DOCSLIB.ORG

Space Mathematics Part 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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. Pieter deVries THE DOMINANT FEATURES OF THE LONG-PERIOD LIBRATIONS OF THE TROJAN MINOR PLANETS 7O P. J. Message OUTLINE OF A THEORY OF NONPERIODIC MOTIONS IN THE NEIGHBORHOOD OF THE LONG-PERIOD LIBRATIONS ABOUT THE EQUILATERAL POINTS OF THE RESTRICTED PROBLEM OF THREE BODIES 79 Eugene Rabe ELEMENTS OF A THEORY OF LIBRATIONAL MOTIONS IN THE ELLIPTICAL RESTRICTED PROBLEM 102 Eugene Rabe THE EQUILIBRIUM SHAPE OF THE EARTH IN THE LIGHT OF RECENT DISCOVERIES IN SPACE SCIENCE 119 John A. O'Keefe THE STABILITY OF A ROTATING LIQUID MASS 155 John A. O'Keefe GEODETIC PROBLEMS AND SATELLITE ORBITS 170 W. H.Guier ELEMENTS OF CALCULUS OF VARIATIONS AND OPTIMUM CONTROL THEORY 212 Magnus R. Hestenes INDEX 255 CONTENTS vii Part3 BASIC FLUID DYNAMICS S. F. Shen SHOCK WAVES IN RAREFIED GASES 54 S. F. Shen MODELS OF GAS FLOWS WITH CHEMICAL AND RADIATIVE EFFECTS 94 F. K. Moore DECAY OF ORBITS 153 P. J. Message THE EFFECT OF RADIATION PRESSURE ON THE MOTION OF AN ARTIFICIAL SATELLITE 167 Gen-ichiro Hori SPECIAL COMPUTATION PROCEDURES FOR DIFFERENTIAL EQUATIONS 179 S. V. Parter THE TWO VARIABLE EXPANSION PROCEDURE FOR THE APPROXIMATE SOLUTION OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS 206 J. Kevorkian RENDEZVOUS PROBLEMS 276 J. C. Houbolt INDEX 301 AUTHOR INDEX FOR THE THREE VOLUMES 305 Foreword In 1962 a committee of the Division of Mathematics of the National Research Council presented to the Space Science Board of the National Academy of Sciences a report on the current and anticipated uses of mathematics in space activities. A key finding was the need for much more intensive work in various mathematical areas, both to develop more powerful results in these areas and to discover ways of using the existing knowledge more effectively. The Summer Seminar of 1963 was planned to fill this need in part. It was the third Summer Seminar in Applied Mathematics sponsored by the American Mathematical Society. In order to keep the Seminar within bounds and to achieve reasonable coherence, many mathematical areas of great importance and urgency for space activity were not considered at the Seminar. The most notable omissions are the area of nonlinear differential equations, which is of use in the study of guidance systems for propulsive space vehicles, and various areas of statistics, such as those involved in the design of experiments to be performed in space, analysis of large amounts of data from experiments aboard space vehicles, etc. Primarily, the topics considered at the Seminar were those having to do with the behavior of nonpropulsive space vehicles. Inevit- ix x FOREWORD ably, this led to heavy emphasis on the theory of orbits. For this reason, the Seminar was combined with the fifth annual Summer Institute in Dynamical Astronomy. Four previous Summer Institutes in Dynamical Astronomy have been sponsored by Yale University, in the summers of 1959 through 1962. They were devoted primarily to topics in the theory of orbits, and were more restricted in scope than the present Seminar. The orbits of most present and planned space vehicles differ so markedly from the orbits of the classical celestial bodies that only parts of the classical theories of celestial mechanics are applicable. Much effort is being expended to determine which of the classical methods are applicable, to find suitable modifications of some of the classical methods to make them more widely applicable, and to find new methods to handle novel situations. Such matters were a main concern of the four previous Summer Institutes in Dynamical Astronomy, and received a considerable amount of attention at the present Seminar; it was to assure this that the Seminar was combined with the fifth Institute. Notes were compiled for the Institute of 1959, but their supply is exhausted. Notes for the Institute of 1960 were edited by William E. Felling and published in 1961 by The McDonnell Aircraft Corporation, St. Louis, Missouri, under the title, "Notes of the Summer Institute in Dynamical Astronomy at Yale University, July 1960". Copies of these notes can be obtained from The McDonnell Aircraft Corporation. The Institute of 1961 was held at Tucson, Arizona, and the Institute of 1962 was held at Yale University. Only scattered notes, mostly in mimeographed form, survive from these meetings. Some of these were collected and have been included in the present Proceedings. Specifically, from the Institute of 1961 come the chapters: Precession and nutation by Alan Fletcher; Lectures on regularization by Paul B. Richards. From the Institute of 1962 come the chapters: Problems of stellar dynamics by G. Contopoulos; Matrix methods by J.M.A. Danby; The effect of radiation pressure on the motion of an artificial satellite by Gen-ichiro Hori; The two variable expansion procedure for the approximate solution of certain nonlinear differential equations by J. Kevorkian; Stability and small oscillations about equilibrium and periodic motions by P. J. Message; The spheroidal method in satellite astronomy by John P. Vinti. FOREWORD xi Some of the lecturesof the earlierInstitutes,and of the present Seminar, have been published elsewhere, either separately or as parts of larger works. Indeed, some lectures excerpted particularly relevant material already in print. Besides the Notes of the 1960 Institute already cited, a listof publications containing material not covered in the present Proceedings which was presented in lecturesat the Institutesand the Seminar follows: R. F. Arenstorf, Periodic solutions of the restricted three body prob- lem presenting analytic continuations of Keplerian motions, Amer. J. Math 85(1963), 27-35. V. I. Arnol'd, The stability of the equilibrium position of a Hamil- tonian system of ordinary differential equations in the general elliptic case, Dokl. Akad. Nauk SSSR 137(1961), 255-257 = Soviet Math. Dokl. 2(1961), 247. __, Generation of quasi-periodic motion from a family of periodic motions, Dokl. Akad. Nauk SSSR 138(1961), 13-15 = Soviet Math. Dokl. 2(1961), 501. __, The classical theory of perturbations and the problem of stability of planetary systems, Dokl. Akad. Nauk SSSR 145(1962), 487-490 = Soviet Math. Dokl. 3(1962), 1008. __, Proof of A. N. Kolrnogorov's theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian, Uspehi Mat. Nauk SSSR 18, Ser. 5 (113), 1963, pp. 13-40. __, Small divisor and stability problems in classical and celestial mechanics, Uspehi Mat. Nauk SSSR 18, Ser. 6(119), 1963, pp. 81-192. R. E. Bellman, Adaptive control processes; a guided tour, Princeton Univ. Press, Princeton, N. J., 1961. R. E. Bellman and S. Dreyfus, Applied dynamic programming, Princeton Univ. Press, Princeton, N. J., 1962. D. Brouwer, Solution of the problem of artificial satellite motion without drag, Astronom. J. 64(1959), 378-397. D. Brouwer and G. M. Clemence, Methods of celestial mechanics, Academic Press, New York, 1961. D. Brouwer and G. Hori, Theoretical evaluation of atmospheric drag effects in the motion of an artificial satellite, Astronom. J. 66(1961), 193-225; 264-265. C. J. Cohen and E. C. Hubbard, A non-singular set of orbit elements, Astronom. J. 67(1962), 10-15. C. C. Conley, A disk mapping associated with the satellite problem, xii FOREWORD Comm. Pure Appl. Math. 17(1964), 237-243. S. Dreyfus, Dynamic programming and the calculus of variations, J. Math. Anal. Appl. 1(1960), 228-239. W. J. Eckert and D. Brouwer, The use of rectangular coordinates in the differential correction of orbits, Astronom. J. 46(1937), 125-132. B. Garfinkel, Motion of a satellite in the vicinity of the critical in- clination, Astronom. J. 65(1960), 624-627. Y. Hagihara, Theories of equilibrium figures of a homogeneous rotat- ing fluid mass, Blaisdell, New York, forthcoming. __, Celestial mechanics, Blaisdell, New York, forthcoming in four volumes. Paul Herget, The computation of orbits, Observatory of the Univer- sity of Cincinnati, 1948. __, Computation of preliminary orbits, Astronom. J. 70(1965), 1-2. G. Hori, Motion of an artificial satellite in the vicinity of the critical inclination, Astronom. J. 65(1960), 291-300.
Recommended publications
  • Analysis and Control of Displaced Periodic Orbits in the Earth-Moon System

    Analysis and Control of Displaced Periodic Orbits in the Earth-Moon System

    IAC-09.C1.2.4 ANALYSIS AND CONTROL OF DISPLACED PERIODIC ORBITS IN THE EARTH-MOON SYSTEM Jules Simo⋆ and Colin R. McInnest Department of Mechanical Engineering, University of Strathclyde Glasgow, G1 1XJ, United Kingdom ⋆E-mail: [email protected] tE-mail: [email protected] ABSTRACT be displaced above the orbital plane of the Earth, so that We consider displaced periodic orbits at linear order in the sail can stay fixed above the Earth at some distance, if the circular restricted Earth-Moon system, where the third the orbital periods are equal. Orbits around the collinear massless body is a solar sail. These highly non-Keplerian points of the Earth-Moon system are also of great interest orbits are achieved using an extremely small sail accel- because their unique positions are advantageous for sev- eration. In this paper we will use solar sail propulsion eral important applications in space mission design (see e.g. Szebehely [3], Roy [4], Vonbun [5], Gomez´ et al. to provide station-keeping at periodic orbits above the L2 point. We start by generating a reference trajectory about [6]). the libration points. By introducing a first-order approx- In the recent years several authors have tried to determine imation, periodic orbits are derived analytically at linear more accurate approximations (quasi-Halo orbits) of such order. These approximate analytical solutions are utilized equilibrium orbits [7]. The orbits were first studied by Far- in a numerical search to determine displaced periodic or- quhar [8], Farquhar and Kamel [7], Breakwell and Brown bits in the full nonlinear model.
  • THE DIMENSION of a VECTOR SPACE 1. Introduction This Handout

    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.
  • MTH 304: General Topology Semester 2, 2017-2018

    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..........................
  • ORBITAL RENDEZVOUS and SPACECRAFT LOITERING in the EARTH-MOON SYSTEM a Thesis Submitted to the Faculty of Purdue University by F

    ORBITAL RENDEZVOUS and SPACECRAFT LOITERING in the EARTH-MOON SYSTEM a Thesis Submitted to the Faculty of Purdue University by F

    ORBITAL RENDEZVOUS AND SPACECRAFT LOITERING IN THE EARTH-MOON SYSTEM A Thesis Submitted to the Faculty of Purdue University by Fouad Khoury In Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautics and Astronautics December 2020 Purdue University West Lafayette, Indiana ii THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF THESIS APPROVAL Dr. Kathleen Howell, Chair School of Aeronautics and Astronautics Dr. Carolin Frueh School of Aeronautics and Astronautics Dr. David Spencer School of Aeronautics and Astronautics Approved by: Dr. Gregory Blaisdell Associate Head of the Graduate School of Aeronautics & Astronautics iii To my parents, Saeb & Lama, and my siblings, Omar & Karmah iv ACKNOWLEDGMENTS "The known is finite, the unknown infinite; intellectually we stand on an islet in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land." - T. H. Huxley This work would not be possible without the support of many of my colleagues and mentors. I am grateful for the experiences and interactions I have had (and hopefully continue to have) with each of them. First, I would like to express my gratitude to my adviser Professor Kathleen Howell for her guidance, patience, and encouragement. It has been my honor to serve as her student, teaching assistant, and researcher. I fur- thermore express my gratitude to my fellow researchers in the Multibody Dynamics Research Group, both past and present. Thank you to Andrew C, Robert, Collin, Brian, Emily, RJ, David, Beom, Ricardo, Rohith, Vivek, Juan, Maaninee, Andrew M, Stephen, Kenza, Bonnie, Nick, Paige, Yuki, and Kenji for your technical advice and feedback.
  • Spacecraft Trajectories in a Sun, Earth, and Moon Ephemeris Model

    Spacecraft Trajectories in a Sun, Earth, and Moon Ephemeris Model

    SPACECRAFT TRAJECTORIES IN A SUN, EARTH, AND MOON EPHEMERIS MODEL A Project Presented to The Faculty of the Department of Aerospace Engineering San José State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Aerospace Engineering by Romalyn Mirador i ABSTRACT SPACECRAFT TRAJECTORIES IN A SUN, EARTH, AND MOON EPHEMERIS MODEL by Romalyn Mirador This project details the process of building, testing, and comparing a tool to simulate spacecraft trajectories using an ephemeris N-Body model. Different trajectory models and methods of solving are reviewed. Using the Ephemeris positions of the Earth, Moon and Sun, a code for higher-fidelity numerical modeling is built and tested using MATLAB. Resulting trajectories are compared to NASA’s GMAT for accuracy. Results reveal that the N-Body model can be used to find complex trajectories but would need to include other perturbations like gravity harmonics to model more accurate trajectories. i ACKNOWLEDGEMENTS I would like to thank my family and friends for their continuous encouragement and support throughout all these years. A special thank you to my advisor, Dr. Capdevila, and my friend, Dhathri, for mentoring me as I work on this project. The knowledge and guidance from the both of you has helped me tremendously and I appreciate everything you both have done to help me get here. ii Table of Contents List of Symbols ............................................................................................................................... v 1.0 INTRODUCTION
  • General Topology

    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).
  • Inner Product Spaces

    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.
  • Tethers in Space Handbook - Second Edition

    Tethers in Space Handbook - Second Edition

    Tethers In Space Handbook - Second Edition - (NASA-- - I SECOND LU IT luN (Spectr Rsdrci ystLSw) 259 P C'C1 ? n: 1 National Aeronautics and Space Administration Office of Space Flight Advanced Program Development NASA Headquarters Code MD Washington, DC 20546 This document is the product of support from many organizations and individuals. SRS Technologies, under contract to NASA Headquarters, compiled, updated, and prepared the final document. Sponsored by: National Aeronautics and Space Administration NASA Headquarters, Code MD Washington, DC 20546 Contract Monitor: Edward J. Brazil!, NASA Headquarters Contract Number: NASW-4341 Contractor: SRS Technologies Washington Operations Division 1500 Wilson Boulevard, Suite 800 Arlington, Virginia 22209 Project Manager: Dr. Rodney W. Johnson, SRS Technologies Handbook Editors: Dr. Paul A. Penzo, Jet Propulsion Laboratory Paul W. Ammann, SRS Technologies Tethers In Space Handbook - Second Edition - May 1989 Prepared For: National Aeronautics and Space Administration Office of Space Flight Advanced Program Development NASA National Aeronautics and Space Administration FOREWORD The Tethers in Space Handbook Second Edition represents an update to the initial volume issued in September 1986. As originally intended, this handbook is designed to serve as a reference manual for policy makers, program managers, educators, engineers, and scientists alike. It contains information for the uninitiated, providing insight into the fundamental behavior of tethers in space. For those familiar with space tethers, it summarizes past and ongoing studies and programs, a complete bibliography of tether publications, and names, addresses, and phone numbers of workers in the field. Perhaps its most valuable asset is the brief description of nearly 50 tether applications which have been proposed and analyzed over the past 10 years.
  • AIAA Houston Horizons Online Magazine for Winter 2006/7

    AIAA Houston Horizons Online Magazine for Winter 2006/7

    Volume 32, Issue 1 AIAA Houston Section www.aiaa-houston.org Winter 2006/7 Advanced Propulsion Concepts Rendering by Adrian Mann AIAA Houston Horizons Winter 2006/7 Page 1 Winter 2006/7 T A B L E O F C O N T E N T S From the Editor 3 HOUSTON Chair’s Corner 4 Horizons is a bi-monthly publication of the Houston section The Future of Space Propulsion: VASIMR 5 of the American Institute of Aeronautics and Astronautics. Rockets, Mach Effects, and Mach Lorentz Thrusters 7 Jon S. Berndt Lunch –n– Learn: Constellation Program Overview and Challenges 10 Editor Lunch –n– Learn: Today’s Unfolding Relationships: Earth,Space,Life 12 AIAA Houston Section Executive Council Staying Informed 13 Dr. Jayant Ramakrishnan Lunch –n– Learn: Earth, Moon, and Spacecraft 14 Chair Membership Page 15 Douglas Yazell Calendar 16 Chair-Elect Cranium Cruncher 15 Steven R. King Past Chair Odds and Ends 18 Tim Propp Conference Presentations/Articles by Houston Section Members 20 Secretary DoD Experiments Launch Aboard Space Shuttle Discovery 21 Dr. Brad Files Treasurer AIAA Local Section News 23 JJ Johnson Ellen Gillespie Vice-Chair, Operations Vice-Chair, Technical Operations Technical Dr. Syri Koelfgen Dr. Al Jackson John McCann Padraig Moloney Dr. Rakesh Bhargava Dr. Zafar Taqvi Nicole Smith Bill Atwell Albert Meza Dr. Ilia Rosenberg Horizons and AIAA Dr. Douglas Schwaab Andy Petro Houston Web Site Svetlana Hanson William West AIAA National Laura Slovey Paul Nielsen Communications Award Michael Begley Prerit Shah Winner Jon Berndt, Editor Dr. Michael Lembeck Steve King Dr. Kamlesh Lulla Gary Cowan Gary Brown Councilors 2005 2006 Mike Oelke JR Reyna Brett Anderson This newsletter is created by members of the Houston section.
  • Vector Space Theory

    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
  • Hilbert Spaces

    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.
  • Space Dimension Renormdynamics

    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.