Calculus of Variations: Higher-Order Variational Problems

Up to CV07, the basic integral functional depended on a function and its first derivative,

(1)

Many important physical models depend on higher derivatives. Bending energy depends on curvature and therefore, in a small-slope beam model, on y′′. Smooth trajectory design can penalize acceleration or jerk. Higher-gradient field theories and regularization methods likewise introduce derivatives beyond first order. The calculus of variations handles these problems by the same central idea used before: vary the function, integrate derivatives off the variation, and apply the Fundamental Lemma. The new feature is that integration by parts must be repeated. Standard treatments are given in [1, 2, 3].

This article develops the classical higher-order Euler–Lagrange equation, with particular attention to the second-order case

(2)

because it already contains the essential structure: a fourth-order differential equation and two independent endpoint variations, η and η′.

1 Why higher derivatives change the problem

Suppose the functional contains y′′. A perturbation

(3)

implies

(4)

Therefore the first variation contains three kinds of terms:

(5)

The term multiplying η′ requires one integration by parts. The term multiplying η′′ requires two. This repeated transfer of derivatives is the source of both the higher-order Euler–Lagrange equation and the enlarged boundary-term structure.

2 Admissible variations for a second-order problem

There are several possible endpoint specifications. The cleanest theorem first fixes both the value and slope at each endpoint:

(6)

An admissible variation must preserve all four conditions. Thus

(7)

and

(8)

This is the higher-order analogue of the fixed-endpoint condition in CV04. Because a fourth-order differential equation will emerge, four scalar boundary conditions are also exactly what is normally needed to determine a unique classical solution.

3 Derivation for F(x,y,y′,y′′)

Start from

(9)

Integrate the F_y′η′ term once:

(10)

Now integrate the F_y′′η′′ term once:

(11)

The remaining integral still contains η′, so integrate by parts again:

(12)

Collect the endpoint terms and the interior terms:

δJ	= ab	(13)
	+ ∫ abη dx.	(14)

This formula is worth separating into two pieces:

(15)

and

(16)

If y and y′ are fixed at both endpoints, then η = η′ = 0 there, so the complete boundary term vanishes. Stationarity for every admissible η gives

(17)

By the Fundamental Lemma,

(18)

This is the second-order Euler–Lagrange equation.

4 Classical theorem for second-order functionals

Theorem. Let

(19)

with F sufficiently smooth, and suppose y_∗ is a sufficiently smooth local extremum among admissible functions satisfying fixed values of y and y′ at a and b. Then y_∗ satisfies

(20)

throughout the interior of the interval.

Proof. The local-extremum hypothesis implies δJ[y_∗; η] = 0 for every admissible variation. The two integrations by parts derived above produce the boundary term and interior term. Fixed y and y′ imply η = η′ = 0 at both endpoints, eliminating the boundary term. The Fundamental Lemma then forces the interior coefficient to vanish pointwise. □

As before, this theorem supplies a necessary stationarity condition. It does not by itself prove that a solution is a minimum.

5 The general nth-order Euler–Lagrange equation

Now consider

(21)

The first variation is

(22)

where y⁽⁰⁾ = y and η⁽⁰⁾ = η.

Integrating the term containing η^(j) by parts exactly j times transfers all derivatives from η onto F_y^(j). The interior result is

(23)

Written out,

(24)

The alternating signs are not arbitrary. Each integration by parts introduces one minus sign.

For completely fixed endpoint data through derivative order n − 1, the admissible variations satisfy

(25)

6 The general boundary structure

Repeated integration by parts also produces the boundary expression

(26)

This formula looks complicated, but its meaning is simple: an nth-order variational problem can contain the independent endpoint variations

(27)

If any corresponding endpoint datum is free, the coefficient multiplying that independent variation must satisfy a natural boundary condition.

7 Natural boundary conditions in the second-order case

Return to

(28)

At a free endpoint the boundary expression is

(29)

If both y and y′ are free there, then η and η′ are independently arbitrary. Therefore stationarity requires

(30)

and

(31)

If only y is free but y′ is fixed, then only the second condition involving the coefficient of η is required. If only y′ is free, then only F_y′′ = 0 is required.

8 Example: pure curvature energy

Consider

(32)

subject to

(33)

Here

(34)

so

(35)

The second-order Euler–Lagrange equation gives

(36)

or

(37)

Therefore every stationary curve is cubic:

(38)

Apply the four endpoint conditions. From y(0) = 0 and y′(0) = 0,

(39)

The two conditions at x = 1 give

C + D	= 1,	(40)
2C + 3D	= 0.	(41)

Solving,

(42)

Hence

(43)

8.1 Why this stationary curve is actually the unique global minimum

Let any admissible competitor be written

(44)

where

(45)

Then

J[y]	= ∫ 01(y ∗′′ + η′′)2dx	(46)
	= J[y∗] + ∫ 01y ∗′′η′′dx + ∫ 01(η′′)2dx.	(47)

Integrating the cross term twice gives

(48)

The boundary term vanishes because η = η′ = 0, and the interior term vanishes because y_∗′′′′ = 0. Thus

(49)

Equality requires η′′ = 0, so η is linear. The four homogeneous endpoint conditions then force η ≡ 0. Therefore the cubic is the unique global minimizer in this admissible class.

9 Connection to Euler–Bernoulli beam bending

For a slender beam in the small-deflection approximation, a standard potential-energy functional is [5]

(50)

where E is Young’s modulus, I is the second moment of area, and q(x) is a distributed transverse load.

Here

(51)

for constant EI. The higher-order Euler–Lagrange equation becomes

(52)

For constant flexural rigidity,

(53)

This fourth-order beam equation is therefore not an unrelated engineering formula; it is a direct Euler–Lagrange equation for bending energy minus load potential.

At a free beam end, both η and η′ are unrestricted. The natural conditions are

(54)

and

(55)

Up to sign convention, these are zero bending moment and zero shear force. This gives a physical interpretation to the abstract higher-order boundary coefficients.

10 Mixed essential and natural boundary conditions

Consider the beam functional above with the left end clamped:

(56)

and the right end free. At x = 0,

(57)

so no natural conditions are generated there. At x = L, η(L) and η′(L) are arbitrary, so

(58)

and

(59)

The fourth-order differential equation therefore receives two essential conditions at the clamped end and two natural conditions at the free end.

This counting principle is useful:

A classical 2nth-order Euler–Lagrange equation normally needs 2n scalar boundary conditions. Essential conditions prescribe endpoint data directly; natural conditions arise from the surviving boundary variation.

11 Example: slope and curvature penalty

Consider

(60)

Then

(61)

Therefore

(62)

or

(63)

The characteristic equation is

(64)

so

(65)

Thus

(66)

Adding a first-derivative penalty changes the cubic minimum-curvature family into a combination of polynomial and hyperbolic terms.

12 Third-order dependence: minimum jerk

A trajectory-design problem may penalize jerk, the third derivative of position. Consider

(67)

The integrand depends only on x′′′, so the n = 3 Euler–Lagrange equation is

(68)

Hence

(69)

The stationary trajectory is therefore a polynomial of degree at most five.

Suppose position, velocity, and acceleration are fixed at both endpoints:

x(0)	= 0,	ẋ(0)	= 0,	ẍ(0)	= 0,	(70)
x(T)	= D,	ẋ(T)	= 0,	ẍ(T)	= 0.	(71)

Introduce normalized time

(72)

and normalized position

(73)

Solving the six endpoint conditions gives

(74)

Thus

(75)

This quintic profile is widely known as the minimum-jerk trajectory. Minimum-jerk models also have an important history in studies of smooth human movement [6].

13 Higher-derivative mechanics

A mechanical action may depend on acceleration,

(76)

The stationarity condition is

(77)

The boundary term is

(78)

This suggests two momentum-like quantities,

(79)

and

(80)

They are conjugate to endpoint variations in position and velocity, respectively. In advanced hamiltonian formulations of nondegenerate higher-derivative mechanics these quantities lead toward the Ostrogradsky construction, which has important stability implications. That subject lies beyond the present classical variational derivation.

14 Regularity and order counting

The appearance of higher derivatives in the functional raises the regularity requirements. A classical derivation assumes enough smoothness that all derivatives generated by repeated integration by parts exist and are continuous as needed.

For

(81)

the Euler–Lagrange equation can contain y′′′′. Thus even though the original integrand explicitly contains derivatives only through second order, the stationarity equation is generally fourth order.

More generally, dependence on y⁽ⁿ⁾ commonly produces a differential equation of order as high as 2n. Degeneracies can reduce the actual order. For example, if the integrand is linear rather than quadratic in the highest derivative, cancellation may occur.

15 Necessary versus sufficient conditions

The generalized Euler–Lagrange equation remains a necessary condition for a smooth local extremum under the stated assumptions. It is not automatically sufficient.

The pure curvature functional

(82)

was easy to classify because it is a positive quadratic functional, and the direct difference argument established a global minimum. A general higher-order functional can be indefinite, nonconvex, or degenerate. Classification requires second-variation and sufficiency theory, developed later in CV11–CV13.

16 Common mistakes

• Stopping after one integration by parts. A term multiplying η′′ must be integrated by parts twice before the Fundamental Lemma can be applied directly.
• Forgetting the η′ boundary term. A second-order functional has two independent endpoint variation channels.
• Fixing y but silently assuming y′ is fixed. These are different admissible classes and lead to different natural boundary conditions.
• Missing the alternating signs. The generalized equation contains (−1)^j because each integration by parts contributes another minus sign.
• Assuming second-order dependence gives a second-order ODE. Quadratic dependence on y′′ typically produces a fourth-order Euler–Lagrange equation.
• Calling every fourth-order equation a beam equation. The beam interpretation depends on the specific energy functional and constitutive model.
• Confusing stationarity with minimality. Higher-order Euler–Lagrange equations still provide necessary conditions unless additional arguments are supplied.

17 A compact workflow

For a higher-order functional, use the following sequence:

1. Identify the highest derivative y⁽ⁿ⁾ appearing in F.
2. Determine which endpoint values among y,y′,…,y⁽ⁿ⁻¹⁾ are fixed and which are free.
3. Form the first variation

4. Integrate the jth term by parts j times.
5. Collect the interior coefficient of η.
6. Apply the Fundamental Lemma to obtain the generalized Euler–Lagrange equation.
7. Keep the complete boundary expression and use endpoint freedom to derive natural conditions.
8. Solve the resulting higher-order boundary-value problem.
9. Separately determine whether the stationary solution is actually a minimum, maximum, or saddle.

18 What CV09 adds

CV08 has enlarged the derivative order while keeping the admissible functions otherwise unconstrained except for endpoint data. CV09 introduces a different complication: integral constraints and isoperimetric conditions. Those problems require variational Lagrange multipliers and an augmented integrand.

The conceptual progression is therefore

(83)

Summary

For

(84)

a smooth stationary function satisfies

(85)

For the important second-order case,

(86)

with boundary term

(87)

These formulas explain why bending-energy problems produce fourth-order equations, why free beam ends produce moment and shear natural conditions, why minimum-curvature interpolation is cubic, and why minimum-jerk trajectories are quintic.

References