Second Variation and Sufficiency Conditions

Setting the first variation to zero is the calculus-of-variations analogue of $f'(x)=0$ : it finds the candidates but says nothing about their character. A geodesic, a least-action trajectory, and a buckled column are all stationary — yet a great-circle arc past the antipode is the longest reasonable route, not the shortest, and a slender column that was rock-steady becomes unstable the instant a load crosses a threshold. What distinguishes a genuine minimum from an impostor is the second-order change in the functional, the second variation $\delta^2 J$ , a quadratic functional of the admissible perturbation $\eta$ . Two obstacles make this harder than the ordinary second-derivative test. First, $\delta^2 J$ lives on an infinite-dimensional function space, so "the Hessian is positive definite" becomes "a Sturm–Liouville quadratic form is positive". Second — and this is the deep surprise — that form can fail to be positive in a purely global way, with nothing wrong pointwise. The pointwise Legendre condition $F_{y'y'}\ge0$ is necessary but never sufficient, because a benign-looking negative potential $Q$ can accumulate over a long enough interval and drag the integral below zero. The decisive object is the Jacobi accessory equation, whose solutions detect conjugate points — the exact locations where neighbouring extremals refocus and a once-minimizing path can be undercut. Absence of a conjugate point inside the interval, together with the strengthened Legendre condition $F_{y'y'}>0$ , is Jacobi's sufficiency theorem: a clean, checkable certificate that a stationary function really is a weak local minimizer. The same machinery counts caustics in optics, Maslov indices in semiclassical mechanics, and the critical loads of elastic stability — one theorem, many faces.

Key takeaways

A vanishing first variation makes a functional stationary; the sign of the second variation $\delta^2 J$ decides whether that stationary function is a weak local minimum, a maximum, or a saddle.
Expanding $J[y+\epsilon\eta]$ to second order and integrating by parts gives the Sturm–Liouville form $\delta^2 J=\int_a^b(P\,\eta'^2+Q\,\eta^2)\,\mathrm{d}x$ , with $P=F_{y'y'}$ and $Q=F_{yy}-\tfrac{\mathrm{d}}{\mathrm{d}x}F_{yy'}$ .
The Legendre condition $F_{y'y'}\ge0$ is necessary; the strengthened form $F_{y'y'}>0$ plus the Jacobi condition (no conjugate point in the open interval) is sufficient for a weak local minimum.
Conjugate points are the interior zeros of the Jacobi accessory equation $-\tfrac{\mathrm{d}}{\mathrm{d}x}(P u')+Q u=0$ ; a nontrivial solution vanishing at both $a$ and some $c<b$ certifies that the extremal no longer minimizes.
Concrete numbers: the harmonic-oscillator action stops being a minimum past a kinetic focus at $t_c=\pi/\omega$ (half a period), and a great-circle arc stops minimizing length past the antipode at angular distance $\pi$ .
Key idea: Stationarity ( $\delta J=0$ ) only finds candidates; the sign of $\delta^2 J=\int(P\eta'^2+Q\eta^2)\mathrm{d}x$ classifies them — and positivity of this quadratic form needs both the pointwise Legendre condition $P>0$ and the global Jacobi condition that no conjugate point lies inside the interval.

Where this lives today

The conjugate point is not a museum piece; it is the working diagnostic of stability and of caustics across physics. In structural engineering the buckling of a loaded elastic column is a second-variation instability: the equilibrium shape minimizes potential energy only up to a critical compressive load, and the first conjugate point of the energy functional coincides exactly with the Euler buckling load $P_{\text{cr}}=\pi^2 EI/L^2$ . Push past it and the straight column is no longer a minimizer — it buckles. In gravitational lensing and geometric optics, conjugate points along light rays are the caustics where images brighten and multiply, and the count a ray has passed sets how many images a source produces. In semiclassical and path-integral quantum mechanics, every kinetic focus a classical trajectory crosses adds $\pi/2$ to the phase of the propagator — the Maslov index — so conjugate-point counting is literally part of the WKB quantization condition and of Gutzwiller's trace formula for quantum chaos. And in Riemannian geometry the same equation of geodesic deviation $\ddot u+K u=0$ links conjugate points to curvature through the Morse index theorem, the backbone of comparison geometry and of the singularity theorems of general relativity.

The physical picture

In ordinary calculus, $f'(x_0)=0$ marks a critical point and $f''(x_0)>0$ certifies a minimum. The calculus of variations swaps the single variable for a whole function and the second derivative for the quadratic functional $\delta^2 J$ . Because a function space is infinite-dimensional, the test "all Hessian eigenvalues are positive" becomes "the Sturm–Liouville form $\int(P\eta'^2+Q\eta^2)\,\mathrm{d}x$ is positive definite". That form pits a stiffness term $P\eta'^2$ , which penalizes wiggling, against a potential term $Q\eta^2$ , which may be destabilizing when $Q<0$ . On a short interval the stiffness term always wins and $\delta^2 J>0$ ; lengthen the interval and the potential term can accumulate enough negative area to flip the sign. The crossover is the first conjugate point, and the first figure shows the geometry that makes it happen — a pencil of extremals leaving one point and refocusing at another.

Figure

A pencil of extremals leaving the source point a spreads apart, then refocuses at the conjugate point c. For a great circle on the unit sphere the accessory equation is u″ + u = 0, whose solution from a next vanishes at angular distance Δφ = π — the antipode. Beyond c the once-shortest arc can be undercut by a neighbour.

The same picture drives the sign of $\delta^2 J$ directly. The Jacobi accessory equation is nothing but the equation for an infinitesimally near extremal — a Jacobi field $u(x)$ measuring the transverse gap between the reference path and its neighbour. When that gap, started at zero at $a$ , returns to zero at some interior $c$ , the neighbours have refocused and a variation built from $u$ makes $\delta^2 J$ vanish; smoothing its corner pushes it negative. So the second figure is the quantity a working analyst actually computes: the Jacobi field itself, plotted for the harmonic-oscillator action, whose first zero pins the conjugate length to a hard number.

Figure

The Jacobi field u(t) = sin ωt for the harmonic-oscillator action (P = 1, Q = −ω²). It leaves zero at the start and returns to zero at the conjugate point t_c = π/ω = T/2 — for ω = 2π s⁻¹ that is t_c = 0.5 s. Left of t_c the second variation is positive (a weak minimum); right of it a variation exists with δ²J < 0 (a saddle).

The core equations

Perturb the stationary function $y$ by $\epsilon\eta$ with $\eta$ vanishing at both endpoints, and expand the functional in powers of $\epsilon$ . The coefficient of $\epsilon$ is the first variation, which is zero by hypothesis; the coefficient of $\epsilon^2$ is the second variation, the leading term that decides everything.

J[y+\epsilon\eta]=J[y]+\epsilon\,\delta J+\tfrac12\epsilon^2\,\delta^2 J+O(\epsilon^3) \qquad (1)

where: $J$ — the functional $\int_a^b F(x,y,y')\,\mathrm{d}x$ ; $y$ — the stationary function under test; $\eta$ — admissible variation with $\eta(a)=\eta(b)=0$ ; $\epsilon$ — small real parameter; $\delta J,\ \delta^2 J$ — first and second variations. At a stationary point $\delta J=0$ , so the sign of $J[y+\epsilon\eta]-J[y]$ is fixed, for small $\epsilon$ , by the sign of $\delta^2 J$ . Differentiating twice in $\epsilon$ at $\epsilon=0$ and using the multivariable chain rule gives the second variation as a quadratic form in $\eta$ and $\eta'$ , with coefficients the second partial derivatives of $F$ evaluated along the extremal.

\delta^2 J=\int_a^b\!\left(F_{yy}\,\eta^2+2F_{yy'}\,\eta\eta'+F_{y'y'}\,\eta'^2\right)\mathrm{d}x \qquad (2)

where: $F_{yy},F_{yy'},F_{y'y'}$ — second partials of $F$ evaluated on $y(x)$ . This is a genuine quadratic functional: replacing $\eta\to\lambda\eta$ scales it by $\lambda^2$ , exactly as a Hessian does in finite dimensions. The cross term $2F_{yy'}\eta\eta'$ obscures the structure. Because $\eta\eta'=\tfrac12(\eta^2)'$ , one integration by parts absorbs it into the $\eta^2$ coefficient and the boundary term dies (since $\eta$ vanishes at the ends), leaving the canonical Sturm–Liouville form.

\delta^2 J=\int_a^b\!\left(P\,\eta'^2+Q\,\eta^2\right)\mathrm{d}x,\qquad P=F_{y'y'},\quad Q=F_{yy}-\frac{\mathrm{d}}{\mathrm{d}x}F_{yy'} \qquad (3)

where: $P$ — the stiffness coefficient multiplying the squared slope; $Q$ — the effective potential coefficient after integration by parts. The whole sufficiency theory is the study of when this one integral is positive for every nonzero admissible $\eta$ . The stiffness term dominates for rapidly varying $\eta$ , so its coefficient controls the local sign. A sharply peaked variation makes $\int P\eta'^2$ blow up like $1/\delta$ while $\int Q\eta^2$ stays finite; if $P$ were negative anywhere the form would go negative. Hence the Legendre necessary condition.

P(x)=F_{y'y'}(x,y,y')\ge 0\quad\text{on }[a,b] \qquad (4)

where: $P(x)$ — the stiffness coefficient; its nonnegativity everywhere on the closed interval is necessary for a weak minimum. The strengthened form $P>0$ (strict) is what the sufficiency theorem requires. Positivity of $P$ is not enough on its own. Treating the second variation as a functional of $\eta$ in its own right and applying Euler–Lagrange to the integrand $P\eta'^2+Q\eta^2$ produces a linear second-order equation — the Jacobi accessory equation — whose solutions are the Jacobi fields that detect refocusing.

-\frac{\mathrm{d}}{\mathrm{d}x}\!\left(P\,u'\right)+Q\,u=0,\qquad u(a)=0,\ \ u\not\equiv0 \qquad (5)

where: $u(x)$ — the accessory variable (Jacobi field); $P,Q$ — the same coefficients as in $\delta^2 J$ . A point $c\in(a,b]$ where this solution next vanishes, $u(c)=0$ , is the point conjugate to $a$ : it is where an infinitesimally near extremal refocuses onto the reference path. Assembling the pieces yields Jacobi's theorem: strict Legendre positivity plus the absence of any interior conjugate point is a sufficient certificate that the second variation is strictly positive, hence that $y$ is a weak local minimizer.

P>0\ \text{on}\ [a,b]\ \ \text{and no }c\in(a,b)\ \text{conjugate to }a\ \ \Longrightarrow\ \ \delta^2 J>0 \qquad (6)

where: $\delta^2 J>0$ — strict positivity for every admissible $\eta\neq0$ , i.e. a weak local minimum in the $C^1$ norm $\|\eta\|_1=\max|\eta|+\max|\eta'|$ . "Weak" is essential: the certificate controls variations small in slope as well as value, and does not by itself rule out strong variations or global competitors.

Derivation AFrom the Taylor expansion to the Sturm–Liouville form

The goal is to turn the abstract statement "expand the functional to second order" into the concrete, computable object $\delta^2 J=\int(P\eta'^2+Q\eta^2)\,\mathrm{d}x$ , isolating the two coefficients $P$ and $Q$ that every subsequent test depends on. We differentiate the perturbed functional twice, read off the raw quadratic form, and then perform the single integration by parts that removes the mixed $\eta\eta'$ term and reveals the Sturm–Liouville structure — the same structure that makes the accessory equation self-adjoint.

Step 1. Write $J[y+\epsilon\eta]=\int_a^b F(x,y+\epsilon\eta,\,y'+\epsilon\eta')\,\mathrm{d}x$ and differentiate the integrand twice with respect to $\epsilon$ . The first derivative is $F_y\,\eta+F_{y'}\,\eta'$ ; differentiating again by the chain rule gives $F_{yy}\eta^2+2F_{yy'}\eta\eta'+F_{y'y'}\eta'^2$ . Why: $\epsilon$ enters only through the two slots $y+\epsilon\eta$ and $y'+\epsilon\eta'$ , so the second derivative is precisely the Hessian of $F$ contracted twice with the perturbation vector $(\eta,\eta')$ .

Step 2. Evaluate at $\epsilon=0$ , where all partials sit on the reference $y(x)$ , and identify $\delta^2 J$ as the coefficient of $\tfrac12\epsilon^2$ in the expansion. Integrating over $[a,b]$ reproduces equation (2), the raw second variation. Why: the expansion $J=J_0+\epsilon\,\delta J+\tfrac12\epsilon^2\,\delta^2 J+\cdots$ defines $\delta^2 J$ as the second $\epsilon$ -derivative, so this step is bookkeeping that fixes normalization.

Step 3. Rewrite the cross term using $\eta\eta'=\tfrac12\frac{\mathrm{d}}{\mathrm{d}x}(\eta^2)$ , so that $2F_{yy'}\eta\eta'=F_{yy'}\frac{\mathrm{d}}{\mathrm{d}x}(\eta^2)$ . Why: turning the mixed term into a total derivative of $\eta^2$ is exactly what lets integration by parts move it onto the $\eta^2$ coefficient, where it can be combined with $F_{yy}$ .

Step 4. Integrate that term by parts: $\int_a^b F_{yy'}(\eta^2)'\,\mathrm{d}x=[F_{yy'}\eta^2]_a^b-\int_a^b(\frac{\mathrm{d}}{\mathrm{d}x}F_{yy'})\eta^2\,\mathrm{d}x$ . The boundary term vanishes because $\eta(a)=\eta(b)=0$ . Collecting the $\eta^2$ coefficients gives $Q=F_{yy}-\frac{\mathrm{d}}{\mathrm{d}x}F_{yy'}$ and leaves $P=F_{y'y'}$ untouched, which is equation (3). Why: the vanishing endpoints kill the boundary contribution, so the manipulation is exact — no approximation enters — and the result is manifestly a self-adjoint Sturm–Liouville form.

Go deeper · Derivation B · 1 minNecessity of the Legendre condition

Derivation BNecessity of the Legendre condition

The goal is to prove that $P=F_{y'y'}\ge0$ everywhere is forced by minimality, using nothing but a cleverly localized variation. The strategy exploits a scaling mismatch: a variation squeezed into a window of width $\delta$ has a slope of order $1/\delta$ , so the stiffness integral $\int P\eta'^2$ scales as $1/\delta$ while the potential integral $\int Q\eta^2$ scales as $\delta$ . As $\delta\to0$ the stiffness term dominates and carries the sign of $P$ at the window's centre, which pins down the necessary condition.

Step 1. Suppose, for contradiction, that $P(x_0)<0$ at some interior $x_0$ . Build a bump supported on $[x_0-\delta/2,\,x_0+\delta/2]$ , for instance $\eta_\delta(x)=\sin^2\!\big(\pi\tfrac{x-x_0}{\delta}\big)$ inside the window and zero outside. It vanishes with its derivative at both edges, so it is an admissible variation. Why: a compactly supported bump lets us probe $\delta^2 J$ at a single point while keeping $\eta$ strictly admissible, isolating the local behaviour of $P$ .

Step 2. Estimate the two contributions. The derivative $\eta_\delta'$ is of order $1/\delta$ over a window of width $\delta$ , so $\int P\eta'^2\sim P(x_0)\,C_1/\delta$ ; the value $\eta_\delta$ is of order $1$ , so $\int Q\eta^2\sim Q(x_0)\,C_2\,\delta$ , with $C_1,C_2$ positive numerical constants. Why: squaring a slope of size $1/\delta$ and integrating over width $\delta$ leaves $1/\delta$ , whereas squaring an order-one value over width $\delta$ leaves $\delta$ — the mismatch is the whole argument.

Step 3. Add them: $\delta^2 J\approx \dfrac{C_1}{\delta}\,P(x_0)+O(\delta)$ . Taking $\delta$ small enough makes the first term dominate, so $\delta^2 J$ inherits the sign of $P(x_0)$ . If $P(x_0)<0$ then $\delta^2 J<0$ . Why: a negative second variation for even one admissible $\eta$ means $y$ can be beaten by a nearby function, contradicting minimality.

Step 4. Since assuming $P(x_0)<0$ produces a contradiction, $P\ge0$ must hold at every interior point, and by continuity on the closed interval. This is the Legendre necessary condition; the sufficiency theorem later upgrades it to the strict $P>0$ . Why: necessity and sufficiency use the same coefficient $P$ but at opposite strengths — nonnegative to survive as a candidate, strictly positive to be certified.

Go deeper · Derivation C · 2 minConjugate points and the death of positivity

Derivation CConjugate points and the death of positivity

The goal is to show why an interior conjugate point is fatal: at such a point a nonzero admissible variation exists on which $\delta^2 J$ vanishes, and any smoothing of it drives the form negative, so the extremal cannot be a minimizer beyond the conjugate point. The engine is an exact-derivative identity: on a solution of the accessory equation the second-variation integrand collapses to a total derivative, so its integral is fixed entirely by boundary values that both happen to vanish.

Step 1. Let $u$ solve the Jacobi accessory equation (5) with $u(a)=0$ , and suppose $u(c)=0$ for some $c\in(a,b)$ . On such a solution, $Qu^2=u\cdot(Pu')'=\frac{\mathrm{d}}{\mathrm{d}x}(Puu')-Pu'^2$ , so the integrand rearranges to $Pu'^2+Qu^2=\frac{\mathrm{d}}{\mathrm{d}x}(Puu')$ . Why: the accessory equation is exactly the condition that makes the quadratic integrand an exact derivative — this is the second-variation analogue of the Beltrami/first-integral trick.

Step 2. Integrate over $[a,c]$ : $\int_a^c(Pu'^2+Qu^2)\,\mathrm{d}x=[Puu']_a^c$ . Both endpoints kill the bracket because $u(a)=0$ and $u(c)=0$ , so the integral is exactly zero. Why: the total-derivative form means the interior details are irrelevant — only the endpoint values of $u$ matter, and both vanish at a conjugate pair.

Step 3. Define the trial variation $\eta^*=u$ on $[a,c]$ and $\eta^*=0$ on $[c,b]$ . It is continuous and admissible (it has a corner at $c$ , which is permitted for weak variations), and $\delta^2 J[\eta^*]=\int_a^c(Pu'^2+Qu^2)\,\mathrm{d}x+0=0$ . Why: extending by zero past $c$ costs nothing in the integral yet produces a genuine, globally admissible variation that lands the second variation exactly on the boundary between positive and negative.

Step 4. The corner at $c$ is a lever. Rounding it — replacing the kink by a smooth patch — strictly lowers $\delta^2 J$ below zero, because the accessory solution already saturates positivity and any admissible modification of a degenerate direction of a quadratic form can only decrease it. Hence positivity fails once a conjugate point lies inside the interval, which is the content of the Jacobi necessary condition. Why: a quadratic form with a nontrivial null vector is at best positive semidefinite, and a null vector inside an open set of admissible directions can always be tilted to make the form negative.

Limiting cases

The character of a stationary function is read off from $P$ , $Q$ , and the location of the first conjugate point:

Case	Result	Meaning
Strengthened Legendre, no conjugate point	$P=F_{y'y'}>0$ and no $c\in(a,b)$	Sufficient for a weak local minimum (Jacobi's theorem).
Legendre violated	$F_{y'y'}<0$ somewhere	A sharply peaked $\eta$ there makes $\delta^2 J<0$ ; not a minimum.
Conjugate point inside interval	$\exists\,c\in(a,b):\ u(c)=0,\ u(a)=0$	A nontrivial accessory solution refutes positivity; not a minimizer beyond $c$ .
Short interval	$b-a\to0^+$	With $P>0$ the stiffness term dominates: every short-enough extremal arc is a genuine local minimizer.
Maximum	$F_{y'y'}<0$ throughout, no conjugate point	Replacing $F\to-F$ , the mirror conditions certify a weak local maximum.
Endpoint exactly conjugate	$c=b$	Boundary case: $\delta^2 J\ge0$ but degenerate — the test is inconclusive (second-order fails).

Common trapThe Legendre condition alone proves a minimum

The most common error is to compute $F_{y'y'}>0$ , declare victory, and stop. The Legendre condition is only necessary (in strict form, locally sufficient on short arcs); it says nothing about the global accumulation of the potential $Q$ . On a long enough interval a conjugate point appears and the stationary function ceases to minimize even though $P>0$ everywhere — the great-circle arc past the antipode and the harmonic-oscillator action past a half period are exactly this failure. Conversely, passing the full Jacobi test buys only a weak local minimum in the $C^1$ norm: it does not exclude lower values of $J$ reached by variations with small displacement but large slope (that is the province of the Weierstrass strong-minimum theory) and it says nothing about global competitors far away in function space.

Worked examples

1 · The straight line really is shortest (confidence builder)

Take the arc-length functional $J[y]=\int_a^b\sqrt{1+y'^2}\,\mathrm{d}x$ , whose extremal is the straight line $y=\text{const}\cdot x$ . Verify the sufficiency conditions and confirm there is no conjugate point on any interval.

Step 1. Compute the coefficients. With $F=\sqrt{1+y'^2}$ one finds $F_{y'y'}=(1+y'^2)^{-3/2}$ , while $F_{yy}=0$ and $F_{yy'}=0$ because $F$ has no explicit $y$ dependence. Hence $P=(1+y'^2)^{-3/2}$ and $Q=0$ . Along a straight line $y'$ is constant, so $P$ is a positive constant — for slope $y'=1$ , $P=2^{-3/2}=\mathbf{0.354}$ , and for $y'=0$ , $P=\mathbf{1}$ . The strengthened Legendre condition $P>0$ holds everywhere.

Step 2. Solve the accessory equation. With $Q=0$ it reads $-(Pu')'=0$ , so $Pu'=\text{const}$ and, since $P>0$ is constant, $u'=\text{const}$ . The solution with $u(a)=0$ is the straight ramp $u=k(x-a)$ . This is monotonic and never returns to zero for any $x>a$ .

Step 3. Read the conclusion. There is no conjugate point on $(a,b)$ for any $b$ , and $P>0$ throughout, so by Jacobi's theorem $\delta^2 J>0$ and the straight line is a weak local minimizer of length on every interval, of any length. This matches intuition — a straight segment is the shortest path between two points no matter how far apart, the flat-space counterpart of the sphere's failure at the antipode. No conjugate point ever appears when Q = 0 and P > 0.

2 · The kinetic focus of the harmonic oscillator (fully worked)

For the action of a unit-mass harmonic oscillator, $J[x]=\int_0^T \tfrac12\big(\dot x^2-\omega^2 x^2\big)\,\mathrm{d}t$ , locate the first conjugate point and evaluate the second variation on a concrete trial function for intervals shorter and longer than it. Use $\omega=2\pi\ \mathrm{s}^{-1}$ (period $T_0=1\ \mathrm{s}$ ).

Step 1. Coefficients. Here $F=\tfrac12(\dot x^2-\omega^2 x^2)$ , so $F_{\dot x\dot x}=1$ , $F_{xx}=-\omega^2$ , $F_{x\dot x}=0$ . Thus $P=1>0$ (Legendre satisfied everywhere) and $Q=F_{xx}-\frac{\mathrm{d}}{\mathrm{d}t}F_{x\dot x}=-\omega^2$ . The negative potential $Q=-\omega^2$ is what will eventually destroy positivity.

Step 2. Accessory equation and conjugate point. Equation (5) becomes $-u''-\omega^2 u=0$ , i.e. $u''+\omega^2 u=0$ . The solution with $u(0)=0$ is $u(t)=\sin(\omega t)$ , which next vanishes at $\omega t=\pi$ , so the conjugate length is $t_c=\pi/\omega$ . Numerically $t_c=\pi/(2\pi)=\mathbf{0.5\ s}$ , exactly half a period — this is the "kinetic focus" of classical mechanics.

Step 3. Second variation on a trial mode. Take the admissible $\eta(t)=\sin(\pi t/T)$ , which vanishes at $0$ and $T$ . Then $\delta^2 J=\int_0^T(\eta'^2-\omega^2\eta^2)\,\mathrm{d}t=\tfrac12\big(\pi^2/T-\omega^2 T\big)=\dfrac{1}{2T}\big(\pi^2-\omega^2 T^2\big)$ . The sign is that of $\pi^2-\omega^2T^2$ , which is positive precisely when $T<\pi/\omega=t_c$ — the accessory prediction, recovered independently.

Step 4. Plug in numbers with $\omega=2\pi$ . For $T=0.4\ \mathrm{s}$ : $\pi^2-\omega^2T^2=9.8696-6.3165=3.553$ , so $\delta^2 J=+\mathbf{4.44}>0$ — a minimum. For $T=0.5\ \mathrm{s}$ : $\pi^2-\omega^2T^2=9.8696-9.8696=\mathbf{0}$ , so $\delta^2 J=\mathbf{0}$ — the degenerate conjugate case. For $T=0.6\ \mathrm{s}$ : $\pi^2-\omega^2T^2=9.8696-14.212=-4.343$ , so $\delta^2 J=-\mathbf{3.62}<0$ — no longer a minimum. The action is a minimum only for $T<0.5\ \mathrm{s}$ ; beyond the kinetic focus it becomes a saddle, which is why the semiclassical propagator picks up a phase (a Maslov index of $1$ ) each time a trajectory crosses a focus.

Physics of the portrait

The sufficiency theory has three fathers, but one name owns its decisive idea. Legendre supplied the necessary $P\ge0$ in 1786 and mistakenly thought it sufficient; Lagrange spotted the gap. It was Carl Gustav Jacob Jacobi who, in a few dense pages, introduced the accessory equation and the conjugate point and closed the argument.

Origins · 1 minCarl Gustav Jacob Jacobi and how this was discovered

Carl Gustav Jacob Jacobi

1804–1851 · the accessory equation and conjugate-point theory of variational sufficiency

Jacobi was one of the great analytic virtuosos of the nineteenth century — the theory of elliptic and theta functions, the Jacobian determinant, the Jacobi identity of Poisson brackets, and the Hamilton–Jacobi equation of mechanics all carry his name. Born in Potsdam in 1804, he became a professor at Königsberg at twenty-two and made it a centre of mathematics before ill health and the failed 1848 revolution unsettled his last years; he died of smallpox in Berlin in 1851, at forty-six. His contribution to the calculus of variations came in an 1837 note in Crelle's Journal, written in his characteristic telegraphic style, that transformed Legendre's incomplete second-variation argument. Legendre had shown in 1786 that $F_{y'y'}\ge0$ is needed for a minimum and then argued — wrongly, as Lagrange noted — that it suffices. Jacobi saw that the missing ingredient was a second differential equation, the accessory (or Jacobi) equation, obtained by varying the second variation itself. Its solutions are the infinitesimally neighbouring extremals, and the points where a solution starting at one end refocuses are the conjugate points that limit how far a minimizing arc can extend. The same accessory equation governs the spread of trajectories around a classical path, tying the sufficiency question to the focusing of orbits and, a century later, to the Morse index and the Maslov phase of semiclassical quantization. Weierstrass and Hilbert would complete the edifice — strong variations, field theory, the invariant integral — but the conjugate point is Jacobi's, and it remains the most surprising fact in the subject: a path can be locally perfect and still fail to minimize, for a reason that lives entirely in the global geometry.

History

1744

— Euler's Methodus inveniendi founds the calculus of variations

1786

— Legendre gives the necessary condition F_y'y' ≥ 0 and overclaims sufficiency

1837

— Jacobi introduces the accessory equation and conjugate points

1868

— Mayer and Clebsch systematize Jacobi's theory for general problems

1879

— Weierstrass develops strong variations and the excess function E

1900

— Hilbert's invariant integral yields a clean sufficiency proof

Going further

Texts. Gelfand & Fomin, Calculus of Variations, ch. 5 (The Second Variation; the Jacobi Condition), §§25–26 · Stone & Goldbart, Mathematics for Physics, ch. 1, §1.4 · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., ch. 22 · Courant & Hilbert, Methods of Mathematical Physics, vol. 1, ch. 4 · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, ch. 22 · Gutzwiller, Chaos in Classical and Quantum Mechanics, ch. 12 (Maslov index). Key checkpoints in the theory: the integration by parts giving $Q=F_{yy}-\frac{\mathrm{d}}{\mathrm{d}x}F_{yy'}$ ; the exact-derivative identity $Pu'^2+Qu^2=(Puu')'$ on accessory solutions; and the $1/\delta$ -vs- $\delta$ scaling of the Legendre-necessity argument. Concrete anchor results: the harmonic-oscillator accessory equation $u''+\omega^2u=0$ gives $t_c=\pi/\omega$ ; the trial-function formula $\delta^2 J=\frac{1}{2T}(\pi^2-\omega^2T^2)$ yields $+4.44$ , $0$ , $-3.62$ at $T=0.4,0.5,0.6\ \mathrm{s}$ for $\omega=2\pi$ ; the arc-length coefficients are $P=(1+y'^2)^{-3/2}$ , $Q=0$ ; and the spherical accessory equation $u''+u=0$ gives the antipode (angular distance $\pi$ ) as the first conjugate point.

Prerequisites. First Variation and Stationarity · Euler–Lagrange Equation · Fundamental Lemma of the Calculus of Variations · Functionals and Function Spaces. Leads to. Geodesics as Variational Problems · Hamilton's Principle of Stationary Action · Constrained Variation and Lagrange Multipliers.

Ask the tutor: Want to walk through computing $P$ , $Q$ , and the first conjugate point for your own functional, step by step?

Why isn't the second-derivative test from calculus enough here?

In finite dimensions the second-derivative test checks whether a Hessian matrix is positive definite — a finite list of eigenvalues. A functional lives on an infinite-dimensional space of functions, so the "Hessian" is the quadratic form $\delta^2 J=\int(P\eta'^2+Q\eta^2)\,\mathrm{d}x$ , an operator with infinitely many modes. Positivity of such a form can fail in a way with no finite-dimensional analogue: everything is fine pointwise (the local "curvature" coefficient $P$ is positive), yet a long-wavelength mode built from the accessory solution accumulates enough negative contribution from $Q$ to make the whole integral zero or negative. That global failure is the conjugate point, and detecting it needs the extra Jacobi machinery that the ordinary test simply does not have.

What exactly is a Jacobi field, physically?

A Jacobi field $u(x)$ is the infinitesimal gap between the reference extremal and an immediately neighbouring one that started from the same point. Solve the accessory equation with $u(a)=0$ and you are tracking how a pencil of extremals fanning out from $a$ spreads. Where that gap grows, neighbours diverge and the reference path is locally the best; where the gap shrinks back to zero — a conjugate point — the neighbours have refocused onto the reference path, and a competing route now exists at essentially the same cost. In mechanics the Jacobi field is literally the linearized deviation of nearby trajectories; in geometry it is the deviation of nearby geodesics, governed by curvature through $\ddot u+Ku=0$ .

Why does the great-circle arc stop being shortest at the antipode?

On the unit sphere the accessory equation for a great-circle geodesic reduces to $u''+u=0$ (with arc length as parameter), because the Gaussian curvature is $K=1$ . A Jacobi field starting at zero is $u=\sin s$ , which returns to zero at $s=\pi$ — exactly the antipodal point. Up to the antipode the great-circle arc is the unique shortest path; beyond it, the neighbouring geodesics have refocused, and the "long way round" the great circle, or a nearby geodesic, is shorter than continuing. The arc is still an extremal — it still satisfies the Euler–Lagrange equation — but it has lost its minimizing status precisely at the conjugate point.

Is a "weak" local minimum a serious limitation?

It is a real caveat worth stating. The sufficiency theorem controls variations that are small in the $C^1$ norm, meaning both $\eta$ and its slope $\eta'$ are small. It therefore does not rule out a variation with tiny displacement but a large, rapidly oscillating slope — a "strong" variation — that could lower the functional. Deciding those cases requires the Weierstrass excess function $E$ and field theory, a strictly stronger set of conditions. For many physical problems (smooth Lagrangians, convex in $y'$ ) the strong and weak conditions coincide, so the distinction is often moot in practice — but it is exactly the loophole through which corner solutions and Lavrentiev-type phenomena slip, so a careful analyst names it.

How does the accessory equation relate to linearizing Euler–Lagrange?

They are the same equation, seen two ways. Take the Euler–Lagrange equation for the original problem and linearize it about the extremal $y$ by writing $y+u$ and keeping terms linear in $u$ ; the result is precisely $-\frac{\mathrm{d}}{\mathrm{d}x}(Pu')+Qu=0$ . Equivalently, treat the second variation $\int(Pu'^2+Qu^2)\,\mathrm{d}x$ as a brand-new variational problem in $u$ and write down its Euler–Lagrange equation — you get the identical accessory equation. This is why conjugate points can be computed either as focal points of the family of extremals or as eigenvalue zero-crossings of the second-variation operator: the two pictures are one equation.

What is the connection to the Maslov index and quantum mechanics?

Each conjugate point (kinetic focus) a classical trajectory passes flips the second variation toward a saddle by adding one negative eigenvalue — the Morse index counts them. In the semiclassical (WKB) propagator and in Feynman's path integral, that count enters the phase: every focus contributes $-\pi/2$ , an integer multiple of which is the Maslov index. So the number of conjugate points along a path is measurable — it appears in interference fringes, in the Gutzwiller trace formula for chaotic spectra, and in the $+\tfrac12$ offsets of Bohr–Sommerfeld quantization. The sufficiency theory of the 1830s is the classical skeleton of the semiclassical phase.