DOCSLIB.ORG

Hilbert Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hilbert Spaces CHAPTER 3 Hilbert spaces There are really three `types' of Hilbert spaces (over C): The finite dimen- sional ones, essentially just Cn; for different integer values of n; with which you are pretty familiar, and two infinite dimensional types corresponding to being separa- ble (having a countable dense subset) or not. As we shall see, there is really only one separable infinite-dimensional Hilbert space (no doubt you realize that the Cn are separable) and that is what we are mostly interested in. Nevertheless we try to state results in general and then give proofs (usually they are the nicest ones) which work in the non-separable cases too. I will first discuss the definition of pre-Hilbert and Hilbert spaces and prove Cauchy's inequality and the parallelogram law. This material can be found in many other places, so the discussion here will be kept succinct. One nice source is the book of G.F. Simmons, \Introduction to topology and modern analysis" [5]. I like it { but I think it is long out of print. RBM:Add description of con- tents when complete and 1. pre-Hilbert spaces mention problems A pre-Hilbert space, H; is a vector space (usually over the complex numbers but there is a real version as well) with a Hermitian inner product h; i : H × H −! C; (3.1) hλ1v1 + λ2v2; wi = λ1hv1; wi + λ2hv2; wi; hw; vi = hv; wi for any v1; v2; v and w 2 H and λ1; λ2 2 C which is positive-definite (3.2) hv; vi ≥ 0; hv; vi = 0 =) v = 0: Note that the reality of hv; vi follows from the `Hermitian symmetry' condition in (3.1), the positivity is an additional assumption as is the positive-definiteness. The combination of the two conditions in (3.1) implies `anti-linearity' in the second variable (3.3) hv; λ1w1 + λ2w2i = λ1hv; w1i + λ2hv; w2i which is used without comment below. The notion of ‘definiteness’ for such an Hermitian inner product exists without the need for positivity { it just means (3.4) hu; vi = 0 8 v 2 H =) u = 0: Lemma 3.1. If H is a pre-Hilbert space with Hermitian inner product h; i then 1 (3.5) kuk = hu; ui 2 is a norm on H: 69 70 3. HILBERT SPACES Proof. The first condition on a norm follows from (3.2). Absolute homogene- ity follows from (3.1) since (3.6) kλuk2 = hλu, λui = jλj2kuk2: So, it is only the triangle inequality we need. This follows from the next lemma, which is the Cauchy-Schwarz inequality in this setting { (3.8). Indeed, using the `sesqui-linearity' to expand out the norm (3.7) ku + vk2 = hu + v; u + vi = kuk2 + hu; vi + hv; ui + kvk2 ≤ kuk2 + 2kukkvk + kvk2 = (kuk + kvk)2: Lemma 3.2. The Cauchy-Schwarz inequality, (3.8) jhu; vij ≤ kukkvk 8 u; v 2 H holds in any pre-Hilbert space. Proof. This inequality is trivial if either u or v vanishes. For any non-zero u; v 2 H and s 2 R positivity of the norm shows that (3.9) 0 ≤ ku + svk2 = kuk2 + 2s Rehu; vi + s2kvk2: This quadratic polynomial in s is non-zero for s large so can have only a single minimum at which point the derivative vanishes, i.e. it is where (3.10) 2skvk2 + 2 Rehu; vi = 0: Substituting this into (3.9) gives (3.11) kuk2 − (Rehu; vi)2=kvk2 ≥ 0 =) j Rehu; vij ≤ kukkvk which is what we want except that it is only the real part. However, we know that, for some z 2 C with jzj = 1; Rehzu; vi = Re zhu; vi = jhu; vij and applying (3.11) with u replaced by zu gives (3.8). Corollary 3.1. The inner product is continuous on the metric space (i.e. with respect to the norm) H × H: Proof. Corollaries really aren't supposed to require proof! If (uj; vj) ! (u; v) then, by definition ku − ujk ! 0 and kv − vjk ! 0 so from (3.12) jhu; vi − huj; vjij ≤ jhu; vi − hu; vjij + jhu; vji − huj; vjij ≤ kukkv − vjk + ku − ujkkvjk continuity follows. Corollary 3.2. The Cauchy-Scwharz inequality is optimal in the sense that (3.13) kuk = sup jhu; vij: v2H;kvk≤1 I really will leave this one to you. 3. ORTHONORMAL SEQUENCES 71 2. Hilbert spaces Definition 3.1. A Hilbert space H is a pre-Hilbert space which is complete with respect to the norm induced by the inner product. As examples we know that Cn with the usual inner product n 0 X 0 (3.14) hz; z i = zjzj j=1 is a Hilbert space { since any finite dimensional normed space is complete. The example we had from the beginning of the course is l2 with the extension of (3.14) 1 X 2 (3.15) ha; bi = ajbj; a; b 2 l : j=1 Completeness was shown earlier. The whole outing into Lebesgue integration was so that we could have the `standard example' at our disposal, namely 2 1 2 1 (3.16) L (R) = fu 2 Lloc(R); juj 2 L (R)g=N where N is the space of null functions. The inner product is Z (3.17) hu; vi = uv: Note that we showed that if u; v 2 L2(R) then uv 2 L1(R): We also showed that if R juj2 = 0 then u = 0 almost everywhere, i.e. u 2 N ; which is the definiteness of the inner product (3.17). It is fair to say that we went to some trouble to prove the completeness of this norm, so L2(R) is indeed a Hilbert space. 3. Orthonormal sequences Two elements of a pre-Hilbert space H are said to be orthogonal if (3.18) hu; vi = 0 which can be written u ? v: A sequence of elements ei 2 H; (finite or infinite) is said to be orthonormal if keik = 1 for all i and hei; eji = 0 for all i 6= j: Proposition 3.1 (Bessel's inequality). If ei; i 2 N; is an orthonormal sequence in a pre-Hilbert space H; then X 2 2 (3.19) jhu; eiij ≤ kuk 8 u 2 H: i Proof. Start with the finite case, i = 1; : : : ; N: Then, for any u 2 H set N X (3.20) v = hu; eiiei: i=1 This is supposed to be `the projection of u onto the span of the ei'. Anyway, computing away we see that N X (3.21) hv; eji = hu; eiihei; eji = hu; eji i=1 72 3. HILBERT SPACES using orthonormality. Thus, u − v ? ej for all j so u − v ? v and hence (3.22) 0 = hu − v; vi = hu; vi − kvk2: Thus kvk2 = jhu; vij and applying the Cauchy-Schwarz inequality we conclude that kvk2 ≤ kvkkuk so either v = 0 or kvk ≤ kuk: Expanding out the norm (and observing that all cross-terms vanish) N 2 X 2 2 kvk = jhu; eiij ≤ kuk i=1 which is (3.19). In case the sequence is infinite this argument applies to any finite subsequence, ei; i = 1;:::;N since it just uses orthonormality, so (3.19) follows by taking the supremum over N: 4. Gram-Schmidt procedure Definition 3.2. An orthonormal sequence, feig; (finite or infinite) in a pre- Hilbert space is said to be maximal if (3.23) u 2 H; hu; eii = 0 8 i =) u = 0: Theorem 3.1. Every separable pre-Hilbert space contains a maximal orthonor- mal sequence. Proof. Take a countable dense subset { which can be arranged as a sequence fvjg and the existence of which is the definition of separability { and orthonormalize it. First if v1 6= 0 set ei = v1=kv1k: Proceeding by induction we can suppose we have found, for a given integer n; elements ei; i = 1; : : : ; m; where m ≤ n; which are orthonormal and such that the linear span (3.24) sp(e1; : : : ; em) = sp(v1; : : : ; vn): We certainly have this for n = 1: To show the inductive step observe that if vn+1 is in the span(s) in (3.24) then the same ei's work for n + 1: So we may as well assume that the next element, vn+1 is not in the span in (3.24). It follows that n X w (3.25) w = v − hv ; e ie 6= 0 so e = n+1 n+1 j j m+1 kwk j=1 makes sense. By construction it is orthogonal to all the earlier ei's so adding em+1 gives the equality of the spans for n + 1: Thus we may continue indefinitely, since in fact the only way the dense set could be finite is if we were dealing with the space with one element, 0; in the first place. There are only two possibilities, either we get a finite set of ei's or an infinite sequence. In either case this must be a maximal orthonormal sequence. That is, we claim (3.26) H 3 u ? ej 8 j =) u = 0: This uses the density of the vj's. There must exist a sequence wk where each wk is a vj; such that wk ! u in H; assumed to satisfy (3.26). Now, each vj; and hence each wk; is a finite linear combination of el's so, by Bessel's inequality 2 X 2 X 2 2 (3.27) kwkk = jhwk; elij = jhu − wk; elij ≤ ku − wkk l l 6. ISOMORPHISM TO l2 73 where hu; eli = 0 for all l has been used.
Load more

Recommended publications
  • THE DIMENSION of a VECTOR SPACE 1. Introduction This Handout
    THE DIMENSION OF A VECTOR SPACE KEITH CONRAD 1. Introduction This handout is a supplementary discussion leading up to the definition of dimension of a vector space and some of its properties. We start by defining the span of a finite set of vectors and linear independence of a finite set of vectors, which are combined to define the all-important concept of a basis. Definition 1.1. Let V be a vector space over a field F . For any finite subset fv1; : : : ; vng of V , its span is the set of all of its linear combinations: Span(v1; : : : ; vn) = fc1v1 + ··· + cnvn : ci 2 F g: Example 1.2. In F 3, Span((1; 0; 0); (0; 1; 0)) is the xy-plane in F 3. Example 1.3. If v is a single vector in V then Span(v) = fcv : c 2 F g = F v is the set of scalar multiples of v, which for nonzero v should be thought of geometrically as a line (through the origin, since it includes 0 · v = 0). Since sums of linear combinations are linear combinations and the scalar multiple of a linear combination is a linear combination, Span(v1; : : : ; vn) is a subspace of V . It may not be all of V , of course. Definition 1.4. If fv1; : : : ; vng satisfies Span(fv1; : : : ; vng) = V , that is, if every vector in V is a linear combination from fv1; : : : ; vng, then we say this set spans V or it is a spanning set for V . Example 1.5. In F 2, the set f(1; 0); (0; 1); (1; 1)g is a spanning set of F 2.
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • General Topology
    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
    [Show full text]
  • Inner Product Spaces
    CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn.
    [Show full text]
  • Vector Space Theory
    Vector Space Theory A course for second year students by Robert Howlett typesetting by TEX Contents Copyright University of Sydney Chapter 1: Preliminaries 1 §1a Logic and commonSchool sense of Mathematics and Statistics 1 §1b Sets and functions 3 §1c Relations 7 §1d Fields 10 Chapter 2: Matrices, row vectors and column vectors 18 §2a Matrix operations 18 §2b Simultaneous equations 24 §2c Partial pivoting 29 §2d Elementary matrices 32 §2e Determinants 35 §2f Introduction to eigenvalues 38 Chapter 3: Introduction to vector spaces 49 §3a Linearity 49 §3b Vector axioms 52 §3c Trivial consequences of the axioms 61 §3d Subspaces 63 §3e Linear combinations 71 Chapter 4: The structure of abstract vector spaces 81 §4a Preliminary lemmas 81 §4b Basis theorems 85 §4c The Replacement Lemma 86 §4d Two properties of linear transformations 91 §4e Coordinates relative to a basis 93 Chapter 5: Inner Product Spaces 99 §5a The inner product axioms 99 §5b Orthogonal projection 106 §5c Orthogonal and unitary transformations 116 §5d Quadratic forms 121 iii Chapter 6: Relationships between spaces 129 §6a Isomorphism 129 §6b Direct sums Copyright University of Sydney 134 §6c Quotient spaces 139 §6d The dual spaceSchool of Mathematics and Statistics 142 Chapter 7: Matrices and Linear Transformations 148 §7a The matrix of a linear transformation 148 §7b Multiplication of transformations and matrices 153 §7c The Main Theorem on Linear Transformations 157 §7d Rank and nullity of matrices 161 Chapter 8: Permutations and determinants 171 §8a Permutations 171 §8b Determinants
    [Show full text]
  • Space Dimension Renormdynamics
    Article Space Dimension Renormdynamics Martin Bures 1,2,* and Nugzar Makhaldiani 1,* 1 Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia 2 Institute of Experimental and Applied Physics, Czech Technical University in Prague, Husova 240/5, 110 00 Prague 1, Czech Republic * Correspondence: [email protected] (M.B.); [email protected] (N.M.) Received: 31 December 2019; Accepted: 15 April 2020; Published: 17 April 2020 Abstract: We aim to construct a potential better suited for studying quarkonium spectroscopy. We put the Cornell potential into a more geometrical setting by smoothly interpolating between the observed small and large distance behaviour of the quarkonium potential. We construct two physical models, where the number of spatial dimensions depends on scale: one for quarkonium with Cornell potential, another for unified field theories with one compactified dimension. We construct point charge potential for different dimensions of space. The same problem is studied using operator fractal calculus. We describe the quarkonium potential in terms of the point charge potential and identify the strong coupling fine structure constant dynamics. We formulate renormdynamics of the structure constant in terms of Hamiltonian dynamics and solve the corresponding motion equations by numerical and graphical methods, we find corresponding asymptotics. Potentials of a nonlinear extension of quantum mechanics are constructed. Such potentials are ingredients of space compactification problems. Mass parameter effects are motivated and estimated. Keywords: space dimension dynamics; coulomb problem; extra dimensions “... there will be no contradiction in our mind if we assume that some natural forces are governed by one special geometry, while other forces by another.” N.
    [Show full text]
  • A Note on Vector Space Axioms∗
    GENERAL ARTICLE A Note on Vector Space Axioms∗ Aniruddha V Deshmukh In this article, we will see why all the axioms of a vector space are important in its definition. During a regular course, when an undergraduate student encounters the definition of vector spaces for the first time, it is natural for the student to think of some axioms as redundant and unnecessary. In this article, we shall deal with only one axiom 1 · v = v and its importance. In the article, we would first try to prove that it is redundant just as an undergraduate student would (in the first attempt), Aniruddha V Deshmukh has and then point out the mistake in the proof, and provide an completed his masters from S example which will be su cient to show the importance of V National Institute of ffi Technology, Surat and is the the axiom. gold medalist of his batch. His interests lie in linear algebra, metric spaces, 1. Definitions and Preliminaries topology, functional analysis, and differential geometry. He All the definitions are taken directly from [1] is also keen on teaching mathematics. Definition 1 (Field). Let F be a non-empty set. Define two op- erations + : F × F → F and · : F × F → F. Eventually, these operations will be called ‘addition’ and ‘multiplication’. Clearly, both are binary operations. Now, (F, +, ·) is a field if 1. ∀x, y, z ∈ F, x + (y + z) = (x + y) + z (Associativity of addition) Keywords Vector spaces, fields, axioms, 2. ∃0 ∈ F such that ∀x ∈ F, 0 + x = x + 0 = x Zorn’s lemma. (Existence of additive identity) 3.
    [Show full text]
  • 17. Inner Product Spaces Definition 17.1. Let V Be a Real Vector Space
    17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function h ; i: V × V −! R; which is • symmetric, that is hu; vi = hv; ui: • bilinear, that is linear (in both factors): hλu, vi = λhu; vi; for all scalars λ and hu1 + u2; vi = hu1; vi + hu2; vi; for all vectors u1, u2 and v. • positive that is hv; vi ≥ 0: • non-degenerate that is if hu; vi = 0 for every v 2 V then u = 0. We say that V is a real inner product space. The associated quadratic form is the function Q: V −! R; defined by Q(v) = hv; vi: Example 17.2. Let A 2 Mn;n(R) be a real matrix. We can define a function n n h ; i: R × R −! R; by the rule hu; vi = utAv: The basic rules of matrix multiplication imply that this function is bi- linear. Note that the entries of A are given by t aij = eiAej = hei; eji: In particular, it is symmetric if and only if A is symmetric that is At = A. It is non-degenerate if and only if A is invertible, that is A has rank n. Positivity is a little harder to characterise. Perhaps the canonical example is to take A = In. In this case if t P u = (r1; r2; : : : ; rn) and v = (s1; s2; : : : ; sn) then u Inv = risi. Note 1 P 2 that if we take u = v then we get ri . The square root of this is the Euclidean distance.
    [Show full text]
  • Vector Space Axioms a Real Vector Space
    Vector Space Axioms A real vector space (that is, a vector space over the real numbers R) is a set V of objects, individually called vectors, and two operations + (plus) and · (scalar multiplication), together satisfying the following axioms. 1. For every pair of vectors u, v ∈ V , the operation + is defined and gives a vector in V . That is, u + v ∈ V for all u, v ∈ V . 2. + is commutative, i.e., u + v = v + u for all u, v ∈ V . 3. + is associative, i.e., u +(v+w)=(u+v) + w for all u, v, w ∈ V . 4. + has at least one identity within V . A vector e is called an identity if, for all u ∈ V , we have u + e = u. In fact, our first theorem from these axioms will be that each vector space has exactly one identity. Therefore, it can be given a unique name, and the usual name is 0. (Use 0¯ when handwriting.) 5. Each v ∈ V has at least one additive inverse within V . That is, for each v ∈ V , there is at least one vector wv ∈ V such that v + wv is an identity. In fact, it is a theorem that every vector in every vector space has exactly one inverse, and so it can be given a special name. The usual name is −v. Think of this as an indivisible symbol; it doesn’t mean minus times v. 6. Scalar multiplication is defined between every real number and every vector in V , and the result is in V . That is, kv ∈ V for all k ∈ R and v ∈ V .
    [Show full text]
  • Vector Spaces
    Vector Spaces Math 240 Definition Properties Set notation Subspaces Vector Spaces Math 240 | Calculus III Summer 2013, Session II Wednesday, July 17, 2013 Vector Spaces Agenda Math 240 Definition Properties Set notation Subspaces 1. Definition 2. Properties of vector spaces 3. Set notation 4. Subspaces Vector Spaces Motivation Math 240 Definition Properties Set notation Subspaces We know a lot about Euclidean space. There is a larger class of n objects that behave like vectors in R . What do all of these objects have in common? Vector addition a way of combining two vectors, u and v, into the single vector u + v Scalar multiplication a way of combining a scalar, k, with a vector, v, to end up with the vector kv A vector space is any set of objects with a notion of addition n and scalar multiplication that behave like vectors in R . Vector Spaces Examples of vector spaces Math 240 Definition Real vector spaces Properties n Set notation I R (the archetype of a vector space) Subspaces I R | the set of real numbers I Mm×n(R) | the set of all m × n matrices with real entries for fixed m and n. If m = n, just write Mn(R). I Pn | the set of polynomials with real coefficients of degree at most n I P | the set of all polynomials with real coefficients k I C (I) | the set of all real-valued functions on the interval I having k continuous derivatives Complex vector spaces n I C, C I Mm×n(C) Vector Spaces Definition Math 240 Definition Properties Definition Set notation A vector space consists of a set of scalars, a nonempty set, V , Subspaces whose elements are called vectors, and the operations of vector addition and scalar multiplication satisfying 1.
    [Show full text]
  • Space Mathematics Part 1
    Volume 5 Lectures in Applied Mathematics SPACE MATHEMATICS PART 1 J. Barkley Rosser, EDITOR MATHEMATICS RESEARCH CENTER THE UNIVERSITY OF WISCONSIN 1966 AMERICAN MATHEMATICAL SOCIETY, PROVIDENCE, RHODE ISLAND Supported by the National Aeronautics and Space Administration under Research Grant NsG 358 Air Force Office of Scientific Research under Grant AF-AFOSR 258-63 Army Research Office (Durham) under Contract DA-3 I- 12 4-ARO(D)-82 Atomic Energy Commission under Contract AT(30-I)-3164 Office of Naval Research under Contract Nonr(G)O0025-63 National Science Foundation under NSF Grant GE-2234 All rights reserved except those granted to the United States Government, otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number 66-20435 Copyright © 1966 by the American Mathematical Society Printed in the United States of America Contents Part 1 FOREWORD ix ELLIPTIC MOTION J. M. A. Danby MATRIX METHODS 32 _-'/ J. M. A. Danby THE LAGRANGE-HAMILTON-JACOBI MECHANICS 40 Boris Garfinkel STABILITY AND SMALL OSCILLATIONS ABOUT EQUILIBRIUM .,,J AND PERIODIC MOTIONS 77 P. J. Message LECTURES ON REGULARIZATION 100 j,- Paul B. Richards THE SPHEROIDAL METHOD IN SATELLITE ASTRONOMY 119 J John P. Vinti PRECESSION AND NUTATION 130 u/" Alan Fletcher ON AN IRREVERSIBLE DYNAMICAL SYSTEM WITH TWO DEGREES J OF FREEDOM: THE RESTRICTED PROBLEM OF THREE BODIES 150 Victor Szebehely PROBLEMS OF STELLAR DYNAMICS 169 j George Contopoulos //,- QUALITATIVE METHODS IN THE n-BODY PROBLEM 259 Harry Pollard INDEX 293 vi CONTENTS Part2 MOTION IN THE VICINITY OF THE TRIANGULAR LIBRATION CENTERS Andr_ Deprit MOTION OF A PARTICLE IN THE VICINITY OF'A TRIANGULAR LIBRATION POINT IN THE EARTH-MOON SYSTEM 31 J.
    [Show full text]
  • Chapter 6. Mathematics and Numbers
    Chapter 6. Mathematics and Numbers EQUATIONS Mathematical equations can present difficult and costly problems of type composi- tion. Because equations often must be retyped and reformatted during composition, errors can be introduced. Keep in mind that typesetters will reproduce what they see rather than what the equation should look like. Therefore, preparation of the manuscript copy and all directions and identification of letters and symbols must be clear, so that those lacking in mathematical expertise can follow the copy. Use keyboard formatting where possible (i.e., bold, super-/subscripts, simple vari- ables, Greek font, etc.), and use MathType or the Word equation editor for display equa- tions. If your equations are drawn from calculations in a computer language, translate the equation syntax of the computer language into standard mathematical syntax. Likewise, translate variables into standard mathematical format. If you need to present computer code, do that in an appendix. Position and Spacing The position and spacing of all elements of an equation must be exactly as they are to appear in printed form. Place superscript and subscript letters and symbols in the correct positions. Put a space before and after most mathematical operators (the main exception is the soli- dus sign for division). For example, plus and minus signs have a space on both sides when they indicate a mathematical operation but have no space between the sign and the number when used to indicate positive or negative position on the number line (e.g., 5 - 2 = 3; a range from -15 to 25 kg). No space is left between variables and their quantities or between multiplied quan- tities when the multiplication sign is not explicitly shown.
    [Show full text]