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By W. C. Reynolds, R. W. MacCormack

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Consider first motion in one dimension. Assume then that the X and X ∗ axes are in relative motion defined by x∗n = xn − ctn , n = 0, 1, 2, 3, . . 36) in which c is a positive constant. If v0,x is the initial velocity of P along the X axis, define v0,x∗ along the X ∗ axis by v0,x∗ = v0,x − c. 37) 2 (x∗ + ct1 ) − (x∗0 + ct0 ) − v0,x = v1,x∗ + c. 38) Hence, for n = 1, v1,x = For n > 1, vn,x   n−1  2  ∗ = (−1)j x∗n−j + (−1)n v0,x xn + (−1)n x∗0 + 2  ∆t  j=1   n−1  2c  + (−1)j tn−j . tn + (−1)n t0 + 2  ∆t  j=1 24 N-Body Problems and Models But, it follows readily that tn + (−1)n t0 + 2 n−1 (−1)j tn−j = j=1 0, n even ∆t, n odd .

Proof. Consider first motion in one dimension. Assume then that the X and X ∗ axes are in relative motion defined by x∗n = xn − ctn , n = 0, 1, 2, 3, . . 36) in which c is a positive constant. If v0,x is the initial velocity of P along the X axis, define v0,x∗ along the X ∗ axis by v0,x∗ = v0,x − c. 37) 2 (x∗ + ct1 ) − (x∗0 + ct0 ) − v0,x = v1,x∗ + c. 38) Hence, for n = 1, v1,x = For n > 1, vn,x   n−1  2  ∗ = (−1)j x∗n−j + (−1)n v0,x xn + (−1)n x∗0 + 2  ∆t  j=1   n−1  2c  + (−1)j tn−j .

M1 (v1,n+1,x − v1,n,x ) + m2 (v2,n+1,x − v2,n,x ) + m3 (v3,n+1,x − v3,n,x ) = 0. 12) in which m1 v1,0,x + m2 v2,0,x + m3 v3,0,x = C1 . 16) m1 v1,0,z + m2 v2,0,z + m3 v3,0,z = C3 . 17) Similarly, in which Thus, 3 Mn = Mi,n = (C1 , C2 , C3 ) = M0 , i=1 n = 1, 2, 3, . . , N -Body Problems with 2 ≤ N ≤ 100 19 which is the classical law of conservation of linear momentum. Note that M0 depends only on the initial data. Thus we have the following theorem. 3. 2 conserves linear momentum, that is, M n = M0 , n = 1, 2, 3, .

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