\partial {y'}^2}$, $Q = \frac{\partial^2 F}{\partial y^2} - \frac{d}{dx}\left(\frac{\partial^2 F}{\partial y \partial y'}\right)$.

Decoding: $P$ is the "coefficient at the square of the increment's derivative," a kind of "effective mass." $Q$ is the "coefficient at the square of the increment itself," effective stiffness.

Legendre Condition (necessary for a minimum): $P(x) = \frac{\partial^2 F}{\partial {y'}^2} \geq 0$ on $[a, b]$. If at any point $P < 0$, then $\delta^2 J$ can be made arbitrarily negative using "sawtooth" perturbations $\eta$—the extremal does not yield a minimum.

Jacobi Condition. Positivity of $P$ is not enough: there must also be no conjugate points. A conjugate point $\bar{x} \in (a, b]$ is a point where a nontrivial solution of the Jacobi equation $(P u')' - Q u = 0$ with the initial condition $u(a) = 0$ vanishes again. If there is no conjugate point in $(a, b)$, the extremal yields a weak minimum.

Field of Extremals and the Weierstrass Theorem

Field of extremals. This is a one-parameter family of extremals covering some region without intersections (as streamlines in hydrodynamics). At each point of the field, the slope $p(x, y)$ is specified—the derivative of the extremal passing through it.

Weierstrass E-function: $E(x, y, p, y') = F(x, y, y') - F(x, y, p) - (y' - p)\cdot \frac{\partial F}{\partial y'}(x, y, p)$.

Geometrically, $E$ is the deviation of $F$ from its tangent plane at $y' = p$. If $F$ is convex in $y'$, then $E \geq 0$.

Theorem (Weierstrass): If an extremal $y_0$ is immersed in the field and $E(x, y, p, y') \geq 0$ for all admissible $y'$, then $y_0$ yields a strong minimum (that is, a minimum among all admissible functions close to $y_0$ in value, even if their derivatives are far apart).

The difference between weak and strong minima is important: a weak minimum takes into account only small perturbations with small $y'$-difference, a strong one—all nearby $y$, including those with rapidly oscillating derivatives.

Connection with Hamiltonian Mechanics

The EL equation is a second-order ODE. It can be rewritten as a first-order system via canonical variables:

$p = \frac{\partial F}{\partial y'}$ (generalized momentum — "how the Lagrangian responds to velocity"), $H(x, y, p) = p y' - F(x, y, y')$ (Hamiltonian — Legendre transform of $F$).

Then EL is equivalent to the Hamiltonian system: $\dot{y} = \frac{\partial H}{\partial p},\ \dot{p} = -\frac{\partial H}{\partial y}$. This reformulation is the precursor of Pontryagin's maximum principle for optimal control problems.

Numerical Example: $\min \int_0^1 (y^2 + {y'}^2),dx,\ y(0) = 1,\ y(1) = 2$

$F = y^2 + {y'}^2$, $\frac{\partial^2 F}{\partial {y'}^2} = 2 > 0$ — Legendre condition is satisfied. Jacobi equation: $2 u'' - 2u = 0 \rightarrow u(x) = \sinh(x)$. $u(0) = 0$, and on $(0, 1]$ $\sinh$ does not vanish — there are no conjugate points. Conclusion: the extremal $y(x) = (\sinh(x)\cdot[2 - \cosh(1)/\sinh(1)] + \cosh(x))/...$ (full form omitted) yields the minimum.

Real Applications

• Mechanical Engineering. The shape of gear teeth is chosen to ensure smooth transfer of forces—this is a variational problem with second order conditions for smoothness of the envelope.
• Financial Mathematics. The optimal Merton strategy (consumption + investment) is verified for sufficiency via analysis of HJB and second order conditions for the value function.
• Production Management. Holt-Winters models for optimal production scheduling use quadratic functionals—the Legendre condition ensures convexity in control, and absence of conjugate points guarantees global optimality of the plan.
• Numerical Optimization. Newton-type algorithms for functionals use the Hessian $\delta^2 J$—if it is positive definite on admissible directions, the Newton step converges correctly to a minimum.

Assignment: For the problem $\min \int_0^1 (y^2 + {y'}^2),dx$, $y(0) = 1$, $y(1) = 2$: (a) write the EL and find the general solution; (b) check the Legendre condition; (c) find conjugate points using the Jacobi equation; (d) confirm that the solution yields a minimum; (e) numerically compare $J$ on the found extremal and on the linear interpolation $y = 1 + x$.