Variational Analysis: Critical Extremals and Sturmian Extensions

By Marston Morse

This text presents extended separation, comparison, and oscillation theorems that replace the classical analysis of Legendre, Jacobi, Hilbert, and others. Its analysis of related quadratic functionals shows how critical extremals can substitute for minimizing extremals.
Author Marston Morse is renowned for his development of a version of variational theory with applications to equilibrium problems in mathematical physics—the theory known as Morse theory, which forms a vital role in global analysis. He begins this treatment of variational analysis with an extended investigation of critical extremals that proceeds to quadratic index forms, advanced and free. Additional topics include focal conditions and Sturm-like theorems, general boundary conditions, and prestructures for characteristic root theory. A helpful pair of appendixes include supplementary information on free linear conditions and subordinate quadratic forms and their complementary forms.

• Language: English
• Publisher: Dover Publications
• Release date: Feb 27, 2013
• ISBN: 9780486153223

Book preview

INDEX

Introduction

We shall single out from each chapter special theorems or concepts which are particularly significant and indicate in what respect they are significant. These comments should be read after reading the chapter in question but before reading the following chapter.

CHAPTER 1. Under the conditions of Theorem 7.4

(1)

Classical proofs of this theorem require that the mappings,

(2)

of [a¹,a²] into R be of class C¹. It suffices that the mappings (2) be continuous. Similar remarks should be made concerning the remaining theorems of Section 7. These Theorems affect the whole body of Sturm-like theorems.

CHAPTER 2. Let Cr be general end point conditions of Section 8 with r > 0. The functional (J,Cr) whose values J(χ,α) are given by (8.19), is the sum of an integral and an external function Θ whose values Θ(α) are determined by the end points (X¹(α), Y¹(α)) and (X²((α), Y²(α)) of the graph of χ.

Theorem 10.1 shows that the second variation of (J,Cr) whose values are given by the right member of (10.13) has a similar structure. The second variation is the sum of an integral and of an external quadratic form bhkuhuk where the r-tuple u = (u1, . . . , ur) is uniquely determined, as (10.14) shows, by the end points of the graph of the variation η.

This similarity of structure of (J,Cr) and its second variation motivated the choice of the representation (8.19) of the functional (J,Cr.

CHAPTER 3. With each critical extremal g of (Jλ,Cr), Definition 14.2 associates a quadratic index form Qλ whose index (Theorem 14.1) equals the count of negative characteristic roots of Conditions (11.8), when r > 0, and of conditions (12.2) when r = 0. This index form is termed derived because its definition depends on an antecedent critical extremal g. It is replaced in Section 15 by a similarly structured free index form whose definition is independent of any antecedent critical extremal g.

In the problem of relating the geodesics g joining two fixed points A and B, on a differentiable manifold Mn, to the homological characteristics of Mn, a derived index form is associated with each geodesic g joining A to B, and enables one to assign local homological characteristics to each g. This will be elaborated in our second volume on variational topology. See the analogous treatment of critical points of a nondegenerate function f on a differentiable manifold. Morse and Cairns [1].

CHAPTER 4. Conditions Wr(λ) combine the Jacobi differential equations with boundary conditions 0 < r ≤ 2m. For each λ ∈ R, conditions Wr(λ) : (15.0) are uniquely determined by giving ωλ(x,η,ζ) as in Appendix I, by prescribing a 2m × r of rank r and an r-square symmetric "comparison" matrixbhk . If λ ranges over R, a system of conditions Wr(λbk remain invariable, is called a canonical system Wr of dimension r.

With each system Wr and value σ ∈ R : (15.2) and a free index form Qσ, whose index and nullity are equal (Theorem 15.3) and are given by Index Theorem 15.2. A nonnull solution of conditions Wr(σ. The index form Qσ is a technical aid in this study. The exceptional case r = 0 is also considered.

CHAPTER 5. The focal points of free focal conditions Vr : (17.6) and (17.7) are introduced in Definition 17.4. The Focal Point Theorem⁴ 17.3 gives the count of focal points of Vr, r determined by Vr in Definition 17.6. When r = 0, Theorem 17.3 (restated as Theorem 17.4) gives the count of conjugate points of x = a¹, at x = a² or in (a¹,a²) in terms of characteristic roots of the canonical system W0 of Conditions (15.1).

In Section 18 the theory of focal points is identified with the theory of von Escherich families of mutually conjugate solutions of the system of DE

(3)

and leads to our Separation Theorem.

CHAPTER 6. The "Extended Separation⁵ Theorem" 20.1 implies the classical Sturm Separation Theorem when m = 1 and replaces it when m > 1. It affirms the following. If two von Escherich families F of the DE (3) have exactly ρ linearly independent solutions in common, then the count of focal points of F in any relatively compact subinterval τ by at most m − ρ.

In Section 21 we compare the focal points in an interval (a¹,d] of two sets of focal conditions Vr , as defined in (17.6) and (17.7). When r = ρ = 0 the comparison reduces to a comparison of conjugate points. In our first comparison theorem, Theorem 21.1, the hypotheses, when m = 1, are similar to classical hypotheses. When m = 1 certain classical theorems such as Reid’s Theorem (Theorem 1.29 of Swanson) are extended in Section 22. Our second comparison theorem is the Nuclear Comparison Theorem 21.4. It has hypotheses which are definitely weaker than those of Comparison Theorem 21.1 but imply the same conclusions. When m = 1 Leighton’s Theorem (Theorem 1.4, page 4 of Swanson) is of the same nature as our Nuclear Comparison Theorem. Leighton’s Example 1, on page 6 of Swanson, is used by us for the same end. See Section 21.

CHAPTER 7. The Oscillation Theorem 24.1 differs in character from the classical oscillation theorems to which we shall turn in Section 37. Theorem 24.1 gives the exact value of the difference

(4)

where index Wr denotes the count of negative characteristic roots of a canonical system Wr for which λ = 0 is not a characteristic root, and index W0 denotes the count of conjugate points of x = a¹ in (a¹,a²) of the underlying DE (3). The proof of Theorem 24.1 depends upon an auxiliary theorem, Theorem 25.1, on quadratic forms, proved in Appendix II. Many other applications of Theorem 25.1 exist. We apply Theorem 24.1 to the periodic case, as defined in Section 27.

CHAPTER 8. In Section 29, general selfadjoint BC (boundary conditions) associated with prescribed selfadjoint DE are defined, as well as the equivalence (Definition 29.1) of any two such sets of BC. Theorem 31.1 then implies that an arbitrary set of selfadjoint BC with an accessory r-plane χr, (Definition 31.1), is equivalent, when r > 0, to some set of BC of form (29.1). When r = 0 it is trivial that selfadjoint BC are equivalent to the conditions, η(a¹) = η(a²) = 0. The Conditions (15.0) and (15.1) are selfadjoint, and (up to an equivalence of BC) are general in form, as we show in Section 31.

CHAPTER 9. r r(λ) introduced in Section 32, include the systems Wr of canonical conditions Wr(λr have infinitely many characteristic roots. It is a corollary that the canonical systems Wr of Section 15 have infinitely many such roots.

In Section 37 Oscillation Criteria are given in a set of theorems whose proofs will be presented in a separate paper. A set of DE of the form (3) is given for x ∈ (0, ∞) and conditioned as in Appendix I. The DE are termed oscillatory if the point x = 1 has infinitely many conjugate points following x = 1. A D¹-mapping x → y(x) : [1, ∞) → Rm is called a thread and the condition that

(5)

a thread condition.

One of the corollaries of the general theorems presented in Section 37 for m ≥ 1, concerns the DE

(6)

where the mapping x → a(x) is continuous. See Swanson, Chapter 2. The corresponding Thread Condition has the form

(7)

The corollary takes the form:

COROLLARY. A necessary and sufficient condition that the DE (6) be oscillatory is that the Thread Condition (7) be satisfied by some thread γ.

The Leighton-Wintner Theorem that the DE (6) is oscillatory if

is implied by the corollary, on taking the thread as the mapping x → y(x) ≡ 1.

Note. The major class of admissible curves is that of x-parameterized curves of class D¹, including broken extremals. This class of curves is adequate for a first simple presentation of the new theorems and structure. It is preferred to a choice of Hilbert Space from the different Banach Spaces which could be profitably used in variational analysis, because it leaves that choice fully open, while preserving a close connection with the classical theory. Index forms are definable in terms of broken extremals and permit an easy transition from the Sturmian properties of a critical extremal to properties which are purely topological. In our presentation of Variational Topology which is to follow, admissible curves will range from the merely continuous to the real analytic and include curves definable in the terminology of Lebesgue.

Part I

CRITICAL EXTREMALS

CHAPTER 1

Minimizing Extremals g: Fixed Endpoints

This chapter gives a review of classical results found in varying forms in the works of Bliss, Carathéodory, Bolza., Hadamard, Tonelli, and other writers on variational theory. Present day notation is introduced and classical proofs are freely modified.

1. The Euler Equations

To properly condition admissible curves several definitions are needed.

Mappings of Class D⁰. Let [a¹,a²J be a closed interval of the axis R of real numbers. Two intervals of the R-axis are termed nonoverlapping if their intersection is empty or a point. A mapping x → h(x) : [a¹,a²] → R will be said to be of class D⁰ if [a¹,a²] is the union of a finite set of closed, nonoverlapping subintervals I, of each of which h is continuous and at each endpoint x0 of which h has a finite limit as x.

Mappings of Class D¹. A mapping x → h(x) : [a¹,a²] → R will be said to be of class D¹ if h is continuous and if [a¹,a²] is the union of a finite set of nonoverlapping closed subintervals I of each of which h’ exists and is continuous, and at each endpoint x0 of which h’ has a finite limit when x.

The Preintegrand f. Let (x, y1, . . . , ym) be written as (x,y) and be the set of rectangular coordinates of a point (x,y) in a euclidean space Em+1 of dimension m + 1. Let X be an open connected subset of Em+1. Let p = (p1, . . . , pm) be an arbitrary point in the m-fold Cartesian product Rm of R. For brevity we write

(1.1)

and consider a real-valued function⁶

(x,y,p) → f(x,y,p) : X × Rm → R

such that f , are of class C². These conditions are satisfied if f is of class C³. It is not sufficient that f be merely of class C² if the solution Y(x : c,a,b) : (2.14) of the Euler equations is to have continuous second order partial derivatives with respect to the initial parameters c, a, b. Such differentiability is essential in several places.

The "Mapping" g and the "Curve" g. Let there be given mappings

(1.2)

each of class D¹. A mapping

x → g(x) : [a¹,a²] → Rm

is thereby defined, and said to be of class D¹. The graph of g is given in X and will be called the curve g. On the curve g

(1.3)

Let there be given a second mapping

(1.3)′

of class D¹. We suppose that the corresponding curve γ joins the endpoints of g in X. We then term γ admissible relative to the base curve g.

The Integral J. We shall compare the Riemann integral

(1.4)

along the curve γ with the integral J(g) along g and prove the following theorem. In J(γ), γ is understood to be either the curve γ or the mapping γ.

THEOREM 1.1. In order that g (of class D¹) afford a minimum to J relative to neighboring admissible curves γ,⁷ it is necessary that there exist constants c1 ... , cm such that

(1.5)

for each x ∈ [a¹,a²] at which g′(x) is defined.

For each i on the range 1, ... , m let x → ηi(x) be a mapping of [a¹,a²] into R of class D¹ such that ηi(a¹) = ηi(a²) = 0. A curve on which

(1.6)

joins g’s endpoints and is in X e sufficiently small. Let J(e) be the value of J along this curve. Observe that

(1.7)⁸

where the superscript x means evaluation for (x,y,p) = (x,g(x),g′(x)) when g′(x) exists, with p the left or right limit of g′(x) at other points of [a¹,a²]. The right member of (1.7) is called the first variation of J along g induced by the variation η. Since J(e) ≥ J(0) by hypothesis, we conclude that J′(0) = 0.

in (1.7) by parts, one finds that

(1.8)

Theorem 1.1 will follow from (1.8) and the modified DuBois Reymond Lemma 1.1.

LEMMA 1.1. If x → ϕ(x) : [a¹,a²] → R is a mapping which is of class D⁰ and if

(1.9)

for each mapping x → ζ(x) : [a¹,a²] → R of class D¹ which vanishes at a¹ and a², then ϕ has the same value c at each point x ∈ [a¹,a²] at which ϕ is continuous.

Let c . The mapping

of [a¹,a²] into R is admissible as ζ in Lemma 1.1. For this ζ, (1.9) takes the form

from which it follows that ϕ(x) = c at points of continuity of ϕ.

To complete the proof of Theorem 1.1 we prefer one of the integers, say k, on the range 1, ... , m. For i ≠ k we take ηi identically zero on [a¹,a²]. The mapping

(1.10)

of [a¹,a²] into R satisfies the conditions on ϕ in Lemma 1.1, so that (1.5) holds for i = k by virtue of Lemma 1.1. Since k can be prescribed on the range 1, ... , m, Theorem 1.1 follows.

COROLLARY 1.1. Under the hypotheses of Theorem 1.1 the following is true:

(i) Each subarc of g which is of class⁹ C¹ satisfies the Euler equations

(1.11)

(ii) At a corner of g with x coordinate x0

(1.12)¹⁰

(iii) If g is of class C¹ and if the m-square determinant

(1.13)

then g is of at least class C².

Statements (i) and (ii) are immediate consequences of Theorem 1.1. Mason and Bliss [1] prove (iii) as follows. Given g, the m conditions,

(1.14)

on the m-tuple z for x ∈ [a¹,a²], have a solution z(x) = g′(x) for x ∈ [a¹,a²]. By virtue of (1.13) the classical implicit function theorems imply that the solution x → z(x) = g′(x) is of class C¹.

At the end of the next section we shall see that g is of class C³ for x ∈ [a¹,a²] when (1.13) holds.

Definition 1.1. An Extremal. A C²-mapping

(1.15)

satisfying the Euler equations (1.11) defines an x-parameterized curve y = y(x) called an extremal. The extremal is the "graph" of the solution (1.15). We shall call the extremal (1.15) primary to distinguish it from secondary extremals of Section 4.

Exercise

1.1. Suppose that g is an extremal and that the variation η, introduced in (1.6), has arbitrary end values η(a²) and η(a¹). Show that the corresponding "first variation" of J along g, as given by (1.7), has the value

(1.16)

2. The Existence of Extremals Near g

There is given an x-parameterized extremal g in X of class C² for x ∈ [a¹,a²].

We shall assume that (1.13) holds along g.

Recall that f(x,y,y’) was defined for triples (x,y,y’) ∈ X × Rm. We shall seek solutions of the Euler equations conditioning triples (x,y,y’) in an open neighborhood Ng in X × Rm of the set of triples (x,y,y′) on g. Triples (x,y,y’) in Ng provide a point (x,y. They are called ordinary triples to distinguish them from canonical triples (x,y,v) which we shall now define.

Canonical triples (x,y,v). We introduce triples

(2.1)

obtained from ordinary triples (x,y,y’) ∈ Ng by setting

(2.2)

We shall write v = fp(x,y,y’) when (2.2) holds and term (x,y,v) a canonical triple.

Under the assumption that (1.13) holds along the extremal g, classical implicit function theorems imply the following.

LEMMA 2.1. If Ng is a sufficiently small open neighborhood in X × Rm of the subset of ordinary triples (x,y,y’) on g, the determinant

(2.3)

and there exists a diff¹¹

(2.4)

of class C² of Ng onto an open neighborhood T(Ng) in X × Rm of the subset of canonical triples (x,y,v) on g such that when (x,y,y′) goes into (x,y,v), (2.2) holds and, equivalently,

(2.5)′

where for each i, Pi is a mapping

(2.5)″

of class C².

The Euler equations transformed. For Ng such that the diff T of Lemma 2.1 is well-defined we introduce the differential equations

(2.6)’

(2.6)″

restricting the triples (x,y,v) to T(Ng). Given a solution x → y(x) = (y1(x), ... , ym(x)) of the Euler equations such that (x,y(x),y’(x)) ∈ Ng, one obtains a solution x → (y(x),v(x)) of the differential equations (2.6) by setting y = y(x) and

(2.7)

Conversely a solution x → (y(x),v(x)) of the equations (2.6) is such that (2.7) holds when (2.6)’ holds and, by virtue of (2.6)″,

(2.8)

with (x,y(x),y′(x)) in Ng.

We summarize as follows.

LEMMA 2.2. If the Euler equations are restricted by the condition that triples (x,y,y′) be in the domain Ng of the diff T, and the equations (2.6) by the condition that triples (x,y,v) be in T(Ng), then the set of ordinary triples (x,y,y′) on a solution x → y(x) of the restricted Euler equations is mapped by T biuniquely onto the set of canonical triples on a solution x → (y(x),v(x)) of the restricted differential equations (2.6).

The mapping of solutions of the Euler equations, restricted as above, into solutions of the differential equations (2.6), restricted as above, is biunique and surjective.

The dependence of extremals upon initial parameters. We shall make use of classical theorems of ordinary differential equation theory to show how extremals near the extremal g depend upon appropriate initial points and slopes. To do this we first show how solutions of (2.6) depend upon initial canonical triples near the initial canonical triple on g. We begin with notation. We first extend the interval [a¹,a²].

The interval ⊃ [a¹,a²]. If (x,y,y′) is an ordinary triple we shall set π(x,y,y′) = x. We refer to the domain Ng of the diff T introduced in Lemma 2.1. Under conditions to be defined let

(2.9)

be an open interval of the x-axis containing the interval [a¹,a²]. Let U be an open neighborhood in Ng ) on g. We suppose U so small that π(U. Set V = T(U). The set V is an open neighborhood in T(Ng) on g. Canonical triples in V will be denoted by (x⁰,y⁰,v⁰). Classical theorems in ordinary differential equation theory imply the following.

A. If the interval differs sufficiently little from [a¹,a²] and if the above neighborhood V ) is sufficiently small with π(U, there then exist mappings

(2.10′)

(2.10)″

of class C² such that, for (x⁰,y⁰,v⁰) fixed in Vinto R induced by the mappings (2.10) afford solutions of the differential equations (2.6) such that

(2.11)′

(2.11)″

Extremals near g. The family (2.10) of solutions of the differential equations (2.6) leads to a family of solutions of the Euler equations as follows. Triples (x,y,y′) ∈ U will be denoted by (c,a,b) or more explicitly, by

(2.12)

For (x : c,a,b× U and i = 1, ... , m, we set

(2.13)

and state the following theorem.

THEOREM 2.1. If and U are conditioned as in A, the mappings

(2.14)

are of class C² and for fixed (c,a,b) ∈ U induce solutions

(2.15)

of the Euler differential equations such that

(2.16)

and the resultant triples (x,Y,Yx) are in Ng.

We have seen that (2.8) is satisfied by a mapping x → y(x) whenever (2.6) is satisfied by a pair of mappings x → y(x), x → v(x). Similarly, for fixed (c,a,b) the partial mapping (2.15) satisfies (2.8), since for fixed (x⁰,y⁰,v⁰), the partial mappings

(2.17)

satisfy (2.6) and the identity

in x is implied by (2.13) if (x⁰,y⁰) = (c,a) and if v⁰ = fp(c,a,b).

That (2.16) holds is verified as follows. A canonical triple (x⁰,y⁰,v⁰) ∈ V is on the solution (2.10) of (2.6) when x = x⁰. If then (c,a,b) is an ordinary triple such that (x⁰,y⁰,v⁰) = T(c,a,b), Lemma 2.2 implies that (c,a,b) is a triple (x,y,y′) on the solution (2.15) of the Euler equation when x = c. That is, (2.16) holds.

Note. The solutions x → y(x), x → v(x) of the differential equations (2.6) given by (2.10) are of at least class C², and since the right hand members of the differential equations (2.6) are of class C², the solutions x → y(x), x → v(x) are of at least class C³. The solutions x → y(x) of the Euler equations affirmed to exist in Theorem 2.1 are accordingly of class C³ by virtue of their definition in (2.13). These solutions have been derived under the assumption that (1.13) holds along each of their graphs.

The mappings Yi of Theorem 2.1 are affirmed to be of class C². It follows that the mappings Yix are also of class C².

The mappings Yi have been derived under the assumption that (1.13) holds along the graph of the solutions (2.15) of the Euler equations. That the mappings Yix are of class C² now follows from the fact that the right hand members of the differential equations (2.6) are of class C².

Definition 2.1. The extension of g. In particular the extremal g with its x-domain [a¹,a²] admits an extension g , for x , as an extremal of class Cso small that along g the determinant,

(2.18)

in Section 2, the condition (2.9), the condition A of Theorem 2.1 and the condition that (2.18) hold for x .

of Appendix I will be taken as an arbitrary open interval.

3. The Necessary Conditions of Weierstrass and Legendre

The Weierstrass E-function is defined by its values

(3.0)

for (x,y) ∈ X and p and q m-tuples in Rm.

We shall prove the following theorem.

THEOREM 3.1. (Weierstrass) If an arc g of form (1.2), of class C¹, affords a minimum to J relative to neighboring x-parameterized curves γ of class D¹, joining the endpoints of g, then

(3.1)

for each triple (x,y,y′) on g and for any m-tuple q ∈ Rm.

Let (x¹,y¹) be a point of g. We treat the case in which x¹ > a¹. The case in which x¹ = a¹ requires at most obvious changes in which a² plays the role of a¹.

The Arc k. Let k be a short open x-parametrized arc in X meeting the point (x¹,y¹) on g. We suppose that k has a representation yi = ki(x), i = 1, ... , m, where the mappings x → ki(x) are of class C¹ on a small open interval for x containing xequal qi where q is prescribed in the Theorem.

We shall compare J along g with J along an x-parameterized curve γα of class D¹ joining the endpoints of g and consisting of three successive arcs each of class¹² C¹, the first of which, λα, will now be defined.¹³

The Arc λα. Let a parameter α be restricted to an arbitrarily small subinterval [x¹ − e, x¹] of the interval (a¹,x¹]. The coordinates yi of points (x,y) on λα shall have the values

(3.2)

for x ∈