Vll l
CONTENTS
2.3.5 Sundma n inequalit y 6 1
2.4 Mor e conjecture s 6 5
2.4.1 Anothe r conjectur e 6 5
2.4.2 Specia l case s 6 7
2.5 Jacob i coordinate s hel p "see " th e dynamic s 6 9
2.5.1 Velocit y decompositio n an d a basis 7 1
2.5.2 Describin g p" wit h th e basi s 7 4
2.5.3 "Seeing " th e gradien t o f U 7 6
2.5.4 A n illustratin g exampl e 7 7
2.5.5 Findin g centra l an d othe r configuration s 7 9
2.5.6 Equation s o f motion fo r constan t / 8 0
2.5.7 Basi s fo r th e coplana r Af-bod y proble m 8 1
3 Findin g Centra l Configuration s 8 3
3.1 Fro m th e ancien t Greek s t o 8 4
3.1.1 Arithmeti c an d geometri c mean s 8 4
3.1.2 Connectio n wit h centra l configuration s 8 8
3.2 Constraint s 9 2
3.2.1 Singularit y structur e o f F 9 4
3.2.2 Som e dynamics 9 7
3.2.3 Stratifie d structur e o f the imag e o f F 9 9
3.3 Geometri c approach—th e rul e o f signs 102
3.3.1 Th e "eonfigurationa l average d length "
£CAL
103
3.3.2 Sign s of gradients-coplanar configuration s 105
3.3.3 Sign s of gradients-three-dimensional configuration s . . 106
3.3.4 Degenerat e configuration s 107
3.4 Consequence s fo r centra l configuration s 109
3.4.1 Surprisin g regularit y 109
3.4.2 Estimate s o n
£CAL
112
3.4.3 Ar e there centra l configuration s o f these types ? . . . . 114
3.5 Wha t can , an d cannot , b e 115
3.5.1 Mor e centra l configuration s 115
3.5.2 Masse s an d collinea r centra l configuration s 119
3.5.3 Masse s an d coplana r configuration s 124
3.6 Ne w kinds o f constraints 126
3.7 Ring s o f Satur n 130
3.7.1 Stabilit y 13
3.7.2 Mor e ring s 133
3.7.3 Saturn , an d som e problem s 136
Previous Page Next Page