By LUN C. K. K. & LIU H. S.

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31) follow readily by mathematical induction. 5. 21) are covariant relative to the x∗ = x − a, Proof. y ∗ = y − b; a, b constants. Deﬁne v0,x = v0,x∗ , v0,y = v0,y∗ . 1, v1,x = 2 (x∗ + a) − (x∗0 + a) − v0,x∗ = v1,x∗ . 1 yield n−1 2 ∗ (xn + a) + (−1)n (x∗0 + a) + 2 (−1)j (x∗n−j + a) vn,x = ∆t j=1 + (−1)n v0,x∗ . 32) implies vn,x = vn,x∗ . Similarly, vn,y = vn,y∗ . Thus, for all n = 0, 1, 2, 3, . . vn,x = vn,x∗ , vn,y = vn,y∗ . Thus, ∗ Fn,x ∗ = Fn,x = m vn+1,x − vn,x vn+1,x∗ − vn,x∗ =m . ∆t ∆t Similarly, ∗ Fn,y ∗ = m and the theorem is proved.

Fn,x ∗ = Fn,x = m vn+1,x∗ − vn,x∗ vn+1,x∗ + c − vn,x∗ − c =m . 40) and the covariance is established. 5. Perihelion Motion In this section and in the next two sections, we show how to apply conservative methodology to problems in physics. As a ﬁrst application, let us examine a planar 3-body problem in which the force of interaction is gravitation. In such problems conservation of energy, linear momentum, and angular momentum are basic. Let Pi , i = 1, 2, 3, be three bodies, with respective masses mi , in motion in the XY plane, in which the force of interaction is gravitation.

Consider the initial value problem x ¨ = x2 , x(0) = 1, x˙ = 1. 64) Choosing φ(x) = − 13 x3 , the system to be solved is 1 xk+1 = xk + (∆t)(vk+1 + vk ) 2 1 vk+1 = vk + (∆t) x2k+1 + xk+1 xk + x2k . 65) yields x2k+1 + 1 − 6 (∆t)2 xk+1 + 1 + 6 6 + (∆t)2 (∆t) = 0. 67) that x21 + 1 − 6 (∆t)2 x1 + 1 + 6 6 + 2 (∆t) (∆t) = 0. 79490525, Eq. 68) has two real roots. Indeed, one must choose the negative sign in the quadratic formula to get the correct root. 01005, while the incorrect solution is x1 = 59998.