Second Variation and Stability

⚡ Key Takeaways

Second Variation and Stability studies whether a critical point of a functional is a genuine minimum by examining the quadratic (second-order) term in its Taylor expansion. In one line: a critical extremal is stable (a local minimizer) only if the second variation is nonnegative, and strictly stable if it is positive definite — conditions encoded by the Jacobi equation, conjugate points, and the Legendre and Jacobi tests. It applies across the calculus of variations, geometry (geodesics, minimal surfaces), optimal control, and physics (action principles). Developed by Legendre, Jacobi, Weierstrass, and Morse from 1786 onward.

💡 The Core Insight

At a critical point the first variation is zero, so optimality is governed by the second-order term. Writing the perturbation as $y + \epsilon\eta$ , the functional expands as

J[y+\epsilon\eta] = J[y] + \epsilon\,\delta J[\eta] + \tfrac{1}{2}\epsilon^2\,\delta^2 J[\eta] + o(\epsilon^2),

and since $\delta J = 0$ , the quadratic functional $\delta^2 J[\eta]$ decides whether $y$ is a local minimizer. The deep point is that $\delta^2 J$ is itself a variational problem — minimizing it leads to an eigenvalue/ODE problem (the Jacobi equation), so stability of one extremal reduces to the existence of conjugate points along it.

DefinitionSecond Variation

Let $J[y] = \int_a^b L(x,y,y')\,dx$ with $y$ admissible and fixed endpoints, so admissible variations $\eta \in C^2[a,b]$ satisfy $\eta(a)=\eta(b)=0$ . The first variation is

1

\quad \delta J[\eta] = \int_a^b \left( L_y\,\eta + L_{y'}\,\eta' \right) dx = \int_a^b \left( L_y - \frac{d}{dx} L_{y'} \right)\eta\, dx,

Derivation hint: differentiate $J[y+\epsilon\eta]$ in $\epsilon$ at $\epsilon=0$ ; integrate the $\eta'$ term by parts using $\eta(a)=\eta(b)=0$ . A critical point satisfies the Euler–Lagrange equation $L_y - \frac{d}{dx}L_{y'} = 0$ . The second variation is the quadratic form

2

\quad \delta^2 J[\eta] = \int_a^b \left( L_{yy}\,\eta^2 + 2L_{yy'}\,\eta\,\eta' + L_{y'y'}\,(\eta')^2 \right) dx,

Derivation hint: take $\frac{d^2}{d\epsilon^2}J[y+\epsilon\eta]\big|_{\epsilon=0}$ , where all second partials of $L$ are evaluated along the extremal $y(x)$ . The symbols: $L_{yy}, L_{yy'}, L_{y'y'}$ are second partial derivatives of $L$ evaluated on the extremal; $\eta$ is the variation field with vanishing endpoints; $J$ has units of [action] depending on context.

DefinitionStability and Strict Stability

An extremal $y$ is stable (a weak local minimizer candidate) if $\delta^2 J[\eta] \ge 0$ for all admissible $\eta$ . It is strictly stable if there exists $c>0$ with $\delta^2 J[\eta] \ge c\|\eta\|^2$ for a suitable norm (e.g. $\|\eta\|_{H^1_0}$ ). If $\delta^2 J[\eta] < 0$ for some $\eta$ , the extremal is unstable and cannot be a minimizer.

TheoremLegendre Necessary Condition

If $y$ is a weak local minimizer of $J$ , then

4

\quad P(x) = L_{y'y'}(x,y(x),y'(x)) \ge 0 \quad \text{for all } x \in [a,b].

Derivation hint: localize $\eta$ near a point $x_0$ via a steep "sawtooth"; the $(\eta')^2$ term dominates and forces $P(x_0)\ge 0$ , else $\delta^2 J$ can be made negative. The strengthened Legendre condition is $P(x) > 0$ on $[a,b]$ .

TheoremJacobi Equation and Conjugate Points

Assume the strengthened Legendre condition $P>0$ . The Euler–Lagrange equation for the quadratic functional $\delta^2 J$ in [3] (treating $\eta$ as the unknown) is the Jacobi equation:

5

\quad -\frac{d}{dx}\!\left( P(x)\,\eta' \right) + Q(x)\,\eta = 0.

Derivation hint: apply the Euler–Lagrange operator to the integrand $\mathcal{L} = P(\eta')^2 + Q\eta^2$ , i.e. $\mathcal{L}_\eta - \frac{d}{dx}\mathcal{L}_{\eta'} = 2Q\eta - 2(P\eta')' = 0$ . A point $c \in (a,b]$ is conjugate to $a$ if there is a nontrivial solution $u$ of [5] with $u(a)=0$ and $u(c)=0$ . The Jacobi necessary condition states: if $y$ is a minimizer, then no point of $(a,b)$ is conjugate to $a$ . The Jacobi sufficient condition (with strengthened Legendre) states: if there is no conjugate point in $(a,b]$ , then $\delta^2 J$ is positive definite and $y$ is a strict weak local minimizer.

Figure

Conjugate point: the Jacobi field vanishes again, signalling loss of stability

ProofSufficiency via Conjugate-Point-Free Factorization

We show: strengthened Legendre ( $P>0$ ) plus no conjugate point in $(a,b]$ implies $\delta^2 J[\eta] > 0$ for all $\eta \not\equiv 0$ with $\eta(a)=\eta(b)=0$ .

Step 1 — The completion-of-squares idea. For any $C^1$ function $w(x)$ and any $\eta$ with vanishing endpoints,

6

\quad \int_a^b \frac{d}{dx}\!\left( w\,\eta^2 \right) dx = \left[ w\,\eta^2 \right]_a^b = 0.

Derivation hint: fundamental theorem of calculus combined with $\eta(a)=\eta(b)=0$ .

Step 2 — Add zero strategically. Expanding the integrand of [6], we have $\int_a^b (w'\eta^2 + 2w\eta\eta')\,dx = 0$ . Add this to $\delta^2 J$ in [3]:

7

\quad \delta^2 J[\eta] = \int_a^b \left( P(\eta')^2 + 2w\,\eta\eta' + (Q + w')\,\eta^2 \right) dx.

Derivation hint: superimpose the identically-zero integral from Step 1 onto [3]; the value is unchanged.

Step 3 — Perfect square. The integrand is a perfect square $P\left(\eta' + \frac{w}{P}\eta\right)^2$ provided the cross and quadratic coefficients match:

8

\quad Q + w' = \frac{w^2}{P} \quad\Longleftrightarrow\quad w' = \frac{w^2}{P} - Q,

Derivation hint: expand $P(\eta' + \frac{w}{P}\eta)^2 = P(\eta')^2 + 2w\eta\eta' + \frac{w^2}{P}\eta^2$ and equate the $\eta^2$ coefficients. This is a Riccati equation for $w$ .

Step 4 — Riccati solvable iff no conjugate point. Substituting $w = -P\,u'/u$ for a solution $u$ of the Jacobi equation [5] linearizes [8] into [5]. A solution $w$ exists on all of $[a,b]$ (no blow-up) precisely when the Jacobi solution $u$ does not vanish on $[a,b]$ — i.e. no conjugate point.

Step 5 — Conclusion. With such $w$ , equation [7] becomes

9

\quad \delta^2 J[\eta] = \int_a^b P\left( \eta' + \frac{w}{P}\eta \right)^2 dx \ge 0,

Derivation hint: substitute the perfect-square form from Step 3 into [7]. and since $P>0$ , this is zero only if $\eta' + \frac{w}{P}\eta \equiv 0$ , a first-order linear ODE whose only solution with $\eta(a)=0$ is $\eta\equiv 0$ . Hence $\delta^2 J > 0$ for nontrivial $\eta$ . $\blacksquare$

∎

mathematician · Carl Gustav Jacob Jacobi | 1804–1851

Jacobi, the prodigious German analyst best known for elliptic functions and the Hamilton–Jacobi equation, supplied in his 1837 lectures the missing piece of the variational puzzle. Legendre (1786) had given a necessary condition but mistakenly thought it sufficient; Lagrange found the gap. Jacobi recognized that the global behavior of solutions to the variation equation — the vanishing of what we now call Jacobi fields — controls whether an extremal genuinely minimizes. His insight that conjugate points mark the breakdown of minimality unified the analytic, geometric, and dynamical strands of the calculus of variations, and seeded Morse's later index theory.

🧠 Intuition

Think of the functional landscape as a surface over the infinite-dimensional space of curves. The Euler–Lagrange equation says you're at a flat spot (critical point). The second variation is the Hessian of that landscape — a quadratic form telling you the local curvature in every direction $\eta$ . If it curves up in all directions (positive definite), you're in a bowl: a minimum. If some direction curves down, you're on a saddle: not a minimum. The conjugate point is the dramatic twist. Imagine geodesics on a sphere leaving the north pole: they all reconverge at the south pole. Up to the south pole, the great-circle arc is the shortest path. Past it, a slightly different route is shorter — the geodesic has stopped minimizing. The south pole is the conjugate point. Jacobi fields are the "spreading" of a family of neighboring extremals; when they refocus (vanish again), the second variation develops a zero eigenvalue, and beyond, a negative one. Stability is thus the statement: neighboring extremals haven't refocused yet.

TA

The Action Principle Lens

In physics the functional is the action $S = \int L\,dt$ and extremals are classical trajectories. The second variation governs the stability of orbits and the semiclassical phase. The Morse index — the number of conjugate points along a path — enters the Van Vleck–Gutzwiller propagator as the Maslov index, contributing a phase $e^{-i\pi\,\text{index}/2}$ to quantum amplitudes. Conjugate points in optics are literal foci where ray families converge; in gravitational lensing they are caustics where image brightness diverges. So "stability" in the variational sense is physically the difference between a focusing and a defocusing trajectory.

ExampleGeodesic on a Sphere — Locating the Conjugate Point

Given: Great-circle arcs on a unit sphere minimize length $J = \int \sqrt{(\theta')^2 + \sin^2\theta\,(\phi')^2}\,d\phi$ . Parametrize an equatorial-type geodesic; the Jacobi equation for normal perturbations $u$ along a unit-speed geodesic on a surface of Gaussian curvature $K=1$ is

10

\quad u'' + K\,u = u'' + u = 0.

Derivation hint: the Jacobi equation on a surface reads $u'' + K u = 0$ with $s$ = arc length; for the unit sphere $K=1$ .

Steps: 1. General solution: $u(s) = A\sin s + B\cos s$ . 2. Impose $u(0)=0$ at the starting point $\Rightarrow B=0$ , so $u(s)=A\sin s$ . 3. Next zero: $\sin s = 0 \Rightarrow s = \pi$ .

Answer: The conjugate point occurs at arc length $\boxed{s = \pi \text{ (antipode)}}$ . A great-circle arc shorter than $\pi$ (i.e. not reaching the antipode) is a strict minimizer; an arc longer than $\pi$ is not length-minimizing — consistent with everyday intuition that you'd go the "short way" around.

ExampleStability of a Harmonic-Oscillator Action Segment

Given: The action for a unit-mass particle, $S = \int_0^T \left( \tfrac12 \dot x^2 - \tfrac12 \omega^2 x^2 \right) dt$ . Here $L = \tfrac12\dot x^2 - \tfrac12\omega^2 x^2$ , so $P = L_{\dot x\dot x} = 1 > 0$ and $Q = L_{xx} = -\omega^2$ .

Steps: 1. Jacobi equation [5]: $-\frac{d}{dt}(1\cdot \eta') + (-\omega^2)\eta = 0 \Rightarrow \eta'' + \omega^2\eta = 0$ . 2. Solution with $\eta(0)=0$ : $\eta(t) = A\sin(\omega t)$ . 3. Conjugate point: first $t>0$ with $\eta=0$ is $\omega t = \pi \Rightarrow t = \pi/\omega$ .

Answer: The classical path is a strict local minimizer of the action iff $\boxed{T < \pi/\omega}$ . For $T \ge \pi/\omega$ a conjugate point appears, $\delta^2 S$ acquires a negative eigenvalue, and the action becomes a saddle — exactly the regime where the WKB propagator picks up a Maslov phase shift. (For half a period the harmonic action ceases to be minimal, though it remains a critical point.)

⚠️ Common mistake

The Legendre condition is not sufficient. A common error is to verify $L_{y'y'} > 0$ everywhere and conclude a minimum. This pointwise condition guards only against high-frequency, locally concentrated perturbations. A global perturbation can still lower $J$ if a conjugate point lies inside $(a,b)$ . Both the strengthened Legendre condition and absence of conjugate points are needed for the sufficiency theorem.

Weak vs. strong minima. The Jacobi/Legendre theory addresses weak local minima (perturbations small in $C^1$ , i.e. both $\eta$ and $\eta'$ small). For strong minima (only $\eta$ small, $\eta'$ unrestricted) one additionally needs the Weierstrass condition $E \ge 0$ on the Weierstrass excess function. A weak minimizer need not be a strong minimizer.

CounterexampleConjugate Point Defeats Legendre

Consider $J[y] = \int_0^L \left( (y')^2 - y^2 \right) dx$ with $y(0)=y(L)=0$ . Here $P = 2 > 0$ (strengthened Legendre holds!) and $Q = -2$ . The Jacobi equation $\eta'' + \eta = 0$ gives $\eta = \sin x$ , with conjugate point at $x=\pi$ . For $L > \pi$ , take $\eta(x) = \sin(\pi x / L)$ :

\delta^2 J = 2\int_0^L \left( \frac{\pi^2}{L^2}\cos^2\frac{\pi x}{L} - \sin^2\frac{\pi x}{L} \right) dx = L\left( \frac{\pi^2}{L^2} - 1 \right) < 0.

So $y\equiv 0$ is not a minimizer despite Legendre being satisfied everywhere — the conjugate point at $\pi < L$ is the culprit. This is the canonical demonstration that pointwise convexity is insufficient.

connection

The second variation is the variational incarnation of the Hessian, linking the Calculus of Variations to finite-dimensional optimization where stability means a positive-definite Hessian. The Jacobi equation is the linearization of the Euler–Lagrange Equation and, in geometry, defines Jacobi Fields along geodesics — connecting to the Index Form of Riemannian Geometry. The counting of negative eigenvalues of $\delta^2 J$ is the Morse index, central to Morse Theory and the Morse Index Theorem, which equates the index to the number of conjugate points. In dynamics, the Riccati substitution in the proof links to Sturm–Liouville Theory (the Jacobi equation is a Sturm–Liouville operator) and to Hamilton–Jacobi Theory via the field-of-extremals construction. In physics the index reappears as the Maslov Index in semiclassical quantization.

🔬 Where this appears

⏳ History

- 1744 — Euler. Establishes the calculus of variations and the first-order (Euler) equation in Methodus inveniendi.

- 1786 — Legendre. Introduces the second variation and the condition

L_{y'y'}\ge 0

, but erroneously claims it sufficient.

- ~1797 — Lagrange. Points out the flaw in Legendre's sufficiency argument; the gap remains open for decades.

- 1837 — Jacobi. In his Königsberg lectures, introduces the Jacobi equation and conjugate points, identifying the true obstruction to minimality.

- 1868 — Weierstrass. Distinguishes weak from strong extrema; introduces the excess function

E

and the field theory of extremals, completing the sufficiency theory for strong minima.

- 1879 — Mayer & Clebsch. Extend conjugate-point theory to multiple integrals and constrained (Lagrange) problems; the Legendre–Clebsch condition emerges.

- 1900s — Hilbert. The invariant integral and field theory give a clean modern proof of sufficiency.

- 1925–1934 — Morse. Creates the calculus of variations in the large (Morse theory); the Morse Index Theorem equates the index of the second variation to the number of conjugate points counted with multiplicity.

- 1965–1970 — Penrose, Hawking. Recast conjugate-point focusing via the Raychaudhuri equation into the singularity theorems of general relativity.

open-problem

Several frontiers complicate the clean classical picture:

Free and broken boundary conditions. With non-fixed endpoints the conjugate point is replaced by focal points, and transversality conditions modify the Jacobi boundary data; the index theory is more delicate.
Multiple integrals. For functionals $\int_\Omega L(x,u,\nabla u)\,dx$ over domains $\Omega\subset\mathbb R^n$ , the Legendre–Hadamard condition (rank-one convexity) replaces $L_{y'y'}>0$ , and strong/quasiconvexity notions (Morrey) determine lower semicontinuity. The relationship between pointwise convexity conditions and existence/regularity remains subtle (cf. Morrey's quasiconvexity, still not fully characterized).
Degenerate second variation (zero eigenvalues). When $\delta^2 J$ is merely semidefinite (a conjugate point exactly at $b$ ), higher variations or center-manifold-type analysis decide stability — a genuinely open analytical regime.
Infinite-dimensional Morse theory & Floer homology. For action functionals that are unbounded above and below (e.g. on the loop space, in symplectic geometry), the classical Morse index is infinite; relative indices and Floer's homological approach are required, an active research area.
Non-smooth and constrained settings. For optimal control with state constraints and bang-bang solutions, second-order conditions involve generalized Jacobi equations and remain technically intricate.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also