Arithmetic Applied Mathematics by Donald Greenspan

By Donald Greenspan

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T · · m • f · (a) • · · · m . • · · · • • · t · m • · • · · m . · · F i g . 9 » (e) f t80 ftIC 38 Arithmetic Applied Mathematics w · · · · · · —*—·—·—·—r—* t · · · · · 1—·—*—*—*—· · # · —■—· # • · . t= t (h) t = t140 (i) t = t160 (j) t --1„ # · ^—· é • · • · · · · · · · Ê· · · • 4—·—*—-— —*—·—·—é « m ¥ · · . * * -. (g) · - F i g . 4 . 9 completed . 2 2 0 39 Conservative Models consecutive pair of points in the than P r. - P Thus, the points P P ? - P in - P set are separated by a distance less are in a stretched position, while the points are compressed.

To do this, at time t. let particle P. of mass m. , have velocity _> v. , 3 + ! and have angular momentum L. , ! iJ that is, L. (r. x v. ) . · L. 3 at t. 42) and this is done as follows. k+i " 2 i,k = m i (? fcH^iV + ( 1 i,k+1 X ( 29 i,k . 45) imply L k+i - L k = (At) (3 V ·46) Now, if T k = 0, k = 0,1,2,... 42), and the discussion would be complete. 47) is valid. To do this observe that -> -> ■ > * T =T +T + T k l,k 2,k 3,k -+ -* . k+ΐΛ ,k> $ 1,k r 12,krl2,k+l(r12,k+r12,k+l) Gm l m 3 (? 50) 13,krl3,k+l(r13,k+rl3,k+l) r | = Gm 2,k l m 2(?

36) Nonconservative Models 55 and the calculations were repeated again. The results now simulated a gravitational collapse in which each particle showed exceptionally strong attraction to all other particles. Indeed, the entire body initially rose upward and gradually settled down into a triangular body similar to, but smaller than, that shown in Fig. 6. To test that the resulting body was cohesive, particle P19 was removed again. Two thousand further iterations showed that P P8 and P14 had moved up 1' the side of the triangle while P23, P26, and P28 had moved down, thereby filling Thus, the resulting body was cohesive, but not in the in the void left by P19.

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