Calculus of Variations: Strong Extrema and Sufficiency Conditions

Calculus of Variations: Strong Extrema and Sufficiency Conditions

The Euler–Lagrange equation tells us which admissible curves are stationary. CV11 and CV12 then used the second variation, the Legendre condition, and the Jacobi equation to determine when a stationary curve behaves locally like a minimum with respect to sufficiently small changes in both the curve and its slope.

A final difficulty remains. A curve can be extremely close to an extremal in position while having a very different derivative. Such competitors are invisible to a purely C¹-local test, yet they matter for a strong minimum. The classical tool that detects them is the Weierstrass excess function, usually called the Weierstrass E-function.

The conceptual progression is

(1)

Classical sufficiency theory is developed in different but equivalent forms in Bolza, Bliss, Gelfand and Fomin, Weinstock, Sagan, and van Brunt [6, 5, 1, 4, 3, 2].

1 Learning objectives

After completing this entry, the reader should be able to:

1. distinguish weak and strong local minima using C¹ and C⁰ neighborhoods;
2. explain why a strong minimum must tolerate competitors whose slopes differ greatly from the extremal slope;
3. derive and interpret the Weierstrass excess function;
4. prove that the Weierstrass condition implies the Legendre condition in the infinitesimal limit;
5. construct a stationary curve that is a weak minimum but not a strong minimum;
6. explain why the Weierstrass condition alone is not sufficient;
7. define a field of extremals and the associated slope field;
8. derive the Hilbert invariant integral and its exactness condition;
9. prove a classical strong-minimum sufficiency theorem using the Hilbert integral and the excess function;
10. connect the no-conjugate-point condition with the local construction of a field of extremals;
11. recognize special convexity structures that give global rather than merely local conclusions.

2 Weak and strong extrema revisited

Let

(2)

on an admissible class of piecewise C¹ curves satisfying fixed endpoint conditions.

A weak local minimum y_∗ is a curve for which there exists ρ > 0 such that

(3)

whenever the competitor satisfies a C¹-type closeness requirement such as

(4)

A strong local minimum requires the same inequality for every admissible competitor that is sufficiently close only in position:

(5)

The derivative of the competitor need not be close to y_∗′. Therefore

(6)

but the converse can fail.

2.1 Why strong closeness is more demanding

A high-frequency perturbation can have tiny amplitude but a large derivative. For example,

(7)

on [0, 1] satisfies

(8)

while

(9)

has derivative amplitude 2 for every n.

Thus y_n approaches the zero function strongly in the C⁰ sense but not weakly in the C¹ sense used here.

3 The Weierstrass excess function

Fix x and y. Let

(10)

be the slope of a candidate extremal and let

(11)

be a competing slope. The tangent line to the function F(x,y,⋅) at p is

(12)

The vertical difference between the true value at q and this tangent approximation is

(13)

This is the Weierstrass excess function.

The basic geometric interpretation is

(14)

If F is convex as a function of its slope argument, then

(15)

for every p and q. Strict convexity gives strict positivity whenever q≠p.

4 The Weierstrass necessary condition

A classical necessary condition for a strong local minimum is the following.

Weierstrass necessary condition

Suppose F is sufficiently smooth and y_∗ is a strong local minimizer. Then along the extremal,

(16)

for every interior point x and every finite comparison slope q for which the local comparison is admissible.

The important phrase is every finite comparison slope. The Legendre condition checks only what happens when q is infinitesimally close to p.

4.1 Why a finite-slope test appears

A strong neighborhood constrains the vertical displacement of the competitor but not its slope. Over a very short interval of length h, one may therefore replace the extremal slope p by a finite slope q while keeping the resulting vertical displacement only of order h.

A short reconnecting segment restores the competitor to the original curve. When the variation is shrunk around a fixed interior point, the leading-order change in the functional is proportional to

(17)

If this number were negative, the small broken-line competitor could be made arbitrarily close in C⁰ while lowering J. That would contradict strong minimality. This is the geometric core of Weierstrass’s argument [6, 5, 2].

5 Legendre is the infinitesimal limit of Weierstrass

Set

(18)

Taylor expansion in the slope argument gives

(19)

Substituting into the definition of E gives

(20)

Therefore, if

(21)

for all sufficiently small positive and negative h, then

(22)

So the Legendre condition is the local quadratic shadow of the Weierstrass condition.

The converse is false. Knowing only

(23)

says that the tangent line is locally supporting near p; it does not guarantee that F(q) remains above that tangent line for slopes far away from p.

6 A curve that is weakly minimizing but not strongly minimizing

Consider

(24)

with fixed endpoints

(25)

The zero function

(26)

is stationary because F depends only on y′ and

(27)

vanishes along y_∗′ = 0.

6.1 Weak local minimality

If a competitor is sufficiently close in C¹ so that

(28)

then

(29)

Hence

(30)

Equality with the fixed endpoints occurs only when y′≡ 0, hence only when y ≡ 0. Thus y_∗ is a strict weak local minimum.

Its second variation is also positive:

(31)

for every nonzero fixed-endpoint variation.

6.2 Failure of strong minimality

For the extremal slope p = 0, the Weierstrass function is

(32)

This becomes negative whenever

(33)

Now use the small-amplitude, high-frequency competitors

(34)

They satisfy the endpoint conditions and

(35)

but

(36)

Therefore

J[yn]	= ∫ 01dx	(37)
	= 4− 16	(38)
	= 2 − 6	(39)
	= −4 .	(40)

Thus arbitrarily C⁰-close competitors have smaller functional value:

(41)

This example is one of the most useful conceptual separations in the subject:

(42)

7 The strengthened Weierstrass condition

For sufficiency arguments one usually imposes a strict version:

(43)

This is called a strengthened Weierstrass condition.

It is stronger than the necessary condition E ≥ 0. It provides strict separation from the tangent line for every competing slope.

If F is strictly convex in y′, then the strengthened Weierstrass condition holds automatically at every slope.

However, this still does not by itself prove that a stationary curve is minimizing. The y-dependence of the functional and the geometry of neighboring extremals still matter.

8 Why Weierstrass alone is not sufficient

Consider the harmonic-oscillator action on [0,T],

(44)

The zero curve is stationary. For any baseline slope p and comparison slope q,

(45)

whenever q≠p.

So the strengthened Weierstrass condition holds globally in the slope variable.

Nevertheless, CV12 showed that the first conjugate point occurs at

(46)

For

(47)

there are directions with negative second variation, and the zero solution is not a minimum.

Therefore

(48)

The Jacobi condition supplies the missing interval information.

9 Fields of extremals

The classical strong-sufficiency proof uses a neighborhood filled by neighboring extremals.

A field of extremals is a one-parameter family

(49)

such that each curve obtained by holding α fixed satisfies the Euler–Lagrange equation, and the family locally fills a region without folding onto itself.

If the relation

(50)

can be inverted locally for

(51)

then the slope of the field extremal passing through (x,y) is

(52)

Thus the field converts a family of extremals into a slope field P(x,y).

10 The Hilbert invariant integral

The field of extremals leads to a remarkable exact differential form.

Define

(53)

and

(54)

Now consider the one-form

(55)

10.1 Why the one-form is closed

Differentiate N with respect to x while holding y fixed:

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Differentiate M with respect to y while holding x fixed:

(57)

Along a field extremal, y′ = P and

(58)

The Euler–Lagrange equation for every member of the field is

(59)

Expanding the total derivative gives

(60)

Substitute this into M_y:

My	= Fy′x + PFy′y + Fy′y′Px + PFy′y′Py − PFy′y − PFy′y′Py	(61)
	= Fy′x + Fy′y′Px	(62)
	= Nx.	(63)

Therefore

(64)

On a simply connected local field region, the one-form ω is exact. Hence its integral depends only on the endpoints.

Define the Hilbert invariant integral

(65)

For two nearby curves with the same endpoints,

(66)

11 The key identity behind strong sufficiency

Along any competitor y(x) inside the field region,

(67)

Therefore

ℋ[y]	= ∫ ab dx	(68)
	= ∫ ab dx.	(69)

By the definition of the Weierstrass excess,

(70)

Hence

(71)

For the distinguished field extremal y_∗,

(72)

so

(73)

and therefore

(74)

Since the Hilbert integral is endpoint dependent only,

(75)

for every nearby competitor with the same endpoints. Subtracting gives the decisive identity

(76)

This formula is one of the cleanest sufficiency statements in the classical theory.

12 A classical strong-minimum sufficiency theorem

One standard formulation is the following [6, 5, 1, 2].

Strong sufficiency theorem, field form

Let y_∗ be a fixed-endpoint extremal for

(77)

Assume there exists a regular field of extremals containing y_∗ in a neighborhood 𝒰 of its graph, and suppose

(78)

for every (x,y) ∈𝒰 and every admissible finite slope q.

Then every admissible comparison curve lying in 𝒰 and having the same endpoints satisfies

(79)

If the excess is strictly positive whenever

(80)

and the local field extremal joining the fixed endpoints is unique, then y_∗ is a strict strong local minimum.

12.1 Proof

The Hilbert-integral identity gives

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If E ≥ 0 pointwise, then the integral is nonnegative. If E > 0 whenever the competitor slope differs from the field slope, equality can hold only if the competitor follows the same extremal field line almost everywhere. Local uniqueness then forces

(82)

Thus the conclusion is strong because no smallness assumption is imposed on y′− y_∗′; only the curve itself must remain inside the field neighborhood.

13 Where Jacobi enters the sufficiency theorem

The field-form theorem assumes that a regular field of extremals already exists. CV12 explains how the Jacobi equation enters when one tries to construct such a field.

Consider a one-parameter family

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of neighboring extremals. The derivative

(84)

is a Jacobi field.

If u vanishes again after the initial point, neighboring extremals refocus. At such a conjugate point the local family fails to remain one-to-one. In contrast, the absence of a conjugate point prevents this first-order focusing and is precisely the nondegeneracy needed in the classical construction of a local extremal field.

Thus, under the standard regularity hypotheses, the practical sufficiency chain is

(85)

Different textbooks package the hypotheses somewhat differently; some state existence of an extremal field directly, while others derive it from Jacobi-type assumptions [5, 6, 2].

14 Example: Dirichlet energy

Consider

(86)

with fixed endpoints. The Euler–Lagrange equation is

(87)

so the extremal is the straight line

(88)

For

(89)

the excess function is

E(p,q)	= q2 −p2 − (q − p)p	(90)
	= (q − p)2 .	(91)

Thus the strengthened Weierstrass condition holds globally.

In fact one can prove more directly that the straight line is the unique global minimizer. Write

(92)

Then

J[y] − J[y∗]	= ∫ abdx	(93)
	= ∫ abmh′dx + ∫ ab(h′)2 dx	(94)
	= ∫ ab(h′)2 dx	(95)
	≥ 0.	(96)

Equality requires h′≡ 0 and the endpoints then give h ≡ 0.

15 Example: planar arc length

For

(97)

let the straight-line extremal have constant slope p. Then

E(p,q)	= − − (q − p)	(98)
	= .	(99)

By the Cauchy–Schwarz inequality,

(100)

so

(101)

with equality only when q = p.

This reproduces the strong local result. Convexity actually permits a global proof: the graph of F(q) = lies above its tangent at p, and the integrated tangent term vanishes because the endpoints determine the average slope.

Thus the straight line is the unique shortest graph joining the two fixed endpoints.

16 A hierarchy of conclusions

The major conditions should not be treated as interchangeable.

A useful memory aid is

(102)

17 Necessary versus sufficient conditions

The logical distinctions are essential.

17.1 Necessary conditions for a strong minimum

Under the usual regularity assumptions, a strong minimum must satisfy conditions such as:

1. Euler–Lagrange stationarity;
2. the Legendre condition;
3. the Jacobi no-interior-conjugate-point requirement in the regular case;
4. the Weierstrass condition E ≥ 0.

Failure of any applicable necessary condition rules out strong minimality.

17.2 Sufficiency needs the conditions to work together

None of the pointwise conditions alone is enough.

• Positive second variation can still fail to control large slope changes, as the y′² − y′⁴ example shows.
• Weierstrass positivity can hold while the Jacobi condition fails, as the long-interval harmonic oscillator shows.
• Euler–Lagrange alone merely says the first variation vanishes.

The field-of-extremals theorem combines these ideas into an actual comparison formula for nearby curves.

18 Common misconceptions

Misconception 1: “If F_y′y′ > 0, the extremal is a strong minimum.”

No. This condition controls only infinitesimal slope changes. The Weierstrass excess can become negative for finite changes of slope.

Misconception 2: “A positive second variation proves strong minimality.”

Not in general. The second variation is a small-C¹ statement. Strong competitors may be C⁰-close while having large derivatives.

Misconception 3: “If E > 0, we no longer need Jacobi.”

No. The harmonic-oscillator example has strictly positive excess in slope but ceases to minimize after the first conjugate point.

Misconception 4: “Strong means the perturbation is smaller.”

The terminology refers to the strength of the minimum property, not to a stronger norm. A strong minimum survives a larger family of competitors because only C⁰ closeness is required.

Misconception 5: “The Hilbert integral is an extra physical conservation law.”

Not generally. Here it is an exact line integral constructed from a field of extremals. Its endpoint invariance is a mathematical comparison device.

19 Practical classification workflow

For a first-order fixed-endpoint problem, a useful procedure is:

1. Derive the Euler–Lagrange equation and identify a stationary candidate y_∗.
2. Compute

   (103)

3. Form the second variation and apply the Jacobi test if interval-wide positivity is not obvious.
4. If only a weak local minimum is required, strengthened Legendre plus the Jacobi condition may already provide the needed local sufficiency result under the standard hypotheses.
5. If a strong minimum is required, compute

   (104)

6. Test finite slope changes, not only q ≈ p.
7. Establish or invoke a regular field of extremals around the candidate.
8. Use the Hilbert-integral identity

   (105)

   for nearby competitors in the field.

9. If a global result is desired, look for global convexity, completion-of-squares identities, geometric inequalities, or other arguments that compare against the whole admissible class.

20 What CV14 adds

CV13 completes the core local-extremum theory for first-order scalar functionals:

(106)

CV14 turns this machinery toward Classical Mechanics. Hamilton’s principle will identify the action functional

(107)

and derive Lagrange’s equations for mechanical systems. The distinction between stationary action and minimum action will be especially important: mechanical trajectories are stationary, but they need not minimize the action over arbitrary time intervals.

Summary

Weak and strong extrema differ because strong comparison curves need only be close in position. Their slopes may differ greatly from the extremal slope.

The Weierstrass excess function

(108)

measures the cost of a finite slope change relative to the tangent approximation at the extremal slope.

For a strong minimum, the Weierstrass necessary condition requires

(109)

Its infinitesimal limit gives the Legendre condition

(110)

A regular field of extremals with slope field P(x,y) generates the Hilbert invariant integral and the identity

(111)

This turns pointwise Weierstrass positivity into a strong local comparison theorem. The Jacobi condition remains essential because it controls the interval-wide focusing of neighboring extremals and supports the construction of a regular extremal field.

Thus the classical sufficiency picture is not one condition but a coordinated hierarchy:

(112)

under the corresponding regularity and field hypotheses.

References