 # Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold. (MN-16)

Marston Morse
Pages: 270
https://www.jstor.org/stable/j.ctt130hkgk

1. Front Matter
(pp. None)
2. Preface
(pp. i-iv)
(pp. v-vii)
4. INTRODUCTION
(pp. viii-x)

We shall give a heuristic account of some of the sections most likely to be difficult by making clear what is the main stream. §21 will be reviewed first because this is the section in which the search for extremals joining A1to A2is reduced to the search for the critical elements of a function of points. It is recommended that the reader make use of this account only after reading the first twenty sections of the book.

Summary of §21. A broken extremal γ joining A1to A2is introduced. The curve γ consists of ν+1 successive elementary...

5. ### PART I The Weierstrass integral J

• Chapter 1 A Riemannian manifold
(pp. 1-15)

§1. A differentiable manifold Mn. The integrals introduced by Weierstrass in his study of the calculus of variations in parametric form (cf. Bolza , p. 189) had the (x,y)-plane as the underlying space. We shall replace the (x,y)-plane by an n-dimensional Riemannian manifold Mn. A Riemannian manifold is a differentiable manifold with a Riemannian structure. We begin accordingly with a brief characterization of a differentiable manifold Mnof dimension n > 1.

The manifold Mnis supposed of class C. A priori, Mnis a connected, topological manifold, coverable by a countable union of open subspaces U, V, etc., which are...

• Chapter 2 Local Weierstrass Integrals
(pp. 16-33)

§3. Weierstrass preintegrands. Weierstrass integrals, as we shall define them on our Reimannian manifold Mn, are generalizations of R-length on Mn. With Weierstrass, was the (x,y)-plane, so that no presentation theory was required. However, general differentiable manifolds presuppose a set of compatible presentations. It is accordingly necessary to begin with a presentation (ϕ,U) in${\cal {D}\text{M}_{\text {n}}}$and replace the R-preintegrands$\caption {(3.0)} (\text{u},\text{r})\to {f}(\text{u},\text{r})={{({{\text{a}}_{\text{ij}}}(\text{u})\ {{\text{r}}^{\text{i}}}{{\text{r}}^{\text{j}}})}^{\frac{1}{2}}}\quad \ ((\text{u},\text{r})\in \text{U}\times {{\dot{\text {R}}}^{\text{n}}})$by more general preintegrands.

To this end there is associated with each presentation$(\phi ,\text{U})\in {\cal {D}\text{M}_{\text {n}}}$a unique mapping$\caption {(3.1)}^* (\text{u},\text{r})\to \text{F}(\text{u},\text{r}):\text{U}\times {\dot{\text{{R}}}^{\text{n}}}\to \text{R}$of class C, subject to the homogeneity condition,

(3.2) F(u,kr) = kF(u,r),

valid for each pair (u,r) in the domain of F...

6. ### PART II The Euler equations

• Chapter 3 The Euler-Riemann Equations
(pp. 34-65)

§6. The Weierstrass nonsingularity condition. Let F be the W-preintegrand associated with a presentation (ϕ, U) in${\cal {D}\text{M}_{\text {n}}}$. As given in (3.1), F has values F(u, r) and domain$\text{U}\times {\dot{\text{{R}}}^{\text{n}}}$. The associated Euler equations$\caption {(6.1)} \frac{\text{d}}{\text{dt}}\text{F}{}_{{}_{\text{r}}\text{i}}(\text{u},\dot{\text {u}})-\text{F}{}_{{}_{\text{u}}\text{i}}(\text{u},\dot{\text {u}})=0\quad \quad \quad ({\text{i}}=1,\ldots,\text{n})$can be understood in two senses. Formally the conditions can be regarded as conditions on a regular C2-mapping t → u(t) : [a, b] → U of the form$\caption {(6.2)}{{\text{F}}_{{{\text{r}}^{\text{i}}}{{\text{r}}^{\text{j}}}}}(\text{u},\dot{\text{u}}){{\ddot{\text{u}}}^{\text{j}}}-{{\text{F}}_{{{\text{r}}^{\text {i}}}{{\text{u}}^{\text{j}}}}}(\text{u},\dot{\text{u}}){{\dot{\text{u}}}^{\text{j}}}-{{\text{F}}_{{{\text{u}}^{\text{i}}}}}(\text{u},\dot{\text{u}})=0\quad \quad (\text{i}=\text{l},\ldots ,\text{n}).$

According to (4.8) the determinant$\caption {(6.3)} |{{\text{F}}_{{{\text{r}}^{\text{i}}}{{\text{r}}^{\text{j}}}}}(\text{u},\text{r})|\equiv 0\quad \quad \quad ((\text{u},\text{r})\in \text{U}\times {{\dot{\text{R}}}^{\text{n}}})$so that Cramer's rule cannot be used to solve the equations (6.2) for the n-tuples$({\ddot{\text{{u}}}^{1}},\ldots,{\ddot{\text{{u}}}^{\text{n}}})$.

For this reason we shall regard the Euler equations a...

• Chapter 4 Conjugate points
(pp. 66-91)

§9. Proper polar families of extremal arcs. To define conjugate points on extremal arcs of the W-integral J on Mnit will be sufficient to define conjugate points on extremals of a W-integral JFin a coordinate domain U. This is because each curve on Mnwhich is both regular and simple admits a representation in the coordinate domain U of a presentation* in${\cal {D}\text{M}_{\text {n}}}$. Extremals of J which are not simple will be treated in Sections 22 and 23 by an extension of the methods of this section.

Let (ϕ, U) be a presentation in${\cal {D}\text{M}_{\text {n}}}$and JF...

7. ### PART III Minimizing arcs

• Chapter 5 Necessary conditions
(pp. 92-108)

§12. Necessity of the Euler condition. Let F be a W-preintegrand associated with a presentation (ϕ, U) inDMn. Concerning F a classical theorem will be proved.

Theorem 12.1. Let$\caption {(12.1)}{\underset \thicksim{\text{z}}}:\text{t}\to \text{z}(\text{t}):[{{\text{t}}_{0}},{{\text{t}}_{1}}]\to \text{U}$be a regular arc on U. If${{\text{J}}_{\text{F}}}({\underset \thicksim{\text{z}}})\le {{\text{J}}_{\text{F}}}({\underset \thicksim{\text{u}}})$for each piecewise regular arc${\underset \thicksim {\text {u}}}$joining the end points of${\underset \thicksim {\text {z}}}$in some open neighborhood N of${\underset \thicksim {\text {z}}}$in U, then for t0≤ t ≤ t1$\caption {(12.2)}\frac{\text{d}}{\text{dt}}{{\text{F}}_{{{\text{r}}^{\text{i}}}}}(\text{z}(\text{t}),\dot{\text{z}}(\text{t}))={{\text{F}}_{{{\text{u}}^{\text{i}}}}}(\text{z}(\text{t}),\text{\dot{z}}(\text{t}))\quad \quad \quad (\text{i}=1,\ldots ,\text{n}).$

Proof. For i = 1, …, n let t → ηi(t) : [t0,t1] → R be a mapping of class D1such that ηi(t0) = ηi(t1) = 0. If e is a...

• Chapter 6 Sufficient Conditions
(pp. 109-124)

§17. A Hilbert integral HF. In Chapter 5 there is given a presentation (ϕ, U) and an associated W-integral JF. An extremal arc$\caption {(17.0)} {\underset \thicksim {\text {z}}}:\text{s}\to \text{z}(\text{s}):[\text{a},\text{b}]\to \text{U}$of JFis given. This arc is R-parameterized in accord with Definition 7.1. The object of Chapter 5 is to establish sufficient conditions that${\underset \thicksim {\text {z}}}$afford a minimum to JFrelative to piecewise regular arcs${\underset \thicksim {\text {u}}}$which join the endpoints of${\underset \thicksim {\text {z}}}$in some neighborhood in U of the carrier$\left| \underset{\sim }{\mathop{\text{z}}}\, \right|$of${\underset \thicksim {\text {z}}}$.

A Hilbert integral HFis a line integral on U associated with JF. It is used in studying the conditions under which...

8. ### PART IV Preparation for Global Theorems

• Chapter 7 Elementary Extremals
(pp. 125-161)

19. Field radii and elementary extremals. We begin with an example. In the [u1,u2]-plane the extremals of the integral of length are straight lines. The extremals issuing from the origin with lengths at most 1 have a representation

(19.1) [u1,u2] = [s cos β, s sin β] [0 ≤ s ≤ 1] with β a parameter. These extremal arcs define a "field" when s > 0, with the origin as "pole" when s = 0. We say that this field has a field radius 1. The parameters β represent points [r1,r2] on the circle

(19.2) C : (r1)2+ (r2)2= 1....

• Chapter 8 Non-simple extremals
(pp. 162-172)

§22. Tubular mappings into Mn. Prior to Part IV, the principal theorems have been restricted to extremal arcs, that is extremals that are simple. Extremals γ of J on Mnwhich are self-intersecting cannot be excluded in a global theory. One must define conjugate points on such an extremal γ and give a precise definition of the classes of curves relative to which a properly conditioned extremal γ gives a proper minimum to J. In such a study a most relevant property of an extremal of J on Mnis that it is regular. We, accordingly, begin with a regular...

9. ### PART V Global Theorems

• Chapter 9 Simplifying concepts
(pp. 173-196)

24. I, extremal nondegeneracy. II, singleton extremals. The object of Part I of this section is to complete the proof of Measure Theorem 11.1.

Let a point A1be prescribed on Mn. Recall that a finite extremal γ of J on Mnissuing from A1is termed degenerate, if the terminal point P of γ is conjugate on γ to A1. We here admit that γ may be nonsimple. Theorem 11.1 may be restated as follows.

Theorem 24.1. For A1prescribed on Mn, let ((A1)) be the set of points P on Mnsuch that A1is conjugate to P on...

• Chapter 10 Reduction to Critical Point Theory
(pp. 197-216)

§26. Extremal homology relations under a finite J-level. In Chapter 10 the study of extremal homology relations will be restricted to the study of extremals joining an ND point pair (A1, A2). Theorem 26.1 concerns the set Sβof extremals of J characterized in Part II of §24. Theorem 26.1 is a consequence of Theorem 7.1 of Landis-Morse . The paper of Landis-Morse was written specially for this application. A principal hypothesis of the Landis-Morse Theorem is formulated in terms of a special deformation called a traction. We shall define a traction in a background of deformation theory.

Deformations. Let...

10. Appendix I. The existence of regular tubular mappings.
(pp. 217-220)
11. Appendix II. Minimizing extremals, phasewise near a minimizing extremal.
(pp. 221-223)
12. Appendix III. The differentiable product manifold (Mn)ν.
(pp. 224-227)
13. Appendix IV. The existence of the tractions Ti of §26.
(pp. 228-243)
14. Bibliography
(pp. 244-251)
15. INDEX OF TERMS
(pp. 252-255)
16. Back Matter
(pp. 256-256)