NEWTON’S GREAT… OVERSIGHT Michael Hugh Knowles 1 of 152 Newton’s Great… Oversight [This page intentionally left blank.] 2 of 152 [email protected] NEWTON’S GREAT… OVERSIGHT Galileo’s Falling Bodies, Newton’s Theory of Gravity, and Lagrange’s Trojan Points and Trojan Asteroids with Their Tadpole and Horseshoe Orbits Michael Hugh Knowles November 7, 2011 3 of 152 Newton’s Great… Oversight Copyright © 1995–2011, Michael Hugh Knowles. All rights reserved under International and Pan-American Copyright Conventions. Cover image: Contour plot of a simple function of the falling rate difference of lighter and heavier bodies (a static, 2-dimensional case). One can see regions relating to the stabilities and instabilities associated with Lagrangian points L1-5. The plot suggests that there might be more to the Lagrangian point story than Lagrange discovered, even more than we currently think. (Plotted with MathCAD 2000.) The basic material for this book—“Newton’s Great… Oversight” (NGO)—was first posted on the Internet in 1995. It has been subsequently revised a number of times, with the contour plots finally added in 2000. Since then the technical content has had no major revisions and only a few minor revisions updating Trojan asteroid info. But the overall presentation has been revamped several times; and the philosophical analyses and other maunderings have been extended and reworked a number of times. This (pre-publication, informal peer review version) ebook is intended for and dedicated to “stoutly gifted children of any age” who will see it as a challenging opportunity for the future evolution of science: “Science, right or wrong; when right, to be kept right; when wrong, to be put right.” FAIR WARNING: this ebook contains much that some people will find provocative, controversial, polemical, even obstreperous, one way and/or another, e.g.: “… Science is still evolving in the direction of ‘Religious’ intolerance… … Science is greatly in need of ‘to be put right’…” FEEDBACK: I am hoping very much for high-quality informal peer review FEEDBACK. Please, send your FEEDBACK, ERROR REPORTS, COMMENTS and SUGGESTIONS to Michael at: [email protected] I am especially hoping for GIFTED FEEDBACK (from gifted kids and their gifted teachers). This ebook, though copyright restrictions may otherwise apply, may be distributed freely for non-profit, non-commercial, educational and/or informal peer review purposes if it is distributed on an “as is” basis, unedited, and in its entirety, as this pdf file. A more recent version of this pdf file may be available through the web page: http://www.mhknowles.net/e-Books/NGOgate.pdf Michael Hugh Knowles Rua Miradouro, 45 Sion Belo Horizonte, MG Brasil 30310-640 [email protected] 4 of 152 [email protected] Science, right or wrong; when right, to be kept right; when wrong, to be put right. “I do not say that John or Jonathan will realize all this; but such is the character of that morrow which mere lapse of time can never make to dawn. The light which puts out our eyes is darkness to us. Only that day dawns to which we are awake. There is more day to dawn. The sun is but a morning star.” Henry David Thoreau (1817-1862) the closing words of Walden; or, Life in the Woods 5 of 152 Newton’s Great… Oversight [This page intentionally left blank.] 6 of 152 [email protected] TABLE OF CONTENTS NEWTON’S GREAT… OVERSIGHT ................................................................................................. 1 NEWTON’S GREAT… OVERSIGHT ................................................................................................ 3 TABLE OF CONTENTS ................................................................................................................... 7 CHAPTER CONTENTS PREVIEW OF THIS BOOK ............................................................................. 9 PRELUDE ................................................................................................................................... 11 PROLOGUE ................................................................................................................................ 12 PRE-SCRIPT ............................................................................................................................... 15 PREFACE ................................................................................................................................... 16 DEDICATION ............................................................................................................................. 20 READERSHIP-AUDIENCE ............................................................................................................ 20 1 INTRODUCTION ................................................................................................................ 23 1.1 VARIOUS WARNINGS: ........................................................................................................... 23 1.2 OF APPLES AND “… OVERSIGHTS” ........................................................................................... 23 1.3 “… OVERSIGHTS” ................................................................................................................. 24 1.4 “WAVE AUF WIEDERSEHEN!” ................................................................................................ 24 2 HISTORY ........................................................................................................................... 29 2.1 ARISTOTLE .......................................................................................................................... 29 2.2 GALILEO ............................................................................................................................. 29 2.3 NEWTON ............................................................................................................................ 31 2.4 LAGRANGE AND HIS “TROJAN PLANETS” ................................................................................... 32 2.5 POINCARÉ’S FURTHER WORK ON THE 3-BODY PROBLEM, AND CHAOS THEORY ................................ 34 2.6 EINSTEIN AND HIS “RELATIVITY” ............................................................................................. 34 3 NEWTON’S GRAVITY AND LAGRANGE’S TROJAN POINTS ................................................... 37 3.1 NEWTON’S THEORY OF GRAVITY ............................................................................................. 37 3.2 THE “GRAVITY” OF FALLING APPLES… ...................................................................................... 37 3.3 “INFINITESIMALS” AND LEVELS OF APPROXIMATION .................................................................... 41 3.4 SOME BASIC EQUATIONS AND SOME SIMPLE EQUATIONS ............................................................ 44 3.5 EQUATIONS FOR A SIMPLE 3-BODY PROBLEM ............................................................................ 47 3.6 A QUICK LOOK AT THE SEPARATE RELEASE CASE… AND EINSTEIN’S “RELATIVITY” ............................. 52 3.7 HOMOGRAPHIC PROPERTY—MAINTAINING SHAPE (GEOMETRIC SIMILARITY) .................................. 52 3.8 GEOMETRY AND THE HOMOGRAPHIC PROPERTY ......................................................................... 54 3.9 EQUATIONS FOR HOMOGRAPHICALLY MAINTAINING AN EQUILATERAL TRIANGLE WITHOUT EXPANSION AND CONTRACTION ................................................................................................................................. 54 3.10 HOMOGRAPHIC-EQUILATERAL EXPANSION AND CONTRACTION ..................................................... 61 3.11 TROJAN POINTS?! ................................................................................................................ 61 3.12 STABILITY?! ........................................................................................................................ 62 4 TROJAN POINTS AND THEIR TADPOLE AND HORSESHOE ORBITS ........................................ 67 4.1 TROJAN ASTEROIDS IN TADPOLE ORBITS ................................................................................... 67 4.2 TROJAN ASTEROIDS IN HORSESHOE ORBITS ............................................................................... 67 4.3 TROJAN SPACE DEBRIS?! TROJAN ATMOSPHERES?! .................................................................... 68 TH ST 5 TROJAN ASTRONOMY IN THE 20 AND EARLY 21 CENTURIES .......................................... 71 5.1 HISTORICAL DIGRESSION—WHY “TROJAN” POINTS? .................................................................. 71 TH ST 5.2 TROJAN POINT ASTRONOMY IN THE 20 AND EARLY 21 CENTURY .............................................. 71 ST 5.3 THE FUTURE OF TROJAN POINT ASTRONOMY IN THE 21 CENTURY ............................................... 73 5.4 (MORE) TROJAN STUDY POSSIBILITIES ...................................................................................... 76 5.5 SIMPLE APPROACH TO TROJANS FOR STUDENTS, AMATEURS AND PROFESSIONALS ............................ 76 6 NEWTON’S—AND SCIENCE’S—GREAT “… OVERSIGHTS” .................................................... 79 6.0 PRE-DIATRIBE… ................................................................................................................... 79 6.1 JUST HOW IMPORTANT IS “NEWTON’S GREAT… OVERSIGHT”?! ................................................... 81 7 of 152 Newton’s Great… Oversight 6.2 NEWTON’S GREAT… OVERSIGHT: HOW?! WHY?! ....................................................................