Hamilton's principle/Mathematics/The Weierstrass excess function

Plain-language version: Strong and weak

Every sufficiency result so far has been for weak minima — minima against competitors that stay close in both position and slope. Widening the comparison class to paths that stay close in position only, with slope unrestricted, requires one further condition. It is Weierstrass's, and it completes the classical theory.

The two comparison classes

A sawtooth competitor close in position and far in slope.

From Functionals and the space of paths, the weak norm controls position and slope, the strong norm controls position only. A strong neighbourhood is the larger set, so a strong minimum is the stronger claim, and every strong minimum is a weak minimum.

The gap is real. A competitor may run alongside the extremal in position while oscillating steeply in slope — a sawtooth of small amplitude and large gradient. Such a path is far from the extremal in the weak norm and close in the strong one, so no weak result says anything about it.

The Legendre and Jacobi conditions of the previous two pages are both derived from the second variation, which is a small-$\epsilon$ object. They cannot see a competitor whose slope differs by $O(1)$.

The excess function

The Weierstrass excess function as a gap above the tangent.

Definition. The Weierstrass excess function is

$E(x,y,p,q)=L(x,y,q)-L(x,y,p)-(q-p)L_{y^{\prime }}(x,y,p)\mathrm{(}\mathrm{1}\mathrm{)}$

Read (1) as a remainder. $L(x,y,p)+(q-p)L_{y^{\prime }}(x,y,p)$ is the tangent line to $L$ in its third slot at $p$, evaluated at $q$. So $E$ is the amount by which $L$ exceeds its own tangent line.

$E\ge 0$ for all $q$ is therefore exactly the statement that $L$ is convex in its slope argument — not merely at $p$, which is Legendre's local condition, but across the whole range of slopes a competitor might use.

The relation between the two is worth setting down. Taylor expansion of (1) in $q$ about $p$ gives

$E=\frac{1}{2}(q-p{)}^2{L}_{y^{\prime }{y}^{\prime }}(x,y,\overline{p})\mathrm{(}\mathrm{2}\mathrm{)}$

for some $\overline{p}$ between $p$ and $q$. So $E\ge 0$ near $q=p$ is Legendre's condition, and $E\ge 0$ for all $q$ is a genuinely stronger demand: $L_{y^{\prime }{y}^{\prime }}\ge 0$ must hold not only along the extremal but at every slope, at each point of a neighbourhood.

The conditions assembled

A corner in an extremal.

Weierstrass's necessary condition. If $y$ is a strong local minimum then

$E(x,y(x),y^{\prime }(x),q)\ge 0\mathrm{(}\mathrm{3}\mathrm{)}$

for every $x\in [a,b]$ and every real $q$.

Sufficiency. If $y$ satisfies the Euler-Lagrange equation, can be embedded in a field of extremals — which the Jacobi condition of the Jacobi condition and conjugate points guarantees — and $E>0$ for all $q\ne p$ at every point of a neighbourhood of the extremal, then $y$ is a strong local minimum.

The necessary form is proved by a needle variation of a different kind from Legendre's: the competitor departs from the extremal at one point with a slope $q$ quite unlike $y^{\prime }$, runs briefly, and rejoins. Such a path is strongly close and weakly far. Computing $J$ along it to first order in the departure length gives exactly $E$ as the leading change, so $E<0$ anywhere produces a cheaper competitor.

The full hierarchy

The four conditions of the sufficiency theory.

condition	scope	necessary for	sufficient with the others for
Euler-Lagrange	pointwise, 1st order	both	—
Legendre $P\ge 0$	pointwise, 2nd order	both	—
Jacobi, no conjugate point	interval, 2nd order	both	weak minimum
Weierstrass $E\ge 0$	pointwise, all slopes	strong only	strong minimum

Four conditions, and only the strengthened combinations in the last column are sufficient for anything. That is the honest summary of the classical theory, and it is the reason this volume has taken seven pages to reach a sufficiency statement.

Worked example: where the strong claim fails

Take

$J[y]={\int }_0^1(y{\text{'}}^2-y{\text{'}}^4)dx\mathrm{(}\mathrm{4}\mathrm{)}$

with $y(0)=y(1)=0$. The extremal $y\equiv 0$ satisfies the Euler-Lagrange equation, and

$L_{y^{\prime }{y}^{\prime }}=2-12y{\text{'}}^2\mathrm{(}\mathrm{5}\mathrm{)}$

which at $y^{\prime }=0$ equals $2>0$. Legendre's strengthened condition holds. $Q=0$, so the accessory equation is ${\eta }^{″}=0$, whose solution vanishing at the origin is $\eta =cx$, with no second zero. No conjugate point. Jacobi's condition holds. The extremal is therefore a weak minimum, and $J[0]=0$.

Now compute the excess function at $p=0$:

$E(x,y,0,q)=(q^2-q^4)-0-q\cdot 0=q^2(1-q^2)\mathrm{(}\mathrm{6}\mathrm{)}$

which is negative for $|q|>1$. Weierstrass's condition fails.

The failure is realised by an explicit competitor. Let $y_n$ be a sawtooth of slope $\pm 2$ with $n$ teeth on $[0,1]$. Its amplitude is $1/n$, so $\Vert y_n{\Vert }_0\to 0$ and it is strongly close to the extremal. But $y_n{\text{'}}^2=4$ and $y_n{\text{'}}^4=16$ everywhere, so

$J[y_n]={\int }_0^1(4-16)dx=-12\mathrm{(}\mathrm{7}\mathrm{)}$

independent of $n$. Every one of these paths beats $J[0]=0$, and they crowd arbitrarily close to the extremal in position.

So $y\equiv 0$ is a weak minimum and is not a strong minimum. Nothing in the second-variation theory could have detected this: the sawtooth is not a small variation in the weak norm, and the whole of the second variation is about small weak variations.

Corners

The strong class admits paths with corners, so the Weierstrass-Erdmann conditions of Boundary terms and natural boundary conditions apply. The relation to (1) is direct: $E\equiv 0$ along a segment means $L$ is affine in the slope there, which is the degeneracy that permits a corner. Strict convexity, $E>0$, forbids corners and recovers the smooth theory.

What this track has and has not settled

O6 closes here. The track has established what a minimum requires and, under strengthened hypotheses, what guarantees one. It has also produced two explicit failures — the conjugate-point failure on the Jacobi condition and conjugate points and the strong-class failure above — in each case with an exhibited competitor rather than an assertion.

Two limits are worth stating plainly. All of this is local: it compares an extremal with nearby paths and says nothing about a distant competitor. And existence has been assumed throughout: every theorem here has the form if an extremal is a minimum then, or if these conditions hold then it is a local minimum. Whether a minimiser exists at all is the direct method, which this volume does not treat and which is named as further reading on Sources.

Next

the Legendre transform opens the third mathematics track, on structure.