Calculus of Variations: The Jacobi Equation and Conjugate Points

Calculus of Variations: The Jacobi Equation and Conjugate Points

CV11 introduced the second variation and showed that the Legendre condition

(1)

is necessary for a weak local minimum. It also exposed a serious limitation: a positive coefficient of (η′)² does not guarantee that the whole quadratic form is positive. The sign of the second variation can depend on the length of the interval.

CV12 develops the classical remedy. The quadratic functional associated with the second variation has its own Euler–Lagrange equation. That differential equation is the Jacobi accessory equation. Its zeros reveal conjugate points, locations at which the second variation loses strict positivity.

The resulting logic is

(2)

Jacobi’s 1837 work supplied the missing interval test that earlier Euler–Lagrange and Legendre conditions did not provide [7, 5, 6]. Modern treatments include [1, 2, 3, 4].

1 Learning objectives

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

1. derive the Jacobi equation from the reduced second variation;
2. explain why the Jacobi equation is an Euler–Lagrange equation for an accessory functional;
3. define a conjugate point and compute the first conjugate point in simple problems;
4. distinguish an interior conjugate point from a conjugate point occurring exactly at the terminal endpoint;
5. use Jacobi’s factorization identity to understand positivity of the second variation;
6. interpret Jacobi fields as infinitesimal separations between neighboring extremals;
7. connect conjugate points with Sturm–Liouville eigenvalues and the loss of positive definiteness;
8. apply the theory to the harmonic-oscillator action;
9. recognize the geometric meaning of conjugate points in the great-circle example.

2 Starting point: the reduced second variation

Consider the first-order fixed-endpoint functional

(3)

and suppose y_∗ is a sufficiently smooth stationary extremal. For a fixed-endpoint variation

(4)

CV11 obtained

(5)

where all derivatives of F are evaluated along

(6)

Integrating the mixed term by parts gives the reduced form

(7)

with

(8)

The strengthened Legendre condition is

(9)

That condition controls the highest-derivative contribution, but it does not determine the sign of

(10)

after integration over an entire interval. If A is negative enough, a sufficiently slowly varying perturbation can make the integral negative.

3 The accessory functional

The reduced second variation is itself a functional of the perturbation η. Define

(11)

on the class

(12)

The integrand of this auxiliary problem is

(13)

Its partial derivatives are

(14)

Therefore the Euler–Lagrange equation for the accessory functional is

(15)

Dividing by 2 and renaming the unknown function u gives

(16)

or equivalently,

(17)

This is the Jacobi equation or Jacobi accessory equation.

3.1 Why the accessory equation matters

The Jacobi equation is not an unrelated differential equation appended to the original variational problem. It is the Euler–Lagrange equation of the quadratic functional that determines the second-order change in J.

Thus there are two Euler–Lagrange equations in the story:

1. the original Euler–Lagrange equation determines stationary candidates y_∗;
2. the Jacobi equation determines critical directions of the second variation around y_∗.

This distinction is fundamental.

4 The distinguished Jacobi field

The Jacobi equation is a homogeneous linear second-order ODE. Under the usual regularity assumptions, there is a unique solution for any two initial data.

To study conjugacy to the left endpoint a, choose the normalized solution

(18)

The normalization u′(a) = 1 is only a convenient scale choice. Any nonzero value of u′(a) produces the same zero locations.

We will call u the distinguished Jacobi field from a.

5 Conjugate points

Definition. A point c > a is conjugate to a along the extremal y_∗ if there exists a nontrivial solution u of the Jacobi equation satisfying

(19)

Equivalently, with the normalized distinguished field defined above, the conjugate points are precisely its zeros after a.

The first conjugate point c₁ is the smallest c > a for which

(20)

If no such point exists, we say that there is no conjugate point to a on the interval under consideration.

6 Jacobi fields from neighboring extremals

The preceding geometric statement can be derived directly. Suppose

(21)

is a smooth one-parameter family of extremals of the original functional. For every α,

(22)

Define the infinitesimal separation field

(23)

Differentiate the Euler–Lagrange equation with respect to α. The first term gives

(24)

The second term gives

(25)

Assuming the mixed derivatives agree,

(26)

we obtain

(27)

Expand the total derivative of the first product:

(28)

The F_yy′u′ terms cancel, leaving

(29)

Using A and R,

(30)

which is exactly the Jacobi equation.

Thus the Jacobi equation is also the linearized Euler–Lagrange equation. It describes how an extremal changes when we move infinitesimally to a neighboring extremal.

7 Jacobi’s factorization identity

The deepest algebraic connection between the Jacobi equation and positivity of the second variation is a factorization identity.

Suppose u is a nonzero solution of

(31)

on an interval where u≠0. Then

(32)

This identity is sometimes presented as a Jacobi or Picone-type transformation.

7.1 Derivation

First,

(33)

Therefore

Ru22	= R2	(34)
	= R(η′)2 − 2Rηη′ + R2η2.	(35)

Next,

= ′η2 + 2Rηη′.

(36)

The mixed terms cancel when the two expressions are added. The remaining coefficient of η² is

(37)

But

R2 + ′	=	(38)
	= A,	(39)

where the Jacobi equation was used in the last step. This proves the identity.

8 Why absence of conjugate points implies positivity

Assume the strengthened Legendre condition

(40)

and suppose the distinguished Jacobi field from a has no zero on

(41)

Because the normalized field satisfies u(a) = 0, the ratio u′∕u is singular at a. A clean proof therefore applies the factorization first on

(42)

and then lets 𝜀 → 0⁺.

For a fixed-endpoint variation, the boundary term vanishes at b because

(43)

The limiting contribution at a also vanishes for sufficiently regular admissible variations with η(a) = 0. Hence

(44)

Under the regular classical hypotheses, if no conjugate point occurs in (a,b], this representation yields strict positivity for every nonzero admissible variation.

The key point is that the strengthened Legendre condition supplies

(45)

while the absence of conjugate points supplies a Jacobi field u that does not vanish inside the interval. Both are needed.

9 What happens at the first conjugate point?

Let c₁ be the first conjugate point to a. Then there is a nontrivial Jacobi field u satisfying

(46)

Multiply the Jacobi equation

(47)

by u and integrate from a to c₁:

(48)

Integrating the first term by parts gives

(49)

Since u vanishes at both endpoints,

(50)

Thus, if the terminal point b is exactly the first conjugate point,

(51)

the second variation has a nontrivial null direction. Strict positivity is lost.

This distinction is important:

• If b < c₁, the second variation is positive definite under the regular Jacobi hypotheses.
• If b = c₁, a nonzero Jacobi field gives δ²J = 0; a non-strict minimum may still be possible.
• If b > c₁, the quadratic form acquires a negative direction; a weak local minimum is impossible under the classical regular hypotheses.

The endpoint case is therefore not the same as having a conjugate point strictly inside (a,b).

10 Jacobi’s necessary condition for a minimum

A classical form of Jacobi’s necessary condition can now be stated.

theorem (Jacobi necessary condition, regular scalar case). Suppose y_∗ is a sufficiently smooth weak local minimizer of

(52)

with fixed endpoints, and suppose the strengthened Legendre condition

(53)

holds along y_∗. Then no point of the open interval

(54)

can be conjugate to a along y_∗.

If b itself is conjugate to a, strict positivity of the second variation fails. Additional information is needed to decide whether a non-strict minimum remains.

This theorem is a genuinely global-in-the-interval condition. It cannot be obtained by inspecting F_y′y′ at one point.

11 A Sturm–Liouville viewpoint

Define the differential operator

(55)

With fixed endpoints,

(56)

integration by parts gives

(57)

The Jacobi equation is exactly the zero-eigenvalue equation

(58)

More generally, consider the Sturm–Liouville eigenvalue problem

(59)

The sign of the smallest eigenvalue controls the sign of the quadratic form:

(60)

for every nonzero admissible variation,

(61)

and

(62)

The first conjugate endpoint is therefore the interval length at which the lowest Dirichlet eigenvalue reaches zero.

12 Constant coefficients: the three basic cases

Suppose

(63)

The Jacobi equation becomes

(64)

Three qualitatively different behaviors occur.

12.1 Case 1: A > 0

Then

(65)

and the distinguished solution is proportional to

(66)

It has no zero after a, so there is no finite conjugate point.

12.2 Case 2: A = 0

Then

(67)

and the distinguished solution is

(68)

Again there is no later zero.

12.3 Case 3: A < 0

Write

(69)

Then

(70)

and the normalized solution is

(71)

Its first later zero occurs at

(72)

This oscillatory case is the model for the harmonic-oscillator action.

13 Worked example: harmonic-oscillator action

Consider

(73)

with fixed endpoints

(74)

The zero path

(75)

is stationary. Since the action is already quadratic,

(76)

Hence

(77)

The Jacobi equation is

(78)

or

(79)

With

(80)

we obtain

(81)

The conjugate times are

(82)

The first is

(83)

13.1 Short intervals: 0 < T < T_c

No conjugate point lies in (0,T]. Under m > 0, Jacobi’s factorization gives

(84)

for every nonzero admissible η. The stationary path is therefore positive to second order.

13.2 Threshold interval: T = T_c

The Jacobi field itself is admissible because

(85)

For this nonzero direction,

(86)

In fact, because the action is exactly quadratic,

(87)

for every amplitude 𝜖. Thus q = 0 is not a strict minimizer, but the appearance of a null direction at the endpoint does not by itself create a negative direction.

13.3 Long intervals: T > T_c

Choose the admissible trial direction

(88)

Then

(89)

Using

(90)

we find

δ2S[0; η]	= 2 −	(91)
	= .	(92)

If

(93)

then

(94)

The stationary path has lost minimizing character.

14 Spectral interpretation of the oscillator threshold

For the harmonic oscillator,

(95)

The Dirichlet eigenfunctions on [0,T] are

(96)

with eigenvalues

(97)

The first eigenvalue is

(98)

It changes sign exactly when

(99)

Thus the first conjugate time and the zero crossing of the lowest eigenvalue are the same event viewed from two complementary perspectives.

15 A Wronskian identity

Let u and v be two Jacobi fields satisfying

(100)

Multiply the first equation by v, the second by u, and subtract. The Auv terms cancel:

(101)

This is

(102)

Therefore

(103)

This weighted Wronskian identity is the one-dimensional analogue of a conserved symplectic pairing between neighboring solutions. It is also useful when comparing Jacobi fields and proving uniqueness properties of their zeros.

16 Riccati form of the Jacobi equation

Where a Jacobi field u is nonzero, define

(104)

Differentiate:

(105)

Using

(106)

and the identity used in the Jacobi factorization,

(107)

so the Jacobi equation gives

(108)

This is a scalar Riccati equation. Riccati equations later reappear in optimal control, linear-quadratic regulation, estimation, and stability theory. Here the singularity of W when u crosses zero is another way to detect a conjugate point.

17 Geometric preview: conjugate points on a sphere

The language of “Jacobi fields” is especially natural in Differential Geometry. Along a geodesic, a Jacobi field measures the infinitesimal separation of neighboring geodesics.

For a sphere of radius r, the Normal Jacobi equation along a unit-speed great circle is

(109)

With

(110)

we obtain

(111)

The first later zero occurs at

(112)

That point is the antipode of the starting point. Great circles leaving one point in slightly different directions reconverge at the antipode. The antipode is therefore conjugate to the starting point.

This example previews the geodesic material developed later in the series: conjugate points express a loss of local minimizing behavior caused by the focusing of neighboring geodesics.

18 What Jacobi’s condition does and does not prove

It is important to separate several logical statements.

18.1 Legendre

The condition

(113)

is a local pointwise necessary condition for a weak minimum.

18.2 Jacobi

The absence of conjugate points is an interval-wide necessary condition. Under the strengthened Legendre condition and standard regularity assumptions, absence of conjugate points through the terminal endpoint also gives positivity of the accessory quadratic form.

18.3 Still not the whole story

Even a positive second variation is fundamentally a local quadratic statement. Establishing a strong minimum, a global minimum, or sufficiency under broad hypotheses requires additional tools. CV13 develops strengthened Legendre, Weierstrass-type excess conditions, and the distinction between weak and strong extrema.

19 Common mistakes

• Confusing the original Euler–Lagrange equation with the Jacobi equation. The first determines stationary candidates; the second governs infinitesimal perturbations about a stationary candidate.
• Using the wrong coefficients. The Jacobi equation uses the reduced second-variation coefficients A = F_yy − dF_yy′∕dx and R = F_y′y′ evaluated along y_∗. They are not arbitrary coefficients from the original integrand.
• Forgetting that R > 0 is part of the regular Jacobi theory. If R vanishes or changes sign, the classical argument requires modification.
• Calling every zero a conjugate point without fixing the left endpoint. Conjugacy is relative to a starting point; one requires a nontrivial Jacobi solution that vanishes at both locations.
• Treating b = c₁ the same as c₁ < b. At the first conjugate endpoint there is a null direction; beyond it a negative direction appears.
• Assuming no conjugate points proves a global minimum. The Jacobi criterion controls the second variation locally around a stationary extremal.
• Forgetting admissibility. Jacobi fields used as test directions must satisfy the appropriate linearized endpoint or constraint conditions.
• Ignoring the interval. The same integrand and same extremal can be minimizing on a short interval and fail to minimize on a longer interval.
• Using the Riccati transform through a zero of u. The quantity u′∕u becomes singular exactly at a conjugate point.

20 A compact Jacobi workflow

For a regular fixed-endpoint scalar problem:

1. Find a stationary extremal y_∗ from the original Euler–Lagrange equation.
2. Compute the reduced second-variation coefficients

   (114)

3. Check the strengthened Legendre condition R > 0 if you intend to use the regular Jacobi test.
4. Form the Jacobi equation

   (115)

5. Solve the normalized initial-value problem

   (116)

6. Locate the first later zero c₁.
7. Compare b with c₁:

   (117)

8. Remember that this is still local classification information; proceed to sufficiency theory when stronger conclusions are needed.

21 Historical perspective

Euler and Lagrange had established the differential equation now called the Euler–Lagrange equation, and Legendre had found a second-order sign condition. Yet those results did not fully decide whether a stationary curve was actually minimizing over an entire interval.

Jacobi addressed this gap in 1837 [7]. His paper, “Zur Theorie der Variations-Rechnung und der Differential-Gleichungen,” used the accessory equation to make the interval length part of the classification problem and supplied the precursor of the modern conjugate-point criterion. Classical developments of the theory can be found in Bliss and Bolza [5, 6].

The conceptual advance can be summarized as

(118)

Jacobi’s equation supplies the missing interval-wide information.

22 What CV13 adds

CV11 and CV12 have now developed the local second-order theory:

(119)

CV13 turns to stronger sufficiency questions. It distinguishes weak and strong minima, introduces the Weierstrass excess function, and explains why controlling only infinitesimal slope changes is not enough for a strong minimum.

Summary

For a stationary extremal, write the reduced second variation as

(120)

Its Euler–Lagrange equation is the Jacobi accessory equation

(121)

A point c > a is conjugate to a if a nontrivial Jacobi field vanishes at both a and c. Under the regular strengthened-Legendre hypotheses, the first conjugate point marks the loss of strict positivity of the second variation:

(122)

Jacobi’s factorization

(123)

explains why the absence of zeros matters. The harmonic oscillator provides the model example: its first conjugate time is

(124)

exactly the threshold at which the lowest eigenvalue of the second-variation operator crosses zero.

References