A quick overview of the central configuration of the Newtonian n-body problem

Transcript

1. Central configuration of the n-body problem: A quick overview of a “problem for the 21st century” Tianran Chen Department of Mathematics Michigan State University Nov. 5, 2012 T.R.Chen (MSU) N-body Nov. 5, 2012 1 / 28
2. Newtonian n-body problem Predicting the motion of celestial bodies governed by gravity. Sun Earth T.R.Chen (MSU) N-body Nov. 5, 2012 2 / 28
3. Newton’s second law m F ma = F T.R.Chen (MSU) N-body Nov. 5, 2012 3 / 28
4. Newton’s second law m F ma = F m F T.R.Chen (MSU) N-body Nov. 5, 2012 3 / 28
5. Newton’s second law m F ma = F m F ma = F T.R.Chen (MSU) N-body Nov. 5, 2012 3 / 28
6. Newton’s second law m F ma = F m F ma = F m F1 F2 F3 T.R.Chen (MSU) N-body Nov. 5, 2012 3 / 28
7. Newton’s second law m F ma = F m F ma = F m F1 F2 F3 ma = F1 + F2 + F3 T.R.Chen (MSU) N-body Nov. 5, 2012 3 / 28
8. Newton’s second law m F ma = F m F ma = F m F1 F2 F3 ma = F1 + F2 + F3 In general ma = j Fj T.R.Chen (MSU) N-body Nov. 5, 2012 3 / 28
9. Newton’s law of universal gravitation r m1 m2 F T.R.Chen (MSU) N-body Nov. 5, 2012 4 / 28
10. Newton’s law of universal gravitation r m1 m2 F F = G m1m2 r2 T.R.Chen (MSU) N-body Nov. 5, 2012 4 / 28
11. Newton’s law of universal gravitation r m1 m2 F F = m1m2 r2 T.R.Chen (MSU) N-body Nov. 5, 2012 4 / 28
12. Newton’s law of universal gravitation r m1 m2 F F = m1m2 r2 r m1 m2 F T.R.Chen (MSU) N-body Nov. 5, 2012 4 / 28
13. Newton’s law of universal gravitation r m1 m2 F F = m1m2 r2 r m1 m2 F F = m1m2 r2 u T.R.Chen (MSU) N-body Nov. 5, 2012 4 / 28
14. Second law + gravitation r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 T.R.Chen (MSU) N-body Nov. 5, 2012 5 / 28
15. Second law + gravitation r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F1 = m1m2 r2 u T.R.Chen (MSU) N-body Nov. 5, 2012 5 / 28
16. Second law + gravitation r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F1 = m1m2 r2 u u = x2 − x1 r T.R.Chen (MSU) N-body Nov. 5, 2012 5 / 28
17. Second law + gravitation r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F1 = m1m2 r2 u u = x2 − x1 r F1 = m1a1 = m1 ¨ x1 T.R.Chen (MSU) N-body Nov. 5, 2012 5 / 28
18. Second law + gravitation r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F1 = m1m2 r2 u u = x2 − x1 r F1 = m1a1 = m1 ¨ x1 m1 ¨ x1 = m1m2 r2 x2 − x1 r = m1m2(x2 − x1) r3 T.R.Chen (MSU) N-body Nov. 5, 2012 5 / 28
19. Second law + gravitation r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F1 = m1m2 r2 u u = x2 − x1 r F1 = m1a1 = m1 ¨ x1 m1 ¨ x1 = m1m2 r2 x2 − x1 r = m1m2(x2 − x1) r3 I.e. m1 ¨ x1 = m1m2(x2 − x1) r3 T.R.Chen (MSU) N-body Nov. 5, 2012 5 / 28
20. Second law + gravitation r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F1 = m1m2 r2 u u = x2 − x1 r F1 = m1a1 = m1 ¨ x1 m1 ¨ x1 = m1m2 r2 x2 − x1 r = m1m2(x2 − x1) r3 I.e. m1 ¨ x1 = m1m2(x2 − x1) r3 Second order ODE T.R.Chen (MSU) N-body Nov. 5, 2012 5 / 28
21. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
22. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 m1 ¨ x1 = m1m2(x2 − x1) r3 m2 ¨ x2 = m1m2(x1 − x2) r3 T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
23. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 m1 ¨ x1 = m1m2(x2 − x1) r3 m2 ¨ x2 = m1m2(x1 − x2) r3 The Newtonian 2-body problem T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
24. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 m1 ¨ x1 = m1m2(x2 − x1) r3 m2 ¨ x2 = m1m2(x1 − x2) r3 The Newtonian 2-body problem 1500s Galilei T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
25. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 m1 ¨ x1 = m1m2(x2 − x1) r3 m2 ¨ x2 = m1m2(x1 − x2) r3 The Newtonian 2-body problem 1500s Galilei 1600s Kepler T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
26. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 m1 ¨ x1 = m1m2(x2 − x1) r3 m2 ¨ x2 = m1m2(x1 − x2) r3 The Newtonian 2-body problem 1500s Galilei 1600s Kepler 1600s Hooke T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
27. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 m1 ¨ x1 = m1m2(x2 − x1) r3 m2 ¨ x2 = m1m2(x1 − x2) r3 The Newtonian 2-body problem 1500s Galilei 1600s Kepler 1600s Hooke 1600s Newton T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
28. 2-body problem r mass: m1 position: x1(t) mass: m2 position: x2(t) F1 F2 m1 ¨ x1 = m1m2(x2 − x1) r3 m2 ¨ x2 = m1m2(x1 − x2) r3 The Newtonian 2-body problem 1500s Galilei 1600s Kepler 1600s Hooke 1600s Newton Elliptic orbits or collision T.R.Chen (MSU) N-body Nov. 5, 2012 6 / 28
29. 3-body problem m1, x1(t) m2, x2(t) m3, x3(t) T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
30. 3-body problem m1, x1(t) m2, x2(t) m3, x3(t) T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
31. 3-body problem m1, x1(t) m2, x2(t) m3, x3(t) T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
32. 3-body problem m1, x1(t) m2, x2(t) m3, x3(t) T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
33. 3-body problem m1, x1(t) m2, x2(t) m3, x3(t) T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
34. 3-body problem r12 r23 r13 m1, x1(t) m2, x2(t) m3, x3(t) T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
35. 3-body problem r12 r23 r13 m1, x1(t) m2, x2(t) m3, x3(t) m1 ¨ x1 T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
36. 3-body problem r12 r23 r13 m1, x1(t) m2, x2(t) m3, x3(t) m1 ¨ x1 = m1m2(x2 − x1) r3 12 T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
37. 3-body problem r12 r23 r13 m1, x1(t) m2, x2(t) m3, x3(t) m1 ¨ x1 = m1m2(x2 − x1) r3 12 + m1m3(x3 − x1) r3 13 T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
38. 3-body problem r12 r23 r13 m1, x1(t) m2, x2(t) m3, x3(t) m1 ¨ x1 = m1m2(x2 − x1) r3 12 + m1m3(x3 − x1) r3 13 m2 ¨ x2 = m2m1(x1 − x2) r3 12 + m2m3(x3 − x2) r3 23 T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
39. 3-body problem r12 r23 r13 m1, x1(t) m2, x2(t) m3, x3(t) m1 ¨ x1 = m1m2(x2 − x1) r3 12 + m1m3(x3 − x1) r3 13 m2 ¨ x2 = m2m1(x1 − x2) r3 12 + m2m3(x3 − x2) r3 23 m3 ¨ x3 = m3m1(x1 − x3) r3 13 + m3m2(x2 − x3) r3 23 T.R.Chen (MSU) N-body Nov. 5, 2012 7 / 28
40. n-body problem m1, x1(t) T.R.Chen (MSU) N-body Nov. 5, 2012 8 / 28
41. n-body problem m1, x1(t) mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk T.R.Chen (MSU) N-body Nov. 5, 2012 8 / 28
42. n-body problem m1, x1(t) mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Contain singularities (“fully” classified) T.R.Chen (MSU) N-body Nov. 5, 2012 8 / 28
43. n-body problem m1, x1(t) mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Contain singularities (“fully” classified) Collision whenever rjk = 0 T.R.Chen (MSU) N-body Nov. 5, 2012 8 / 28
44. n-body problem m1, x1(t) mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Contain singularities (“fully” classified) Collision whenever rjk = 0 Separation rjk → ∞ as t → ∞ T.R.Chen (MSU) N-body Nov. 5, 2012 8 / 28
45. n-body problem m1, x1(t) mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Contain singularities (“fully” classified) Collision whenever rjk = 0 Separation rjk → ∞ as t → ∞ Collision at ∞ rjk → ∞ in finite time T.R.Chen (MSU) N-body Nov. 5, 2012 8 / 28
46. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
47. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
48. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
49. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
50. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
51. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
52. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
53. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems Orbits can be chaotic for n 3 T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
54. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems Orbits can be chaotic for n 3 Long term simulation can be hard T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
55. n-body problem mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk Numerical simulations can be done Power series solutions exist away from singularities Singularities are “fully” classified Puiseux series solutions exist near certain singularities What more do we want??? General n-body problems Orbits can be chaotic for n 3 Long term simulation can be hard Orbits near singularity are hard to model T.R.Chen (MSU) N-body Nov. 5, 2012 9 / 28
56. Central configurations A special class of solutions deserves attention: Central Configuration T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
57. Central configurations A special class of solutions deserves attention: Central Configuration “Acceleration of each body line up with the center of mass” T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
58. Central configurations A special class of solutions deserves attention: Central Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
59. Central configurations A special class of solutions deserves attention: Central Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass The combined force of all other bodies mimic a single central force T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
60. Central configurations A special class of solutions deserves attention: Central Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass The combined force of all other bodies mimic a single central force F1 F2 F3 m1, x1(t) T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
61. Central configurations A special class of solutions deserves attention: Central Configuration “Acceleration of each body line up with the center of mass” ¨ xj = λj(c − xj) for some λj > 0 c = center of mass The combined force of all other bodies mimic a single central force F1 F2 F3 m1, x1(t) c F1 + F2 + F3 m1, x1(t) T.R.Chen (MSU) N-body Nov. 5, 2012 10 / 28
62. Central configurations: interesting properties ¨ xj = λj(c − xj) T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
63. Central configurations: interesting properties ¨ xj = λj(c − xj) A central configuration will remain a central configuration for all t T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
64. Central configurations: interesting properties ¨ xj = λj(c − xj) A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
65. Central configurations: interesting properties ¨ xj = λj(c − xj) A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
66. Central configurations: interesting properties ¨ xj = λj(c − xj) A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) Orbits are uniform rotations around the center of mass xj(t) = R(t)xj(0) T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
67. Central configurations: interesting properties ¨ xj = λj(c − xj) A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) Orbits are uniform rotations around the center of mass xj(t) = R(t)xj(0) Prediction can be made with arbitrary accuracy T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
68. Central configurations: interesting properties ¨ xj = λj(c − xj) A central configuration will remain a central configuration for all t The geometric shape formed by x1(t), . . . , xn(t) remain congruent to the initial shape for all t Planar case: xj(t) ∈ R2 (relative equilibrium) Orbits are uniform rotations around the center of mass xj(t) = R(t)xj(0) Prediction can be made with arbitrary accuracy The system is “stable” (as long as no collision) T.R.Chen (MSU) N-body Nov. 5, 2012 11 / 28
69. Central configurations: applications ¨ xj = λj(c − xj) T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
70. Central configurations: applications ¨ xj = λj(c − xj) Analyze collision T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
71. Central configurations: applications ¨ xj = λj(c − xj) Analyze collision Analyze separation T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
72. Central configurations: applications ¨ xj = λj(c − xj) Analyze collision Analyze separation We see everywhere T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
73. Central configurations: applications ¨ xj = λj(c − xj) Analyze collision Analyze separation We see everywhere Generalizations: T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
74. Central configurations: applications ¨ xj = λj(c − xj) Analyze collision Analyze separation We see everywhere Generalizations: Einstein’s n-body problem T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
75. Central configurations: applications ¨ xj = λj(c − xj) Analyze collision Analyze separation We see everywhere Generalizations: Einstein’s n-body problem Point charges problem T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
76. Central configurations: applications ¨ xj = λj(c − xj) Analyze collision Analyze separation We see everywhere Generalizations: Einstein’s n-body problem Point charges problem Vortex problem T.R.Chen (MSU) N-body Nov. 5, 2012 12 / 28
77. The basic questions T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
78. The basic questions Does central configurations exist? T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
79. The basic questions Does central configurations exist? If yes, is there only finite many of them? T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
80. The basic questions Does central configurations exist? If yes, is there only finite many of them? If finite, how many? T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
81. The basic questions Does central configurations exist? If yes, is there only finite many of them? If finite, how many? What are they? T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
82. The basic questions Does central configurations exist? If yes, is there only finite many of them? If finite, how many? What are they? How do they behave? T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
83. The basic questions Does central configurations exist? If yes, is there only finite many of them? If finite, how many? What are they? How do they behave? Do they exist in nature? T.R.Chen (MSU) N-body Nov. 5, 2012 13 / 28
84. Equivalent classes ¨ xj = λj(c − xj) remain invariant under. . . T.R.Chen (MSU) N-body Nov. 5, 2012 14 / 28
85. Equivalent classes ¨ xj = λj(c − xj) remain invariant under. . . Scaling T.R.Chen (MSU) N-body Nov. 5, 2012 14 / 28
86. Equivalent classes ¨ xj = λj(c − xj) remain invariant under. . . Scaling Affine translation T.R.Chen (MSU) N-body Nov. 5, 2012 14 / 28
87. Equivalent classes ¨ xj = λj(c − xj) remain invariant under. . . Scaling Affine translation Permutation T.R.Chen (MSU) N-body Nov. 5, 2012 14 / 28
88. Equivalent classes ¨ xj = λj(c − xj) remain invariant under. . . Scaling Affine translation Permutation Rotation T.R.Chen (MSU) N-body Nov. 5, 2012 14 / 28
89. Equivalent classes ¨ xj = λj(c − xj) remain invariant under. . . Scaling Affine translation Permutation Rotation Equivalent classes of central configurations T.R.Chen (MSU) N-body Nov. 5, 2012 14 / 28
90. Equivalent classes ¨ xj = λj(c − xj) remain invariant under. . . Scaling Affine translation Permutation Rotation Equivalent classes of central configurations x1 x2 x3 is equivalent to x1 x2 x3 T.R.Chen (MSU) N-body Nov. 5, 2012 14 / 28
91. 2-body problem Trivial! T.R.Chen (MSU) N-body Nov. 5, 2012 15 / 28
92. 2-body problem Trivial! Every configuration is a central configuration T.R.Chen (MSU) N-body Nov. 5, 2012 15 / 28
93. 2-body problem Trivial! Every configuration is a central configuration All configurations are equivalent: 1 equivalent class T.R.Chen (MSU) N-body Nov. 5, 2012 15 / 28
94. 2-body problem Trivial! Every configuration is a central configuration All configurations are equivalent: 1 equivalent class x1(t) x2(t) ¨ x1 ¨ x2 T.R.Chen (MSU) N-body Nov. 5, 2012 15 / 28
95. 2-body problem Trivial! Every configuration is a central configuration All configurations are equivalent: 1 equivalent class x1(t) x2(t) ¨ x1 ¨ x2 Orbits completely understood by Newton (Principia: 1687) T.R.Chen (MSU) N-body Nov. 5, 2012 15 / 28
96. 2-body problem Trivial! Every configuration is a central configuration All configurations are equivalent: 1 equivalent class x1(t) x2(t) ¨ x1 ¨ x2 Orbits completely understood by Newton (Principia: 1687) Exist in nature (everywhere) T.R.Chen (MSU) N-body Nov. 5, 2012 15 / 28
97. 3-body problem Central configurations for 3-body problem (relative equilibria) T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
98. 3-body problem Central configurations for 3-body problem (relative equilibria) Exists? Yes. T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
99. 3-body problem Central configurations for 3-body problem (relative equilibria) Exists? Yes. Finite? Yes. T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
100. 3-body problem Central configurations for 3-body problem (relative equilibria) Exists? Yes. Finite? Yes. How many? Exactly 5 equivalent classes: T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
101. 3-body problem Central configurations for 3-body problem (relative equilibria) Exists? Yes. Finite? Yes. How many? Exactly 5 equivalent classes: Lagrangian points T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
102. 3-body problem Central configurations for 3-body problem (relative equilibria) Exists? Yes. Finite? Yes. How many? Exactly 5 equivalent classes: Lagrangian points Newton 1 collinear class (hypothesis) T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
103. 3-body problem Central configurations for 3-body problem (relative equilibria) Exists? Yes. Finite? Yes. How many? Exactly 5 equivalent classes: Lagrangian points Newton 1 collinear class (hypothesis) Euler 3 collinear class T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
104. 3-body problem Central configurations for 3-body problem (relative equilibria) Exists? Yes. Finite? Yes. How many? Exactly 5 equivalent classes: Lagrangian points Newton 1 collinear class (hypothesis) Euler 3 collinear class Lagrange All 5 classes: Lagrangian points (roots of a 5th degree polynomial) T.R.Chen (MSU) N-body Nov. 5, 2012 16 / 28
105. Euler’s central configurations Earth Moon T.R.Chen (MSU) N-body Nov. 5, 2012 17 / 28
106. Euler’s central configurations Earth Moon T.R.Chen (MSU) N-body Nov. 5, 2012 17 / 28
107. Euler’s central configurations Earth Moon T.R.Chen (MSU) N-body Nov. 5, 2012 17 / 28
108. Euler’s central configurations Earth Moon T.R.Chen (MSU) N-body Nov. 5, 2012 17 / 28
109. Lagrange’s central configurations Earth Moon T.R.Chen (MSU) N-body Nov. 5, 2012 18 / 28
110. Lagrange’s central configurations Earth Moon T.R.Chen (MSU) N-body Nov. 5, 2012 18 / 28
111. Lagrange’s central configurations Earth Moon T.R.Chen (MSU) N-body Nov. 5, 2012 18 / 28
112. Lagrangian points in nature 1772 Lagarange: theoretical proof T.R.Chen (MSU) N-body Nov. 5, 2012 19 / 28
113. Lagrangian points in nature 1772 Lagarange: theoretical proof 1906 Max Wolf: Jupiter’s Trojan astroids on both equilateral triangles T.R.Chen (MSU) N-body Nov. 5, 2012 19 / 28
114. Lagrangian points in nature 1772 Lagarange: theoretical proof 1906 Max Wolf: Jupiter’s Trojan astroids on both equilateral triangles . . . . . . T.R.Chen (MSU) N-body Nov. 5, 2012 19 / 28
115. Lagrangian points in nature 1772 Lagarange: theoretical proof 1906 Max Wolf: Jupiter’s Trojan astroids on both equilateral triangles . . . . . . 1990 Levy and Holt: Mars’ Eureka trojan astroid (trails Mars) T.R.Chen (MSU) N-body Nov. 5, 2012 19 / 28
116. Lagrangian points in nature 1772 Lagarange: theoretical proof 1906 Max Wolf: Jupiter’s Trojan astroids on both equilateral triangles . . . . . . 1990 Levy and Holt: Mars’ Eureka trojan astroid (trails Mars) 2010 NEOWISE: Earth’s trojan astroid (ahead of earth) T.R.Chen (MSU) N-body Nov. 5, 2012 19 / 28
117. Lagrangian points in nature 1772 Lagarange: theoretical proof 1906 Max Wolf: Jupiter’s Trojan astroids on both equilateral triangles . . . . . . 1990 Levy and Holt: Mars’ Eureka trojan astroid (trails Mars) 2010 NEOWISE: Earth’s trojan astroid (ahead of earth) Now Many man-made objects T.R.Chen (MSU) N-body Nov. 5, 2012 19 / 28
118. Beyond 3-body Does a central configuration alway exist for n-body problem? T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
119. Beyond 3-body Does a central configuration alway exist for n-body problem? Yes. T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
120. Beyond 3-body Does a central configuration alway exist for n-body problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
121. Beyond 3-body Does a central configuration alway exist for n-body problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
122. Beyond 3-body Does a central configuration alway exist for n-body problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
123. Beyond 3-body Does a central configuration alway exist for n-body problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses Are there only finite many of them? T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
124. Beyond 3-body Does a central configuration alway exist for n-body problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses Are there only finite many of them? No one knows! T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
125. Beyond 3-body Does a central configuration alway exist for n-body problem? Yes. Theorem (Moulton 1910) There are exactly n!/2 collinear central configurations. Are there other ones? Yes for equal masses Are there only finite many of them? No one knows! Problem (Smale’s 6th problem) Is the number of equivalent classes for planar central configurations (relative equilibria) for the n-body problem finite? T.R.Chen (MSU) N-body Nov. 5, 2012 20 / 28
126. Why is it important? T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
127. Why is it important? An very old problem T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
128. Why is it important? An very old problem Many people have tried (and failed) T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
129. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
130. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
131. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
132. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
133. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
134. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
135. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
136. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
137. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? Dark matter problem T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
138. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? Dark matter problem The vortex problem T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
139. Why is it important? An very old problem Many people have tried (and failed) Involves many kind of mathematics Differential equation Topology Geometry Several complex variables Algebraic geometry Many implications in Physics Is Saturn’s rings a central configuraion? Dark matter problem The vortex problem Many application in computer simulation T.R.Chen (MSU) N-body Nov. 5, 2012 21 / 28
140. Known results T.R.Chen (MSU) N-body Nov. 5, 2012 22 / 28
141. Known results Planar 4-body 2005: Hampton and Moeckel T.R.Chen (MSU) N-body Nov. 5, 2012 22 / 28
142. Known results Planar 4-body 2005: Hampton and Moeckel Planar 5-body 2012: Albouy and Kaloshin T.R.Chen (MSU) N-body Nov. 5, 2012 22 / 28
143. Known results Planar 4-body 2005: Hampton and Moeckel Planar 5-body 2012: Albouy and Kaloshin Algebraic geometry / Tropical geometry T.R.Chen (MSU) N-body Nov. 5, 2012 22 / 28
144. Known results Planar 4-body 2005: Hampton and Moeckel Planar 5-body 2012: Albouy and Kaloshin Algebraic geometry / Tropical geometry Computer assisted proofs T.R.Chen (MSU) N-body Nov. 5, 2012 22 / 28
145. Proving the finiteness ¨ xj = λj(c − xj) mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
146. Proving the finiteness ¨ xj = λj(c − xj) mj ¨ xj = n k=1,k j mjmk (xk − xj) r3 jk T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
147. Proving the finiteness ¨ xj = λj(c − xj) ¨ xj = n k=1,k j mk (xk − xj) r3 jk T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
148. Proving the finiteness ¨ xj = λj(c − xj) ¨ xj = n k=1,k j mk (xk − xj) r3 jk T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
149. Proving the finiteness ¨ xj = λj(c − xj) ¨ xj = n k=1,k j mk (xk − xj) r3 jk λj(c − xj) = n k=1,k j mk (xk − xj) r3 jk T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
150. Proving the finiteness ¨ xj = λj(c − xj) ¨ xj = n k=1,k j mk (xk − xj) r3 jk λj(c − xj) = n k=1,k j mk (xk − xj) r3 jk No derivative respect to t: T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
151. Proving the finiteness ¨ xj = λj(c − xj) ¨ xj = n k=1,k j mk (xk − xj) r3 jk λj(c − xj) = n k=1,k j mk (xk − xj) r3 jk No derivative respect to t: we can fix t = 0 (initial configuration) T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
152. Proving the finiteness ¨ xj = λj(c − xj) ¨ xj = n k=1,k j mk (xk − xj) r3 jk λj(c − xj) = n k=1,k j mk (xk − xj) r3 jk No derivative respect to t: we can fix t = 0 (initial configuration) Still need to capture solutions form equivalent classes Scaling Translation Permutation Rotation T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
153. Proving the finiteness ¨ xj = λj(c − xj) ¨ xj = n k=1,k j mk (xk − xj) r3 jk λj(c − xj) = n k=1,k j mk (xk − xj) r3 jk No derivative respect to t: we can fix t = 0 (initial configuration) Still need to capture solutions form equivalent classes Scaling Translation Permutation Rotation can be done via change of variables T.R.Chen (MSU) N-body Nov. 5, 2012 23 / 28
154. An equivalent problem Original ODE −→ · · · −→ Algebraic equations T.R.Chen (MSU) N-body Nov. 5, 2012 24 / 28
155. An equivalent problem Original ODE −→ · · · −→ Algebraic equations I.e. get a polynomial system P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) T.R.Chen (MSU) N-body Nov. 5, 2012 24 / 28
156. An equivalent problem Original ODE −→ · · · −→ Algebraic equations I.e. get a polynomial system P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) Equivalent classes of planar central configurations have a one-to-one correspondence to (positive) solutions of P = 0. T.R.Chen (MSU) N-body Nov. 5, 2012 24 / 28
157. An equivalent problem Original ODE −→ · · · −→ Algebraic equations I.e. get a polynomial system P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) Equivalent classes of planar central configurations have a one-to-one correspondence to (positive) solutions of P = 0. Equivalent question Is the number of positive solutions of P = 0 finite? T.R.Chen (MSU) N-body Nov. 5, 2012 24 / 28
158. An equivalent problem Original ODE −→ · · · −→ Algebraic equations I.e. get a polynomial system P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) Equivalent classes of planar central configurations have a one-to-one correspondence to (positive) solutions of P = 0. Equivalent question Is the number of positive solutions of P = 0 finite? Stronger question Is the number of complex solutions of P = 0 finite? T.R.Chen (MSU) N-body Nov. 5, 2012 24 / 28
159. A stronger problem Is the number of complex solutions P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) finite? T.R.Chen (MSU) N-body Nov. 5, 2012 25 / 28
160. A stronger problem Is the number of complex solutions P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) finite? If yes, then we have the finiteness of planar central configurations. T.R.Chen (MSU) N-body Nov. 5, 2012 25 / 28
161. A stronger problem Is the number of complex solutions P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) finite? If yes, then we have the finiteness of planar central configurations. Tropism: . . . T.R.Chen (MSU) N-body Nov. 5, 2012 25 / 28
162. A stronger problem Is the number of complex solutions P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) finite? If yes, then we have the finiteness of planar central configurations. Tropism: . . . P(z1, . . . , zm) = 0 T.R.Chen (MSU) N-body Nov. 5, 2012 25 / 28
163. A stronger problem Is the number of complex solutions P(z1, . . . , zm) =        P1(z1, . . . , zm) . . . Pm(z1, . . . , zm) finite? If yes, then we have the finiteness of planar central configurations. Tropism: . . . P(z1, . . . , zm) = 0 has tropisms no tropisms =⇒ Finite number of solutions T.R.Chen (MSU) N-body Nov. 5, 2012 25 / 28
164. A stronger problem Original ODE T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
165. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
166. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 Finiteness of central configurations T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
167. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
168. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
169. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
170. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 Classical algebraic geometry: probably impossible for n > 5 T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
171. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 Classical algebraic geometry: probably impossible for n > 5 More sophisticated numerical algorithms are needed T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
172. A stronger problem Original ODE −→ · · · −→ Polynomial system P(z) = 0 Finiteness of central configurations ⇐= P(z) has no tropism The key is to find “tropism” Brutal force: probably impossible for n > 5 Classical algebraic geometry: probably impossible for n > 5 More sophisticated numerical algorithms are needed There may be a connection to the homotopy continuation methods T.R.Chen (MSU) N-body Nov. 5, 2012 26 / 28
173. Summary T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
174. Summary Newtonian n-body problem T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
175. Summary Newtonian n-body problem Central configurations of n-body problem T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
176. Summary Newtonian n-body problem Central configurations of n-body problem An important problem T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
177. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
178. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
179. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem An open problem, very little is known T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
180. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
181. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
182. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
183. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? Can we numerically find central configurations? (Open for n > 4) T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
184. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? Can we numerically find central configurations? (Open for n > 4) Can we use numerical methods to understand the orbits? T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
185. Summary Newtonian n-body problem Central configurations of n-body problem An important problem An old problem An interesting problem An open problem, very little is known Involves many different kinds of mathematics The computational aspect: Can numerical results prove or disprove the conjecture? Can we numerically find central configurations? (Open for n > 4) Can we use numerical methods to understand the orbits? The applications: T.R.Chen (MSU) N-body Nov. 5, 2012 27 / 28
186. Thank You! T.R.Chen (MSU) N-body Nov. 5, 2012 28 / 28