CONTENTS i x
4 Collision s bot h rea l an d imaginar y 137
4.1 On e bod y proble m 138
4.1.1 Levi-Civita' s approac h 140
4.1.2 Kustaanheim o an d Stiefel' s approac h 141
4.1.3 Topologica l obstruction s an d hair y ball s 143
4.1.4 Sundman' s solutio n o f the three-bod y proble m . . . . 144
4.2 Sundma n an d th e three-bod y proble m 147
4.2.1 Comple x singularities ? 147
4.2.2 Avoidin g comple x singularitie s 148
4.2.3 Singularitie s retaliat e 149
4.3 Generalize d Weierstrass-Sundma n theore m 150
4.3.1 A simple case-th e centra l forc e proble m 151
4.3.2 Large r p-values an d "Blac k Holes " 151
4.3.3 Lagrange-Jacob i equatio n 153
4.3.4 Proo f o f the Weierstrass-Sundma n Theore m 154
4.3.5 Bounde d abov e mean s bounde d belo w 159
4.3.6 Problem s 161
4.3.7 A n interestin g historica l footnot e 161
4.4 Singularitie s - a n overvie w 162
4.4.1 Behavio r o f a singularity 163
4.4.2 Non-collisio n singularitie s 165
4.4.3 Of f t o infinit y 166
4.4.4 Problem s 170
4.5 Rat e o f approach o f collisions 172
4.5.1 Genera l collision s 172
4.5.2 Tauberia n Theorem s 173
4.5.3 Proo f o f the theore m 179
4.6 Sharpe r asymptoti c result s 184
4.7 Spin , o r n o spin? 185
4.7.1 Usin g the angula r momentu m 187
4.7.2 Th e collinea r cas e 189
4.8 Manifold s define d b y collision s 191
4.8.1 Structur e o f collision set s 192
4.8.2 Proo f o f Theorem 4.18 193
4.9 Proo f o f the slowl y varying assertio n 195
4.9.1 Center s o f mass 198
4.9.2 Bac k t o th e proo f 20 1
4.9.3 Th e las t step s 20 3
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