Calculus of Variations: Functionals, Admissible Functions, and Extrema

The calculus of variations does not ask merely for a number that minimizes a function. It asks for an entire function, curve, field, or trajectory that optimizes a scalar quantity. Before differentiating a functional, one must therefore answer three questions:

1. What objects is the functional allowed to act on?
2. Which of those objects are physically and mathematically admissible?
3. What does it mean for two admissible functions to be “near” one another?

The third question is easy to overlook. In ordinary single-variable calculus, closeness is inherited from the usual distance |x−x_∗|. In a function space there are many inequivalent ways to measure distance. Two curves can be very close in height while having very different slopes. Consequently, the phrase “local minimum” is incomplete until the relevant neighborhood of functions has been specified.

This entry develops that foundation. It introduces admissible classes, function spaces, norms, relative and global extrema, and the classical distinction between weak and strong extrema. CV02 will then use these ideas to define variations and the first variation.

1 Learning objectives

After this entry, the reader should be able to

1. identify the domain of a functional and distinguish it from the domain of an ordinary function;
2. construct an admissible class from smoothness, endpoint, integral, or inequality constraints;
3. explain why an admissible class need not be a vector space;
4. distinguish C⁰, C¹, and C² regularity at an introductory level;
5. define the uniform C⁰ norm and a standard C¹ norm;
6. explain why a local extremum is defined relative to both an admissible set and a notion of neighborhood;
7. distinguish global, local, weak, and strong minima and maxima;
8. prove that a global minimum is a strong local minimum and that a strong local minimum is a weak local minimum in the classical setting;
9. understand by explicit example how a weak minimum can fail to be a strong minimum; and
10. distinguish attainment of a minimum from the mere existence of an infimum.

2 A functional includes its domain

A functional is a mapping

(1)

where 𝒜 is the domain of admissible objects. In the classical first-order problem,

(2)

𝒜 is usually a class of sufficiently smooth real-valued functions on [a,b] satisfying specified constraints.

It is mathematically incomplete to specify only the formula for J[y] and leave 𝒜 implicit. The same integral expression can define different optimization problems on different admissible classes.

For example, consider

(3)

One possible domain is

(4)

Another is

(5)

The formula for J is identical, but the optimization problems are not. The first class forces the curve to connect (0, 0) to (1, 1); the second forces it to return to zero.

3 Admissible functions and admissible classes

An admissible function is a function permitted by all constraints of the variational problem. The collection of all such functions is the admissible class 𝒜.

The constraints defining 𝒜 can come from mathematics, physics, or both.

3.1 Regularity constraints

The integrand may require derivatives to exist. If

(6)

then a classical treatment commonly assumes y ∈ C¹([a,b]). If

(7)

then C² regularity may be assumed in an elementary treatment.

Later analysis often weakens these assumptions. For example, finite-energy problems naturally lead to Sobolev spaces in which derivatives may exist only in a weak sense. The present course begins with classical C^k spaces so the variational logic is visible before the functional-analytic generalization is introduced.

3.2 Endpoint constraints

A fixed-endpoint class has the form

(8)

The endpoint conditions are not decorative boundary data appended after the variational calculation. They determine which comparison curves are allowed and, in CV02, which variations are allowed.

3.3 Periodic and mixed constraints

A periodic class might require

(9)

A mixed problem might fix one endpoint but leave the other free:

(10)

Such changes alter the endpoint terms that survive integration by parts. Natural boundary conditions and transversality conditions are developed in CV05.

3.4 Integral constraints

An isoperimetric problem may impose an additional condition such as

(11)

The admissible class is then restricted to functions satisfying both the endpoint conditions and the integral constraint. CV09 develops the variational Lagrange-multiplier method for this situation.

3.5 Inequality and obstacle constraints

A physical path may be forbidden from entering part of the domain. An obstacle problem can contain a condition such as

(12)

This is an important reminder that an admissible class need not be a vector space. If y₁ and y₂ are admissible, an arbitrary linear combination

(13)

need not satisfy the same endpoint, integral, or inequality constraints.

4 Function spaces used in the classical theory

To speak about neighborhoods of functions, we first need a space in which the functions live.

4.1 The space C⁰([a,b])

The notation

(14)

denotes the continuous real-valued functions on [a,b].

A natural measure of size is the uniform or supremum norm

(15)

For two functions y and z, the induced distance is

(16)

Thus y and z are close in C⁰ if their graphs remain vertically close at every point.

4.2 The space C¹([a,b])

The space C¹([a,b]) contains functions whose first derivatives exist and are continuous. For the present series, define

(17)

Then

(18)

requires both the functions and their first derivatives to be close.

Another common convention is

(19)

These two formulas define equivalent norms on C¹([a,b]) and therefore the same notion of local convergence. The max convention is used here because it makes the inclusion of neighborhoods especially transparent.

4.3 The space C²([a,b])

Similarly, C²([a,b]) contains functions with continuous first and second derivatives. A corresponding norm can be defined by

(20)

The higher the norm order, the more derivative information is controlled by a small neighborhood.

5 Neighborhoods of functions

Let y_∗ be a candidate function. Its open C⁰ neighborhood of radius ρ > 0 is

(21)

Its open C¹ neighborhood is

(22)

Because

(23)

we have

(24)

This inclusion is the geometric reason a strong variational minimum is a stronger property than a weak minimum, despite the initially counterintuitive terminology.

A C¹ neighborhood is smaller because it excludes curves whose heights are close but whose slopes change rapidly. A C⁰ neighborhood is larger because it allows such highly oscillatory competitors.

6 Global extrema

Let J : 𝒜→ ℝ.

6.1 Global minimum

A function y_∗ ∈𝒜 is a global minimum or global minimizer if

(25)

It is a strict global minimum if

(26)

A global maximum is defined by reversing the inequalities.

The defining feature is that a global optimum is compared with the entire admissible class. No notion of distance is needed.

7 Local or relative extrema

A local minimum compares y_∗ only with admissible functions in a neighborhood of y_∗. The mathematically precise statement requires both an admissible set and a distance or topology.

Let d be a distance on the ambient function space. The function y_∗ ∈𝒜 is a local minimum relative to 𝒜 if there exists ρ > 0 such that

(27)

for every

(28)

The word relative means relative to the admissible set 𝒜. It does not mean weak or strong. Weak and strong extrema are two different ways of choosing the neighborhood in which the relative comparison is made.

This distinction mirrors finite-dimensional constrained optimization. If a point is constrained to a surface, “local” means local among points on that surface, not among all points of the surrounding space.

7.1 Why local extrema require a topology

In ordinary calculus, the statement

(29)

has an obvious meaning. For functions there is no single universally correct analogue. One may control values, derivatives, integral averages, or other properties. Different choices generate different neighborhoods and therefore potentially different notions of local optimality.

The classical first-order calculus of variations gives special names to two important choices: weak extrema and strong extrema.

8 Weak local extrema

For a first-order functional on a C¹ admissible class, y _∗ is a weak local minimum if there exists ρ > 0 such that

(30)

for every admissible y satisfying

(31)

Equivalently, both

(32)

and

(33)

must hold under the max-norm convention.

Thus weak minimality tests the candidate against curves that are close in both position and slope.

A weak local maximum is defined by reversing the inequality.

9 Strong local extrema

The function y_∗ is a strong local minimum if there exists ρ > 0 such that

(34)

for every admissible y satisfying only

(35)

The derivatives of y and y_∗ are not required to be close.

This means a strong minimum must survive comparison with a larger set of nearby curves, including small-amplitude, high-frequency oscillations that can have large slope changes.

The term “strong” therefore refers to the strength of the optimality claim, not to a stronger closeness requirement.

10 Hierarchy of minimum concepts

The classical implications are

(36)

10.1 Proposition: global implies strong, and strong implies weak

proposition. Let J be defined on an admissible class 𝒜⊂ C¹([a,b]). Using the norms defined above:

1. every global minimum is a strong local minimum;
2. every strong local minimum is a weak local minimum.

Proof. If y_∗ is a global minimum, then

(37)

for every y ∈𝒜. Therefore the same inequality certainly holds for the smaller collection of admissible functions satisfying ∥y − y_∗∥₀ < ρ, for any ρ > 0. Hence y_∗ is a strong local minimum.

Now suppose y_∗ is a strong local minimum. Then for some ρ > 0,

(38)

for every admissible y in N₀(y_∗; ρ). But

(39)

Therefore the same inequality holds for every admissible y in the smaller C¹ neighborhood N₁(y_∗; ρ). Hence y_∗ is a weak local minimum. □

The converses are false in general. The next example explicitly proves the failure of the strong-to-weak converse.

11 Example: a global minimizer proved without Euler–Lagrange

Consider

(40)

on

(41)

We claim that

(42)

is the unique global minimizer.

Every admissible function can be written as

(43)

where

(44)

Then

(45)

and therefore

(46)

Expanding,

(47)

By the Fundamental theorem of Calculus,

(48)

Hence

(49)

But

(50)

Thus

(51)

Equality requires

(52)

for every x, and the endpoint conditions then force u = 0. Therefore y_∗(x) = x is the unique global minimizer.

This proof is stronger than a stationarity calculation. It does not merely produce a necessary differential equation; it directly compares the candidate with every admissible function.

12 Why weak and strong minima are genuinely different

A curve can be arbitrarily close to another curve in C⁰ while remaining far away in C¹.

Consider

(53)

Then

(54)

Thus y_n → 0 uniformly. However,

(55)

so

(56)

for every n. The derivatives do not approach zero at all.

This phenomenon allows a weak minimum to fail to be a strong minimum.

13 Worked counterexample: weak minimum but not strong minimum

Define

(57)

on the fixed-endpoint class

(58)

Consider the candidate

(59)

Its functional value is

(60)

13.1 Step 1: prove weak local minimality

Choose any radius ρ satisfying

(61)

If

(62)

then in particular

(63)

Therefore

(64)

Integrating gives

(65)

Thus y_∗ = 0 is a weak local minimum.

13.2 Step 2: construct C⁰-near curves with lower cost

Use the admissible sequence

(66)

Each y_n satisfies

(67)

and

(68)

So every C⁰ neighborhood of y _∗ contains y_n for sufficiently large n.

But

(69)

Therefore

(70)

Using

(71)

and

(72)

we obtain

(73)

Since

(74)

we have

(75)

for every n.

Thus arbitrarily C⁰-close admissible curves have lower functional value. The zero function is therefore not a strong local minimum.

This example proves concretely that

(76)

The failure is caused by high-frequency perturbations: their amplitudes are small, but their slopes are not.

14 Why the terminology feels reversed at first

It is tempting to think that a “strong neighborhood” should control more quantities than a “weak neighborhood.” Classical variational terminology uses the words in the opposite logical sense.

A weak minimum is required to defeat only competitors that are close in both y and y′. That is a relatively small comparison set.

A strong minimum is required to defeat every competitor whose graph is close in y, even if its derivative differs greatly. That is a larger comparison set and therefore a stronger claim.

One useful memory aid is

(77)

rather than “strong means more derivatives are close.”

15 Strict local extrema

A weak or strong local minimum is called strict if the inequality is strict for every distinct admissible competitor in the relevant neighborhood. For example, a strict strong local minimum satisfies

(78)

for every admissible y≠y_∗ sufficiently close in C⁰.

Strictness concerns uniqueness inside the local neighborhood. It is separate from the weak-versus-strong distinction.

16 Maximum, minimum, and saddle-type behavior

All definitions above have corresponding maximum forms obtained by reversing inequalities.

A stationary function, once stationarity is defined in CV02, need not be a minimum or a maximum. It may have saddle-type behavior: some admissible perturbations increase the functional while others decrease it.

The ordinary-calculus analogy remains useful:

(79)

is not enough to classify x_∗. Likewise, the eventual Euler–Lagrange equation identifies stationary candidates but does not by itself prove minimum or maximum behavior.

This is why the series later develops the second variation, Legendre condition, Jacobi equation, conjugate points, and Weierstrass condition.

17 Extremal versus minimizer: terminology discipline

Classical texts sometimes use the word extremal in more than one way. It may mean an actual function giving a local extremum, or it may mean a solution of the Euler–Lagrange equation even before minimum or maximum character has been established.

To avoid that ambiguity, this series uses the following language whenever possible:

• minimizer or maximizer: a function satisfying the actual optimization inequality;
• stationary function: a function whose first variation vanishes for every admissible first-order variation;
• Euler–Lagrange extremal: used only when referring to the common classical terminology for a solution of the Euler–Lagrange equation.

Thus a later statement that “y_∗ is stationary” should never be silently upgraded to “y_∗ minimizes J.”

18 A minimum value need not be attained

Optimization contains another distinction that is independent of weak and strong local behavior: a functional can have an infimum without possessing a minimizer.

Consider the admissible class of positive constant functions on [0, 1],

(80)

and the functional

(81)

For y_c,

(82)

The values can be made arbitrarily close to zero by choosing c small, so

(83)

However, c = 0 is excluded from 𝒜. No admissible function attains the value zero. Therefore no global minimizer exists.

A minimizing sequence is

(84)

for which

(85)

The limiting function y = 0 lies on the boundary of the admissible class but not inside it.

This simple example foreshadows a major branch of modern variational analysis: existence theory. The direct method of the calculus of variations studies conditions such as compactness, coercivity, closedness of the admissible set, and lower semicontinuity that prevent a minimizing sequence from escaping without converging to an admissible minimizer.

19 The admissible set is part of the physics

In physical applications, the admissible class encodes what trajectories are physically possible before any optimization principle is applied.

19.1 Classical mechanics

Hamilton’s principle uses histories q(t) connecting prescribed endpoint configurations. In its standard fixed-endpoint form,

(86)

The comparison histories are not required to satisfy the equations of motion. They are kinematically admissible alternatives used to test stationarity of the action.

19.2 Geometrical optics

A ray path must connect source and observation points and remain within the physical optical medium. Interfaces or forbidden regions alter the admissible set and can generate additional corner or transversality conditions.

19.3 Elasticity and field theory

Boundary conditions can prescribe displacement, traction, field values, or periodicity. The natural function space is often selected by finiteness of the physical energy rather than by an arbitrary desire for smoothness.

For example, an energy containing

(87)

suggests controlling first derivatives in an integral sense. This is one of the reasons Sobolev spaces become natural in continuum and field theories.

20 Neighborhood choice and physical perturbations

The weak-versus-strong distinction has a physical interpretation. A weak neighborhood excludes perturbations with large slope or velocity changes. A strong neighborhood can include high-frequency, small-amplitude perturbations.

Whether those perturbations are physically meaningful depends on the model. A continuum model that ignores small-scale microstructure may cease to be valid for arbitrarily rapid oscillations. Conversely, if the physical theory does permit such perturbations, a proposed minimum should survive them.

The mathematical topology therefore encodes a statement about which kinds of nearby behavior are being tested.

21 Common misconceptions

21.1 Misconception 1: the formula for J defines the whole problem

It does not. The admissible class, regularity, and constraints are part of the definition.

21.2 Misconception 2: every admissible class is a vector space

Endpoint, integral, and inequality constraints can destroy closure under addition and scalar multiplication.

21.3 Misconception 3: “relative extremum” means weak extremum

It does not. Relative means relative to the admissible set. Weak and strong specify the type of function-space neighborhood used for local comparison.

21.4 Misconception 4: strong minimum means y′ must also be close

The classical definition is the opposite. Strong minimality controls only the graph distance in C⁰ and therefore tests against a larger class of nearby curves.

21.5 Misconception 5: a weak minimum is almost the same as a strong one

The oscillatory counterexample in this entry shows that the difference can be fundamental. Small-amplitude curves can have large derivatives and can reveal directions of decrease invisible inside a small C¹ neighborhood.

21.6 Misconception 6: if the infimum is finite, a minimizer exists

Not necessarily. A minimizing sequence can converge to an object outside the admissible class or fail to converge in the required topology.

21.7 Misconception 7: solving Euler–Lagrange will prove the minimum

Euler–Lagrange is a stationarity condition. Classification and existence are separate questions.

22 A compact definition table

For a functional J : 𝒜→ ℝ and candidate y_∗ ∈𝒜:

23 What CV02 adds

The present entry has defined what it means for a function to be locally optimal, but it has not yet derived a practical necessary condition for such an optimum.

CV02 introduces an admissible perturbation

(88)

and converts the functional into an ordinary scalar function of 𝜖,

(89)

The first variation will be defined by

(90)

The logical bridge will then be the infinite-dimensional analogue of Fermat’s necessary condition:

If an admissible function is a sufficiently regular local extremum, then the first-order change must vanish in every admissible direction.

That statement is the beginning of stationarity. CV03 will prove the Fundamental Lemma needed to turn stationarity into a pointwise equation, and CV04 will derive the Euler–Lagrange equation.

24 Summary

A variational problem consists of more than an integral formula. It requires an admissible class 𝒜, a functional J, and a precise notion of comparison.

For classical first-order problems, the most important neighborhood norms are

(91)

and

(92)

A weak local minimum is tested against sufficiently C¹-close curves. A strong local minimum is tested against the larger set of sufficiently C⁰-close curves. Consequently,

(93)

while the converses need not hold.

The phrase “relative extremum” refers to optimization relative to the admissible set; it is not a synonym for weak extremum. Finally, even a finite infimum need not be attained, so existence of a minimizer is logically separate from stationarity and classification.

These definitions supply the topology and optimization language needed for CV02, where nearby admissible functions become one-parameter variations and the first variation is derived rigorously.

25 References and further reading

1. I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover Publications.
2. R. Weinstock, Calculus of Variations with Applications to Physics and Engineering, Dover Publications.
3. M. Giaquinta and S. Hildebrandt, Calculus of Variations I: The Lagrangian Formalism, Springer.
4. B. Dacorogna, Direct Methods in the Calculus of Variations, Springer.
5. C. Lanczos, The Variational Principles of Mechanics, Dover Publications.