Second-Order Conditions and the Jacobi Condition

Motivation: When Is an Extremal a Minimum?

The Euler-Lagrange equation is a first-order condition, the analogue of “derivative equals zero.” Just like in ordinary calculus, a critical point can be a minimum, maximum, or a saddle point. Second-order conditions are needed to determine the type of extremal. This is especially important in optimal control problems: one needs not just to find an extremal, but to make sure it actually minimizes the functional and is not, for example, a saddle.

The Second Variation

Consider the functional $J[y] = \int F(x, y, y'), dx$ and its second derivative with respect to the parameter $\varepsilon$ under the perturbation $y^* + \varepsilon\eta$:

Second variation: $\delta^2 J[y^; \eta] = \frac{d^2}{d\varepsilon^2}\Big|_{\varepsilon=0} J[y^ + \varepsilon\eta]$

Calculation (Taylor expansion in $\varepsilon$ up to $\varepsilon^2$ inclusive):

$ \delta^2 J = \int \left[ F_{yy}, \eta^2 + 2 F_{yy'}, \eta, \eta' + F_{y'y'}, (\eta')^2 \right], dx $

Using integration by parts: $\delta^2 J = \int (P, (\eta')^2 + Q, \eta^2), dx$, where:

• $P = F_{y'y'}$ (second derivative of $F$ with respect to $y'$)
• $Q = F_{yy} - \frac{d}{dx} F_{yy'}$ (combined term)

Necessary condition for a minimum: $\delta^2 J[y^*; \eta] \geq 0$ for all admissible $\eta$.

The Legendre Condition

Legendre condition (necessary): for a minimum, it is necessary that $P(x) = F_{y'y'}(x, y^(x), y^{\prime}(x)) \geq 0$ on the entire segment $[x_0, x_1]$.

Meaning: $F$ must be “convex in $y'$” along the extremal. If $P(x) < 0$ at even a single point, the extremal is not a minimum.

Strengthened Legendre Condition (sufficient): $P(x) > 0$ strictly on the entire $[x_0, x_1]$.

Example: $J[y] = \int \frac{1}{2}(y'^2 - y^2), dx$ (the pendulum). $F = \frac{1}{2}(p^2 - y^2)$. $F_{y'y'} = 1 > 0$—the Legendre condition is satisfied. This alone is not sufficient for a minimum; the Jacobi condition must also be checked.

The Jacobi Equation and Conjugate Points

The Legendre condition checks definiteness “at each point” separately. The Jacobi condition is a global condition.

Quadratic functional: $\delta^2 J = \int (P, (\eta')^2 + Q, \eta^2), dx$—this is a functional of $\eta$. For which $\eta$ is it minimal?

Sturm-Liouville equation for eigenvalues: $-\left( P u' \right)' + Q u = \lambda u$.

Jacobi equation (zero eigenvalue): $-\left( P u' \right)' + Q u = 0$

with initial conditions $u(x_0) = 0$, $u'(x_0) = 1$.

Conjugate point $\bar{x}$ to $x_0$: the nearest zero of the solution $u(x)$ of the Jacobi equation, i.e., the nearest $\bar{x} > x_0$ at which $u(\bar{x}) = 0$.

Jacobi condition for a minimum: the conjugate point $\bar{x}$ must not lie in the open interval $(x_0, x_1)$. If $\bar{x} \in (x_0, x_1)$, the extremal is not a minimum.

Complete Analysis: The Pendulum (Conjugate Points Problem)

Functional: $J[y] = \frac{1}{2}\int_0^T (y'^2 - y^2), dx$ (small oscillations of the pendulum, $y$—the angle).

$F = \frac{1}{2}(p^2 - y^2)$. $P = F_{y'y'} = 1$, $Q = F_{yy} - \frac{d}{dx} F_{yy'} = -1$.

Jacobi equation: $-u'' - u = 0 \Rightarrow u'' + u = 0$. This is the harmonic oscillator equation!

Solution with $u(0) = 0$, $u'(0) = 1$: $u(x) = \sin(x)$.

Conjugate points: $u(\bar{x}) = \sin(\bar{x}) = 0 \Rightarrow \bar{x} = \pi, 2\pi, 3\pi, ...$

Conclusion: If $T < \pi$, then the conjugate point $\bar{x} = \pi > T$—the Jacobi condition is satisfied, the extremal is a minimum. If $T > \pi$, then $\bar{x} = \pi \in (0, T)$—the Jacobi condition is violated, the extremal is NOT a minimum.

Physical meaning: when $T < \pi$, the pendulum has not yet completed a quarter period—the trajectory is optimal. When $T > \pi$, the pendulum has “swung over”—there is a shorter path.

Morse Theorem and Topological Consequences

Morse index of the quadratic functional $\delta^2 J$ = the number of “negative directions,” i.e., the number of conjugate points $x_0$ in $(x_0, x_1)$.

Morse theorem: connects critical points of the functional (extremals) with the topology of the path space. The number of geodesics between two points on a compact manifold is ≥ the sum of the Betti numbers of the path manifold. This gives lower bounds on the number of solutions to variational problems.

Ritz Method: Approximate Solution

Often the Euler-Lagrange equation is hard to solve analytically. Ritz Method (1908): seek a solution in the form:

$ y_n(x) = \sum_{k=1}^n a_k \varphi_k(x) + \varphi_0(x) $

where $\varphi_k$ are basis functions (polynomials, sines, finite elements), $\varphi_0$ satisfies the boundary conditions.

$J[y_n] = J(a_1, ..., a_n)$—a function of $n$ numbers $\rightarrow \frac{\partial J}{\partial a_k} = 0$ (system of $n$ equations).

Finite element method (FEM)—a variant of Ritz's method with piecewise linear basis functions. This is the foundation of ANSYS, COMSOL, and other engineering analysis systems. FEM reduces an infinite-dimensional variational problem to a finite-dimensional system of linear equations $KU = F$.

Weierstrass Conditions for Strong Minimum

The Jacobi condition guarantees a weak minimum—optimality among smooth perturbations. For a strong minimum (among all admissible functions, including those with discontinuous derivatives), stronger conditions are required. The Weierstrass condition requires that the Weierstrass $E$-function:

$ E(x, y, y', q) = F(x, y, q) - F(x, y, y') - (q - y') F_{y'}(x, y, y') $

be nonnegative for all $q$ (not just $q \approx y'$). This means that a deviation of the derivative even by a finite amount does not decrease the functional.

Singular Extremals

In some problems, the optimal trajectory passes through singular points (singular arcs), where the Legendre condition degenerates: $F_{y'y'} = 0$. On such segments, the Euler-Lagrange equation loses its status as a second-order equation and additional conditions are required. Singular extremals frequently occur in optimal control problems with control constraints: on some segments the control “hits” the boundaries (bang-bang), on others—inside the admissible set (singular regime).

Applications

• Rocket control: calculating the optimal trajectory requires checking all second-order conditions to guarantee minimality
• Aerodynamics: wing profile optimization as minimization of drag—one must check the Jacobi condition to ensure that the found profile is actually a minimum and not a saddle
• Finance: optimal Merton consumption strategy—extremal of the HJB equation; second-order conditions guarantee maximum utility
• Robotics: manipulator trajectory planning with energy minimization—checking second-order conditions at segment junction points