A05-132 2010 2 11 1
1 3 1.1.......................................... 3 1.2..................................... 3 1.3..................................... 3 2 4 2.1............................... 4 2.2 Runge-Kutta.................................... 8 2.3.................................... 10 2.4................................ 10 3 13 3.1................................. 13 3.2..................................... 13 4 14 4.1............................... 14 4.2............................ 15 4.3................................ 16 4.3.1.................................. 16 4.3.2............................. 18 4.3.3 XY/XZ........................... 19 4.3.4 /.............................. 20 4.3.5 /................................. 21 4.4.................................... 25 5 26 2
1 1.1 8 100 1 8 [1] 1.2 xyz 3 2 x-y x-z 1.3 2 3 4 5 3
2 2.1 3 M m m d2 r dt 2 = GMm r 2 (2.1) G r(t) (t ) m (2.1) N 1 N M 1 d 2 r dt 2 = GM 1M 2 r 2 G M 1M 3 r 2 (2.2) M 1 M 2 M 3 4
1: N M 1 M 2 M 3 V n i ( m i ) d 2 r m i dt = n 2 j=1 G m im j r 2 ij (2.3) r ij i j i j i,j=1,2,3... n 3 x-y x-z 3 [2] 2 2: 5
x, y, z 3 (x,y or z) x θ ( 3) 3: (x,y or z) x θ cos θ x = x r, sin θ y = y r, sin θ z = z r (2.4) m i d 2 x dt 2 m i d 2 y dt 2 = n j=1 = n j=1 d 2 z m i dt = n 2 j=1 G m im j r 2 ij G m im j r 2 ij G m im j r 2 ij x r y r z r (2.5) (2.6) (2.7) x, y, z, r x = x j x i (2.8) y = y j y i (2.9) z = z j z i (2.10) r = (x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 (2.11) m i d 2 x i dt 2 = n j=1 Gm i m j (x j x i ) ((x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 ) 3 2 6 (2.12)
m i d 2 y i dt 2 m i d 2 z i dt 2 = n j=1 = n j=1 Gm i m j (y j y i ) ((x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 ) 3 2 Gm i m j (z j z i ) ((x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 ) 3 2 (2.13) (2.14) 7
2.2 Runge-Kutta Runge-Kutta Runge-Kutta Euler 1 x 2 Runge-Kutta dy = f(x, y) (2.15) dx (x n,y n ) x n+1 =x n + x y n+1 k 1 = x f (x n, y n ), k 2 = x f (x n + x, y n + k 1 ) y n+1 = y n + 1 2 (k 1 + k 2 ) (2.16) k 1 x n y n+1 k 2 k 1 2 O(( x 3 )) 2 1 y n Runge-Kutta s k i = x f(x n + c i x, y n + x a ij k j ) j =1 y n+1 = y n + 1 2 (k 1 + k 2 ) (2.17) a ij, b i, c i s 4 Runge-Kutta k 1 = x f (x n, y n ) k 2 = x f(x n + x 2, y n + k 1 2 ) k 3 = x f(x n + x 2, y n + k 2 2 ) k 4 = x f(x n + x, y n + k 3 ) y n+1 = 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ) (2.18) 4 x n x n k 1,k 2,k 3,k 4 y n+1 y n x x 8
4: 4 Runge-Kutta x 0 x 0 +h 4 1. (x 0,y 0 ) f(x 0,y 0 ) h y k 1 2. 1 (x 0 + h/2,y 0 + k 1 /2) (x 0,y 0 ) h y k 2 3. (x 0,y 0 ) (x 0 + h/2,y 0 + k 2 /2) h k 3 4. (x 0,y 0 ) (x 0,y 0 ) (x 0 + h/2,y 0 + k 3 /2) h k 4 5. (x 1,y 1 ) Runge-Kutta 1 2 (2.12)(2.13)(2.14) Runge-Kutta dx dt = v x dv n x m i dt = dy dt = v y j=1 dv n y m i dt = j=1 Gm i m j (x j x i ) ((x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 ) 3 2 Gm i m j (y j y i ) ((x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 ) 3 2 (2.19) (2.20) (2.21) (2.22) 9
dz dt = v z dv n z m i dt = j=1 Gm i m j (z j z i ) ((x j x i ) 2 + (y j y i ) 2 + (z j z i ) 2 ) 3 2 (2.23) (2.24) 2 1 Runge-Kutta 2.3 G ( ) 1969 6 27 [4] - / Runge-Kutta h 0.002 G = 6.672 10 11 m 3 s 2 kg 1 A = 1.49597870 10 11 m 2.4 0.002 3142 1 [3] 0 029 % 17 10
( ) 1.0 1.0 / 6023600 1.0 / 408523.5 1.0 / 332946 1.0 / 3098710 1.0 / 1047.355 1.0 / 3498.5 1.0 / 22869 1.0 / 19314 1.0 / 15384615 (1.0 / 332946) 0.01230002 1: ( ) ( / ) x (A[m]) y (A[m]) z (A[m]) 0.0 0.0 0.0 3.572602 10 1-9.154906 10 2-8.598100 10 2 6.082494 10 1-3.491324 10 1-1.955433 10 1 1.160148 10 1-9.266055 10 1-4.018062 10 1-1.146886 10 1-1.328366-6.061551 10 1-5.384208-8.312495 10 1-2.250979 10 1 7.889889 4.595709 1.558429-1.826989 10-1.162732-2.503762 10 1-1.605950 10-2.394293 10-9.400426-3.048800 10-8.732159 10 1 8.911354-8.081773 10 4-1.994630 10 3-1.087262 10 3 2: x, y, z 1969 6 27 6 11
x (A[m/ ]) y (A[m/ ]) z (A[m/ ]) 0.0 0.0 0.0 3.367845 10 3-2.488934 10 2 1.294407 10 2 1.095242 10 2-1.561250 10 2 6.328876 10 3 1.681162 10 2-1.743130 10 3 7.599750 10 4 1.448200 10 2-2.372847 10 4-2.837487 10 4 1.092366 10 3 6.523294 10 3-2.823012 10 3-3.217205 10 3-4.330631 10 3 1.926416 10 3 2.215450 10 4 3.767654 10 3-1.653244 10 3 2.643125 10 3 1.503487 10 3-6.812687 10 4 3.225418 10 4 3.148760 10 3-1.080185 10 3 6.010848 10 4 1.674454 10 4-8.556208 10 5 3: x, y, z 1969 6 27 6 12
3 3.1 x-y 2 x-y-z 3 x-z 23 ( 5) x-y 5: 3.2 500 ( 3142 ) 3 3 13
4 4.1 6: 1. 2. x-z 3. 4. 14
5. / 6. XY/XZ x-z 7. / 8. GO 9. 10. / / x y / 4.2 (0, 0) [2] 2.1 2 15
4.3 4.3.1 7: 7 GO 16
8: 8 7 17
4.3.2 9: 9 6 18
4.3.3 XY/XZ 10: XZ XZ XY/XZ 10 x-z 10 x 23 19
4.3.4 / 11: 1 11 XZ 20
4.3.5 / 12: / 12 / / 4.1 X Y / 21
13: / / 13 / 11 22
14: 1 14 23
15: 3 15 24
4.4 16: 25
5 3 3 3 [1],,, (2008) [2], (2009) [3], (2010) [4] :,pp.12-13,20-21 (2000) 26