B ver B

B ver. 2017.02.24 B

Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2....................... 14 1.3............................ 16 1.3.1..................... 16 1.3.2 1................. 16 1.3.3................ 17 1.3.4................... 21 1.4.................................. 21 1.4.1 1............................. 21 1.4.2 2............................. 22 2 23 2.1....................... 23 2.2...................... 25 2.3................ 26 2.4.................................. 29 2.4.1.................. 29 2.4.2..................... 31 2.4.3..................... 33 2.4.4 2.................... 35 2.4.5.................. 37 2.4.6 2....................... 39 3 43 3.1.............................. 43 3.2........................... 45 3.3........................... 48 3.4............................ 50 3.5...................... 51 3

CONTENTS 3.5.1........................ 51 3.5.2......................... 53 3.5.3........................ 54 3.5.4.................. 54 3.6............................ 55 3.7..................... 56 4 59 4.1................................. 59 4.2................................. 63 4.2.1............................ 63 4.2.2.......................... 64 4.2.3....................... 65 4.3....................... 68 4.4............................ 70 4.5.................................. 72 4.5.1.................... 72 4.5.2 2....................... 73 4.5.3.................... 77 5 79 5.1................ 79 5.2........................... 81 5.3............................ 83 5.4.............................. 84 5.4.1 1........................... 84 5.4.2.......................... 85 5.4.3............................. 86 5.4.4 :..................... 87 5.5..................... 88 5.6........................ 89 5.7........................... 90 5.8.................................. 91 5.8.1 1............................. 91 5.8.2 2............................. 92 5.8.3 3............................. 95 5.8.4 4............................. 96 5.8.5 5............................. 98 6 101 6.1........................... 101 6.2......................... 103 6.3....................... 104 6.4............................ 106 6.5................................. 108 6.6....................... 113 4

CONTENTS 6.7............................ 114 6.8................. 117 6.9.................. 118 6.10........................ 119 6.11............................ 120 6.12............................ 122 6.13......................... 124 6.14................................ 124 7 127 7.1............................ 127 7.2........................... 131 7.3..................... 134 7.4...................... 136 7.5 =.................... 137 8 141 8.1...................... 141 8.2 1.................... 142 8.3................................ 146 8.4..................... 148 8.4.1.................. 148 8.4.2.................. 149 8.4.3................ 150 8.5........................... 152 8.6................................ 155 A 157 A.1 1............................ 157 A.2 2 =......................... 160 A.3 4 =......................... 161 A.4......................... 163 A.5........................... 164 A.6 2........................... 164 5

1 y = x 2 + 1 y = x 2 1 m 1: k l 0 7

CONTENTS ( ) ( ) k s (1 + 4x 2 0)ẍ 0 = 4x 0 ẋ 2 l0 0 ( x + 2x 0 y) (1) (1 + 4x 2 1)ẍ 1 = 4x 1 ẋ 2 1 m s ) ( s l0 ( k m s ) ( x + 2x 1 y) (2) x 0 x 1 x x = x 0 x 1 y = x 2 0 + x 2 1 + 2 s = x 2 + y 2 (1) (2) x 0 x 1 l 0 2 l 0 2 t t t 0, t 1, t 2, = (1) (2) 4 = Processing https://processing.org 8

CONTENTS Processing two_balls_on_parabolas.pde web page k/m = 8, l 0 = 2.9 x 1 = 1.5, x 2 = 0.8 x 1 x 2 2 x 1 -x 2 two_balls_on_parabolas_pde_x1_x2_map.pde web page 2: x 1 x 2 http://www.research.kobe-u.ac.jp/csi-viz/members/kageyama/lectures/h28_latter/analytical_mechanics/index.ja.html 9

Chapter 1 1.1 1.1.1 3 (x, y, z) (x 1, x 2, x 3 ) x P t x(t) P v = dx dt v = ẋ 2 d 2 x dt 2 3 ẍ v v = (v x, v y, v z ) = (v 1, v 2, v 3 ) 11

1 a b = 3 a j b j m v K 1 j=1 K = 1 2 mẋ2 3 K = 1 2 m ( ẋ 2 1 + ẋ 2 2 + ẋ 2 ) m 3 = 2 3 j=1 ẋ 2 j 1.1.2 x-y y = x 2 y x s s = -1 x = - 0.763929 s = 1 x 1 x x = 0.763929... 1 12

1.1 2 x (x 0, x 1 ) (s 0, s 1 ) x F (x, ẋ, ẍ) = 0 x 2 x s F (s, ṡ, s) = 0 s 2 F = 0 x s 13

1 1.2 1.2.1 F U F = U F i = U x i F U F 2 (x, y) f(x, y) f(x, y) f f f(x, y) f 1.2.2 k l 0 m x F x = k(x l 0 ) (1.1) U(x) = k 2 (x l 0) 2 (1.2) 14

1.2 (r, rφ) r φ k m (r, φ) 2 (r, ϕ) r U(r) = k 2 r2 (1.3) z g m F = (0, 0, mg) U(x, y, z) = mgz U(x 1, x 2, x 3 ) = mgx 3 F i = GMm x i r 3 U = GMm 1 r r n = nr n 1 x i x i r 15

1 1.3 1.3.1 L K U L = K U 1.3.2 1 g m t q(t) dq/dt = q z m K = 1 2 m q2 U = mgq 16

1.3 L(q, q) = m 2 q2 mgq (1.4) L SI J q q x l 0, k k m x q q (= dq/dt) = K = 1 2 m q2, U = k 2 (q l 0) 2 L L(q, q) = m 2 q2 k 2 (q l 0) 2 0 L(q, q) = m 2 q2 k 2 q2 1.3.3 2 x 1 -x 2 k l 0 = 0 m 17

1 x 2 (x 1, x 2 ) k m (x 1, x 2 ) x 1 (x 1, x 2 ) (ẋ 1 (t), ẋ 2 (t)) K = m 2 (ẋ2 1 + ẋ 2 2). U = k 2 (x2 1 + x 2 2). L(x 1, x 2, ẋ 1, ẋ 2 ) = m 2 (ẋ2 1 + ẋ 2 2) k 2 (x2 1 + x 2 2) (1.5) M m 18

1.3 x 3 (x 1, x 2, x 3 ) r m (x 1, x 2, x 3 ) M x 2 x 1 x 1 -x 2 -x 3 (x 1, x 2, x 3 ) (ẋ 1, ẋ 2, ẋ 3 ) K = m 2 v2 = m 2 ( v 2 1 + v 2 2 + v 2 3) = 3 j=1 m 2 ẋjẋ j = m 2 ẋjẋ j U = GMm r G GMm = 3 j=1 x jx j G = 6.67384 10 11 (m 3 kg 1 s 2 ) L L(x 1, x 2, x 3, ẋ 1, ẋ 2, ẋ 3 ) = m 2 ẋjẋ j + GMm 3 j=1 x jx j p. 15 x 1 x 2 m k/2 0 19

1 k / 2 x 2 (x 1, x 2 ) m (x 1, x 2 ) k / 2-1 1 x 1 x 1 (x 1, x 2 ) = ( 1, 0) (x 1, x 2 ) = (+1, 0) (x 1, x 2 ) (ẋ 1, ẋ 2 ) K = m 2 (ẋ2 1 + ẋ 2 2) U U 1 = k { (x1 1) 2 + x 2 4 2) } U 2 = k 4 { (x1 + 1) 2 + x 2 2) } U = k 4 [{ (x1 1) 2 + x 2 2) } + { (x 1 + 1) 2 + x 2 2) }] L(x 1, x 2, ẋ 1, ẋ 2 ) = m 2 (ẋ2 1 + ẋ 2 2) k [{ (x1 1) 2 + x 2 4 2) } + { (x 1 + 1) 2 + x 2 2) }] (1.6) 20

1.4 1.3.4 (1.5) (1.6) L a (x 1, x 2, ẋ 1, ẋ 2 ) = m 2 (ẋ2 1 + ẋ 2 2) k 2 (x2 1 + x 2 2) (1.7) (1.6) L b (x 1, x 2, ẋ 1, ẋ 2 ) = m 2 (ẋ2 1 + ẋ 2 2) k 2 (x2 1 + x 2 2) k 2 (1.8) k 2 1.4 1.4.1 1 y l ₀ x x m k x x-y y = l 0 m k l 0 x x ẋ L(x, ẋ) 21

1 K = m 2 ẋ2 U = k 2 (l l 0) 2 l = x 2 + l 2 0 L(x, ẋ) = K U = m 2 ẋ2 k 2 ( x 2 + l 2 0 l 0) 2 (1.9) U 2 1.4.2 2 y 1 s x x-y y = x + 1 m k l 0 l 0 = 0 y s L(s, ṡ) 2.4.1 p. 29 (a) 22

Chapter 2 2.1 x U m d U (mv) + dt x = 0, (2.1) v K v = ẋ (2.2) K = m 2 v2 (2.3) v v K v = mv (2.4) = (2.1) (2.4) (2.1) ( ) d K + U = 0, (2.5) dt ẋ x v ẋ (2.2) U x ẋ U = 0, (2.6) ẋ K x 23 = 0, (2.7)

2 (2.5) L(x, ẋ) = K(ẋ) U(x) (2.8) d dt ( ) L L = 0, (2.9) ẋ x 3 v i = ẋ i i K = 3 j=1 U m 2 ẋjẋ j (2.10) U = U(x 1, x 2, x 3 ) (2.11) L(x 1, x 2, x 3, ẋ 1, ẋ 2, ẋ 3 ) = K(ẋ 1, ẋ 2, ẋ 3 ) U(x 1, x 2, x 3 ) (2.12) d dt (mẋ i) + U = 0, (i = 1, 2, 3), (2.13) x i K (2.10) (ẋ 1, ẋ 2, ẋ 3 ) ẋ i i K ẋ i = mẋ i (2.14) (2.14) (2.13) d dt ( K ẋ i ) + U x i = 0, (i = 1, 2, 3) (2.15) (2.6) (2.7) (2.15) (2.12) d dt ( L ẋ i ) L x i = 0, (2.16) 24

2.2 2.2 (2.9) (2.16) 1 3 N (q 1, q 2,..., q N ) L(q 1, q 2,..., q N, q 1, q 2,..., q N ) d dt ( L q i ) L q i = 0 (i = 1, 2,..., N) (2.17) 1 L(q, q) d dt ( ) L L q q = 0 (2.18) 25

2 2.3 q g m q (1.4) L(q, q) = m 2 q2 mgq. (2.19) L q = m q d dt ( ) L = d (m q) = m q q dt L q = mg {m q} { mg} = 0 m q = mg 26

2.3 k m x p. 17 q 1 L(q, q) = m 2 q2 k 2 q2 (2.20) d dt k/m = ω 2 L q = m q ( ) L = d (m q) = m q q dt L q = kq c 1, c 2 {m q} { kq} = 0 (2.21) q + ω 2 q = 0 (2.22) q(t) = c 1 cos (ωt + c 2 ), (2.23) q(t) = c 1 cos (ωt) + c 2 sin (ωt) (2.24) p. 21 x-y y = l 0 m k l 0 x L(x, ẋ) (1.9) L(x, ẋ) = K U = m 2 ẋ2 k 2 ( x 2 + l 2 0 l 0) 2 (2.25) 27

2 y l ₀ x x m k x L ẋ = mẋ d dt ( ) L = d (mẋ) = mẍ ẋ dt (??) ( ) L x = k x 2 + l 2 0 l 0 ( = kx 1 ) l 0 l 2 0 + x 2 x x2 + l 2 0 {mẍ} { kx ( 1 l 0 l 2 0 + x 2 )} = 0, mẍ + kx ( 1 l 0 l 2 0 + x 2 ) = 0 28

2.4 2.4 2.4.1 y 1 s l x (x, x+1) x x x-y y = x + 1 m k l 0 l 0 = 0 y s (a) L(s, ṡ) (b) (a) K = m 2 ṡ2 l U = k 2 l2 l 2 s s = 2 x (x, y) = (x, x + 1) l 2 = x 2 + y 2 = x 2 + (x + 1) 2 = 2x 2 + 2x + 1 = s 2 + 2s + 1 29

2 L(s, ṡ) = K U = m 2 ṡ2 k 2 ( s 2 + ) 2s + 1 L(s, ṡ) = K U = m 2 ṡ2 k (s 2 s + ) 2 (b) d dt L ṡ = mṡ ( ) L = m s ṡ ( L s = k s + 1 ) 2 { ( {m s} k s + 1 )} = 0, 2 s + k m ( s + 1 2 ) = 0. s = s + 1 2 s + k m s = 0 s = c 1 cos (ωt + c 2 ) c 1 c 2 ω = k/m k (x, y) = ( 1/2, 1/2) 4 = partilce_on_slanted_line.pde web page 30

2.4 x y s ṡ 2.4.2! g φ m 2 " 2 m g ( x y 31

2 ϕ (a) L(ϕ, ϕ) (b) ϕ(t) (c) ϕ 1 (a) (x, y) (x, y) = (sin ϕ, cos ϕ) (ẋ, ẏ) = (cos ϕ ϕ, sin ϕ ϕ) K K = m 2 (ẋ2 + ẏ 2 ) = m 2 ϕ 2 U U = mgy = mg cos ϕ L(ϕ, ϕ) = m 2 ϕ 2 mg cos ϕ (b) L = m ϕ ϕ ( ) d L dt ϕ = m ϕ L = mg sin ϕ ϕ ( ) d L dt ϕ L ϕ = 0 { m ϕ } {mg sin ϕ} = 0 ϕ ϕ g sin ϕ = 0 32

2.4 (c) ϕ 1 sin ϕ ϕ ϕ = gϕ ϕ(t) = c 1 e g t + c 2 e g t 2.4.3 z (1,0,1) y θ x m 1 x 2 + y 2 = 1 (x, y, z) = (1, 0, 1) k 0 (a) x θ θ = 0 (x, y) = (1, 0). θ = π/2 (x, y) = (0, 1) L(θ, θ) (b) (c) θ 1 (a) K = m 2 θ 2 (x, y, z) = (cos θ, sin θ, 0) l l 2 = (cos θ 1) 2 + sin 2 θ + 1 = 3 2 cos θ U = k 2 l2 = k (3 2 cos θ) 2 33

2 L(θ, θ) = m 2 θ 2 k (3 2 cos θ) 2 (b) L = m θ θ d dt ( ) L θ = m θ L θ = k sin θ { } m θ { k sin θ} = 0 m θ + k sin θ = 0 ω 2 := k m θ + ω 2 sin θ = 0 (c) θ 1 sin θ θ θ = ω 2 θ θ(t) = c 1 cos (ωt) + c 2 sin (ωt) c 1 c 2 θ(t) = c 1 cos (ωt + c 2 ) 34

2.4 2.4.4 2 q 2 β q 1 β 1 2 m 1 1 m 2 2 1 2 k 0 1 1 q 1 2 2 q 2 (a) L(q 1, q 2, q 1, q 2 ) (b) (c) (a) K K = m 2 ( q 2 1 + q 2 2) U L(q 1, q 2, q 1, q 2 ) = m 2 (2.26) U = k 2 l2 (2.27) l 2 = q 2 1 + q 2 2 2q 1 q 2 cos β (2.28) ( q 2 1 + q 2) 2 k ( q 2 2 1 + q2 2 2q 1 q 2 cos β ) (2.29) 35

2 (b) d dt L q i = m q i (i = 1, 2) (2.30) ( ) L = m q i (i = 1, 2) (2.31) q i q 1 q 2 L q 1 = k (q 1 q 2 cos β) (2.32) L q 2 = k (q 2 q 1 cos β) (2.33) {m q 1 } { k (q 1 q 2 cos β)} = 0 (2.34) {m q 2 } { k (q 2 q 1 cos β)} = 0 (2.35) ω 2 = k/m q 1 + ω 2 (q 1 q 2 cos β) = 0 (2.36) q 2 + ω 2 (q 2 q 1 cos β) = 0 (2.37) (c) s p = q 1 + q 2 2 s m = q 1 q 2 2 (2.36) (2.37) (2.38) (2.39) s p + ω 2 ms p = 0 (2.40) s m + ω 2 ps m = 0 (2.41) ω p = ω 1 + cos β (2.42) ω m = ω 1 cos β (2.43) (2.40) (2.41) s p = c 1 cos (ω m t + c 2 ) (2.44) 36

2.4 s m = c 3 cos (ω p t + c 4 ) (2.45) c 1, c 2, c 3, c 4 (2.38) (2.39) q 1 q 2 q 1 (t) = s p + s m = c 1 cos (ω m t + c 2 ) + c 3 cos (ω p t + c 4 ) (2.46) q 2 (t) = s p s m = c 1 cos (ω m t + c 2 ) c 3 cos (ω p t + c 4 ) (2.47) 2.4.5 θ θ 1 x-y x x g y y = 0 m t = 0 (y = 0) x x = 0 y θ (a) x θ (b) x, y θ (c) K(θ, θ) (d) L(θ, θ) (e) (a) x x = θ 37

2 (b) x = θ sin θ y = 1 cos θ (c) x y t ẋ = dx dt = dx dθ dθ dt = (1 cos θ) θ ẏ = dy dt = dy dθ = sin θ θ dθ dt K(θ, θ) = m 2 (ẋ2 + ẏ 2 ) = m 2 { (1 cos θ) 2 + sin 2 θ } θ2 = m(1 cos θ) θ 2 (d) U = 0 U = mgy L = K U L(θ, θ) = m(1 cos θ) θ 2 mg(1 cos θ) L(θ, θ) ( = m (1 cos θ) θ2 g) (e) L = 2m(1 cos θ) θ θ ( ) d L dt θ = d { 2m(1 cos θ) dt θ } = 2m(1 cos θ) θ + 2m sin θ θ 2 L ( ) θ = m sin θ θ2 g { 2m (1 cos θ) θ + 2m sin θ θ 2} { ( m sin θ θ2 g)} = 0 38

2.4 2m(1 cos θ) θ + m sin θ θ 2 + mg sin θ = 0 m 2(1 cos θ) θ + sin θ θ 2 + g sin θ = 0 2 1 cos θ sin θ θ + θ 2 + g = 0 2.4.6 2 (r, rφ) m r φ k (r, φ) k l 0 = 0 m 1.3.3 p. 17 (a) (r, ϕ) L(r, ϕ, ṙ, ϕ) (b) 39

2 (a) U = k 2 r2 (2.48) (x 1, x 2 ) (r, φ) x 1 = r cos ϕ (2.49) x 2 = r sin ϕ (2.50) v r = ṙ (2.51) v ϕ = r ϕ (2.52) K = m 2 ( v 2 r + v 2 ϕ) = m 2 (ṙ2 + r 2 ϕ2 ) (2.53) (2.51) (2.52) (2.49) (2.50) v x = v 1 = ẋ 1 = d dt (r cos ϕ) = ṙ cos ϕ r ϕ sin ϕ (2.54) v y = v 2 = ẋ 2 = d dt (r sin ϕ) = ṙ sin ϕ + r ϕ cos ϕ (2.55) v 2 = v 2 1+v 2 2 = (ṙ cos ϕ r ϕ sin ϕ) 2 +(ṙ sin ϕ+r ϕ cos ϕ) 2 = ṙ 2 +r 2 ϕ2 (2.56) (2.53) L(r, ϕ, ṙ, ϕ) = K U = m 2 (ṙ2 + r 2 ϕ2 ) k 2 r2 (2.57) (b) (2.57) r d dt L ṙ = mṙ ( ) L = m r ṙ 40

2.4 r {m r} L r = mr ϕ 2 kr { mr ϕ } 2 kr = 0 m r mr ϕ 2 + kr = 0 2 ϕ d dt L ϕ = mr2 ϕ ( ) L ϕ = d dt ( ) mr 2 ϕ L ϕ L ϕ = 0 ϕ { d ( mr ϕ) } 2 {0} = 0 dt d ( ) mr 2 ϕ = 0 dt mr 2 ϕ = const. 41

Chapter 3 3.1 3 3 1. 2. 3. 3 zxz 43

3 z φ x θ z ψ cos φ sin φ 0 R 1 (φ) = sin φ cos φ 0 (3.1) 0 0 1 R 2 (θ) = R 3 (ψ) = 1 0 0 0 cos θ sin θ 0 sin θ cos θ cos ψ sin ψ 0 sin ψ cos ψ 0 0 0 1 (3.2) (3.3) R(φ, θ, ψ) = R 3 (ψ)r 2 (θ)r 1 (φ) (3.4) cos ψ cos φ cos θ sin ψ sin φ cos θ sin ψ cos φ + cos ψ sin φ sin θ sin ψ = cos θ cos ψ sin φ sin ψ cos φ cos θ cos ψ cos φ sin ψ sin φ sin θ cos ψ sin θ sin φ sin θ cos φ cos θ (3.5) e x e x e y = R e y (3.6) e z e z R R t R = RR t = I (3.7) 44

3.2 a a a = ( a x a y a z ) e x e y e z (3.8) (3.6) a = ( a x a y a ) z R e x e y e z (3.9) a a = (a x a y a z ) e x e y e z (3.10) (a x a y a z ) = ( a x a y a z) R (3.11) ( a x a y a ) z = (ax a y a z ) R t (3.12) a x a y a z = R a x a y a z (3.13) 3.2 G G G G G R(φ(t), θ(t), ψ(t)) x G G x(t) = (x(t) y(t) z(t)) = (x y z ) e x(t) e y(t) e z(t) e x e y e z (3.14) (3.15) 45

3 v (3.15) v G (3.19) (3.20) v(t) = ẋ(t) (3.16) ė x(t) = (x y z ) ė y(t) (3.17) ė z(t) e x = (x y z ) Ṙ(t) e y (3.18) e z = (x y z) R t (t)ṙ(t) v = (v x v y v z ) e x e y e z e x e y e z (3.19) (3.20) (v x v y v z ) = (x y z) Ω (3.21) Ω = R t Ṙ (3.22) ( R Ω = R t Ṙ = R t R φ + φ θ θ + R ) ψ ψ = R t R R φ + Rt φ θ θ + R t R ψ ψ (3.23) φ R t R = I (3.24) R t R φ + Rt φ R = 0 (3.25) R t R ( φ + R t R ) t = 0 (3.26) φ R t ( R/ φ) R t ( R/ θ) R t ( R/ ψ) (3.23) Ω Ω Ṙ = Ṙ 11 Ṙ 12 Ṙ 13 Ṙ 21 Ṙ 22 Ṙ 23 Ṙ 31 Ṙ 32 Ṙ 33 = R R φ + φ θ θ + R ψ ψ (3.27) 46

3.2 Ṙ 11 = θ sin θ sin ψ sin φ φ(cos θ sin ψ cos φ + cos ψ sin φ) ψ(cos θ cos ψ sin φ + sin ψ cos φ) (3.28) Ṙ 12 = θ sin θ sin ψ cos φ + φ(cos ψ cos φ cos θ sin ψ sin φ) + ψ(cos θ cos ψ cos φ sin ψ sin φ) (3.29) Ṙ 13 = θ cos θ sin ψ + ψ sin θ cos ψ (3.30) Ṙ 21 = θ sin θ cos ψ sin φ + φ(sin ψ sin φ cos θ cos ψ cos φ) + ψ(cos θ sin ψ sin φ cos ψ cos φ) (3.31) Ṙ 22 = θ sin θ cos ψ cos φ φ(cos θ cos ψ sin φ + sin ψ cos φ) ψ(cos θ sin ψ cos φ + cos ψ sin φ) (3.32) Ṙ 23 = θ cos θ cos ψ ψ sin θ sin ψ (3.33) Ṙ 31 = θ cos θ sin φ + φ sin θ cos φ (3.34) Ṙ 32 = φ sin θ sin φ θ cos θ cos φ (3.35) Ṙ 33 = θ sin θ (3.36) R t 0 ψ cos θ + φ θ sin φ + ψ sin θ cos φ Ω = ψ cos θ φ 0 θ cos φ + ψ sin θ sin φ θ sin φ ψ sin θ cos φ θ cos φ ψ sin θ sin φ 0 (3.37) ω x = θ cos φ + ψ sin θ sin φ (3.38) ω y = θ sin φ ψ sin θ cos φ (3.39) ω z = ψ cos θ + φ (3.40) G Ω = ṘRt = 0 ω z ω y ω z 0 ω x (3.41) ω y ω x 0 (3.21) v x = ω y z ω z x (3.42) v y = ω z x ω x y (3.43) v z = ω x y ω y z (3.44) v = ω x (3.45) 47

3 ω N i x i v i (3.42) (3.44) v ix (t) = ω y (t)z i (t) ω z (t)x i (t) (3.46) v iy (t) = ω z (t)x i (t) ω x (t)y i (t) (3.47) v iz (t) = ω x (t)y i (t) ω y (t)z i (t) (3.48) x i (t) G x 3.3 (3.7) (3.18) v G v = (x y z ) ṘRt R = (x y z ) ṘRt v = ( v x v y v z ) e x e y e x e y e z e x e y e z e z (3.49) (3.50) (3.51) (3.50) (3.51) ( v x v y v z ) = (x y z ) Λ (3.52) Λ = ṘRt (3.53) 0 φ cos θ + ψ θ sin ψ φ sin θ cos ψ Λ = φ cos θ ψ 0 θ cos ψ + φ sin θ sin ψ θ sin ψ + φ sin θ cos ψ θ cos ψ φ sin θ sin ψ 0 (3.54) 0 ω z ω y = ω z 0 ω x (3.55) ω y ω x 0 48

3.3 ω x = θ cos ψ + φ sin θ sin ψ (3.56) ω y = θ sin ψ + φ sin θ cos ψ (3.57) ω z = φ cos θ + ψ (3.58) (φ, θ, ψ) φ = csc θ(ω x sin ψ + ω y cos ψ) (3.59) θ = ω x cos ψ ω y sin ψ (3.60) ψ = ω z cot θ(ω x sin ψ + ω y cos ψ) (3.61) (3.56) (3.58) (3.52) v x = ω yz ω zx (3.62) v y = ω zx ω xy (3.63) v z = ω xy ω yz (3.64) G (3.38) (3.40) G (3.62) (3.64) ω x ω y = R ω x ω y (3.65) ω z ω z ω (3.13) (3.62) (3.64) v x v y v z = Q = ( θ sin ψ + φ sin θ cos ψ)z ( φ cos θ + ψ)y ( φ cos θ + ψ)x ( θ cos ψ + φ sin θ sin ψ)z ( θ cos ψ + φ sin θ sin ψ)y ( θ sin ψ + φ sin θ cos ψ)x z sin θ cos ψ y cos θ z sin ψ y x cos θ z sin θ sin ψ z cos ψ x y sin θ sin ψ x sin θ cos ψ x sin ψ + y cos ψ 0 = Q (3.66) (3.67) φ θ ψ (φ, θ, ψ) (δφ, δθ, δψ) (x, y, z ) (δx, δy, δz ) δx δy δz = Q δφ δθ (3.68) δψ 49

3 (x, y, z ) (φ, θ, ψ) (x 1, x 2, x 3) (ϕ 1, ϕ 2, ϕ 3 ) (3.68) δx α = 3 Q αβ δϕ β (3.69) β=1 x α ϕ β = Q αβ (3.70) 3.4 (φ, θ, ψ) a = a i e i = a i e i a i = R ij a j (3.71) a i = R ji a j (3.72) R (3.5) Ṙ (3.28) (3.36) R ij R kj = R ji R jk = δ ik (3.73) a kṙkjr ij = ϵ ijk ω ja k (3.53) (3.55) (3.74) ω j (3.56) (3.58) a(t) = a i (t)e i = a i (t)e i(t) b(t) = ȧ(t) = b i (t)e i = b i (t)e i (t) b i e i = b = ȧ = ȧ ie i + a iė i (3.75) = ȧ ie i + a iṙije j (3.76) ( ) = ȧ jr ji + a jṙji e i (3.77) b i = ȧ jr ji + a jṙji (3.78) = ȧ jr ji + a kṙklδ li (3.79) = ȧ jr ji + a kṙklr jl R ji (3.73) (3.80) ) = (ȧ j + a kṙklr jl R ji (3.81) = ( ȧ j + ϵ jl ω la m) Rji (3.74) (3.82) 50

3.5 b (3.72) b j = ȧ j + ϵ jlm ω la m (3.83) ȧ1 ȧ 2 ȧ 3 = ȧ 1 ȧ 2 ȧ 3 + ω 1 ω 2 ω 3 a 1 a 2 a 3 (3.84) a = x (3.62) (3.64) 3.5 3.5.1 N (φ, θ, ψ) i G v ix(t) = ω y(t)z i ω z(t)x i (3.85) v iy(t) = ω z(t)x i ω x(t)y i (3.86) v iz(t) = ω x(t)y i ω y(t)z i (3.87) 3 3 v α = ϵ αβγ ω βx γ (3.88) β=1 γ=1 51

3 (v x, v y, v z) = (v 1, v 2, v 3) K = = = = = = = = 1 2 = 1 2 N i=1 N i=1 α=1 N m i 2 v2 i (3.89) 3 3 m i 2 v 2 iα (3.90) 3 3 3 3 i=1 α=1 β=1 γ=1 β =1 γ =1 N 3 3 3 3 i=1 β=1 γ=1 β =1 γ =1 N 3 3 i=1 β=1 γ=1 N i=1 N i=1 m i 2 m i 2 3 β=1 γ=1 3 β=1 γ=1 m i 2 ϵ αβγϵ αβ γ ω βx iγω β x iγ (3.91) m i 2 (δ ββ δ γγ δ βγ δ γβ )ω βx iγω β x iγ (3.92) m i 2 (ω βx iγω βx iγ ω βx iγω γx iβ) (3.93) 3 3 (x iµ) 2 (ω β) 2 µ=1 β=1 3 3 (x iµ) 2 µ=1 β=1 γ=1 3 β=1 γ=1 3 ω βω γδ β γ 3 ω βω γx iβx iγ (3.94) 3 β=1 γ=1 3 ω βω γx iβx iγ (3.95) [ 3 N 3 ] ω βω γ m i (x iµ) 2 δ β γ x iβx iγ (3.96) i=1 µ=1 3 ω βω γi β γ (3.97) I β γ = N N I = i=1 m i i=1 m i [ 3 µ=1 (x iµ) 2 δ β γ x iβx iγ y 2 i + z 2 i x i y i x i z i y i x i x 2 i + z i 2 ] y i z i z i x i z i y i x 2 i + y i 2 (3.98) (3.99) I I G e x, e y, e z I = I 1 0 0 0 I 2 0 0 0 I 3, I βγ = I β δ βγ (3.100) 52

3.5 I 1, I 2, I 3 K = 1 2 3 I β (ω β) 2 (3.101) β=1 3.5.2 (φ, θ, ψ) = (ϕ 1, ϕ 2, ϕ 3 ) U U i (x i, y i, z i ) = (x i;1, x i;2, x i;3 ) U ϕ β = = = N 3 i=1 γ=1 N 3 i=1 γ=1 N 3 i=1 γ=1 U x i;γ x iγ ϕ β (3.102) U x i;γ Q i;γβ (3.103) Q t i;βγ U (3.104) x i;γ (3.103) (3.70) (3.104) Q t i Q i Q t i = z i sin θ cos ψ y i cos θ x i cos θ z sin θ sin ψ y i sin θ sin ψ x i sin θ cos ψ z i sin ψ z i cos ψ x i sin ψ + y i cos ψ y i x i 0 (3.105) i f i;x f i;y f i;z = U/ x i U/ y i U/ z i (3.106) f φ f θ f ψ = U/ φ U/ θ U/ ψ (3.107) (3.103) N 3 f ϕβ = Q t i;βγf i;γ (3.108) i=1 γ=1 53

3 3.5.3 L(φ, θ, ψ, φ, θ, ψ) =K U (3.109) = 1 { I1 (ω 2 1) 2 + L 2 (ω 2) 2 + L 3 (ω 3) 2} U(φ, θ, ψ) (3.110) = 1 { I 1 ( 2 θ cos ψ + φ sin θ sin ψ) 2 + I 2 ( θ sin ψ + φ sin θ cos ψ) 2 +I 3 ( φ cos θ + ψ) 2} U(φ, θ, ψ) (3.111) = 1 2 φ2 ( I 1 sin 2 θ sin 2 ψ + I 2 sin 2 θ cos 2 ψ + I 3 cos 2 θ ) + 1 2 θ 2 ( I 1 cos 2 ψ + I 2 sin 2 ψ ) + 1 2 I 3 ψ 2 + φ θ sin θ cos ψ sin ψ (I 1 I 2 ) + I 3 ψ φ cos θ U(φ, θ, ψ) (3.112) 3.5.4 L φ = I 1ω 1 sin θ sin ψ + I 2 ω 2 sin θ cos ψ + I 3 ω 3 cos θ (3.113) L φ = U φ = f φ (3.114) L θ = I 1ω 1 cos ψ I 2 ω 2 sin ψ (3.115) L θ = I 1ω 1 φ cos θ sin ψ + I 2 ω 2 φ cos θ cos ψ I 3 ω 3 φ sin θ + f θ (3.116) L ψ = I 3ω 3 (3.117) L ψ = I 1ω 1( θ sin ψ + φ sin θ cos ψ) I 2 ω 2( θ cos ψ + φ sin θ sin ψ) + f ψ (3.118) l α = I α ω α (3.119) d dt (l 1 sin θ sin ψ + l 2 sin θ cos ψ + l 3 cos θ) = f φ (3.120) d dt (l 1 cos ψ l 2 sin ψ) = l 1 φ cos θ sin ψ +l 2 φ cos θ cos ψ l 3 φ sin θ +f θ (3.121) d dt l 3 = l 1ω 2 l 2ω 1 + f ψ (3.122) 54

3.6 3.6 (3.122) (3.120) (3.121) (3.122) d dt l 1 = l 2ω 3 l 3ω 2 + f θ cos ψ + (f φ csc θ f ψ cot θ) sin ψ (3.123) d dt l 2 = l 3ω 1 l 1ω 3 f θ sin ψ + (f φ csc θ f ψ cot θ) cos ψ (3.124) (g 1, g 2, g 3) = (g x, g y, g z) g 1 g 2 g 3 = P f φ f θ f ψ, P = csc θ sin ψ cos ψ cot θ sin ψ csc θ cos ψ sin ψ cot θ cos ψ 0 0 1 (3.125) d dt l 1 = l 2ω 3 l 3ω 2 + g 1 (3.126) d dt l 2 = l 3ω 1 l 1ω 3 + g 2 (3.127) d dt l 3 = l 1ω 2 l 2ω 1 + g 3 (3.128) (3.122) (3.124) l 1 l 2 l 3 = l 1 l 2 l 3 ω 1 ω 2 ω 3 + g 1 g 2 g 3 (3.129) (ω x, ω y, ω z) ω x = (I 2 I 3 ) ω I yω z + g x 1 I 1 (3.130) ω y = (I 3 I 1 ) ω I zω x + g y 2 I 2 (3.131) ω z = (I 1 I 2 ) ω I xω y + g z 3 I 3 (3.132) (3.59) (3.61) 55

3 (g 1, g 2, g 3) (3.125) (3.108) g α = = = 3 P αβ f β (3.133) β=1 3 N P αβ β=1 i=1 γ=1 N i=1 γ=1 3 Q t i;βγf i;γ (3.134) 3 (P Q t i) αγ f i;γ (3.135) P Q t i P Q i 0 z P Q t i y i i = z i 0 x i (3.136) y i x i 0 g 1 g 2 g 3 g α = = N ϵ αβγ x i;βf i;γ (3.137) i=1 N i=1 x i;1 x i;2 x i;3 f i;1 f i;2 f i;3 (3.138) (g 1, g 2, g 3) G 3.7 (3.45) v = ω x (3.139) ω x v l l = = = N m i x i v i (3.140) i=1 N m i x i (ω x i ) (3.141) i=1 N m i {(x i x i ) ω (x i ω) x i } (3.142) i=1 56

3.7 G x,y,z (x i, y i, z i ) = (x i;1, y i;2, z i;3 ) N 3 3 l α = m i x i;γ x i;γ Ω α x α x β Ω β (3.143) = = i=1 N i=1 3 m i γ=1 i=1 γ=1 3 x i;γ x i;γ γ=1 { N m i δ αβ γ=1 β=1 3 δ αβ Ω β x α β=1 3 x β Ω β β=1 (3.144) } 3 x i;γ x i;γ x α x β ω β (3.145) x iα G (x i, y i, z i ) = (x i;1, y i;2, z i;3 ) l α = = 3 γ=1 i=1 { N m i δ αβ γ=1 } 3 x i;γx i;γ x αx β ω β (3.146) 3 I αβ ω β (3.147) β=1 I αβ I αβ l α(t) = I α ω α(t) (3.148) G (3.101) K = 1 3 I β (ω 2 β) 2 (3.149) β=1 l g G l = g (3.150) a b = ȧ b b 1 b 2 b 3 = ȧ 1 ȧ 2 ȧ 3 a 1 a 2 a 3 (3.5) R(t) ω 1 ω 2 ω 3 (3.151) 57

3 (g 1, g 2, g 3) (3.150) l 1 l 2 ω 1 ω l 2 = g 1 g 3 ω 3 2 (3.152) g 3 l 1 l 2 l 3 (3.150) 58

Chapter 4 4.1 q i (i = 1, 2,, N) N Q i (i = 1, 2,, N) Q 1 = Q 1 (q 1, q 2,, q N ) Q 2 = Q 2 (q 1, q 2,, q N ). Q N = Q N (q 1, q 2,, q N ) q 1 = q 1 (Q 1, Q 2,, Q N ) q 2 = q 2 (Q 1, Q 2,, Q N ). q N = q N (Q 1, Q 2,, Q N ) ( ) d L dt Q L = k Q k N i=1 { d dt ( L q i ) L } qi (4.1) q i Q k q i (i = 1, 2,, N) Q i (i = 1, 2,, N) Q i 59

4 q i = N j=1 q i Q j Q j (4.2) q i Q j = q i Q j (4.3) (4.3) (4.2) ( ) d qi dt Q = k q i Q k = N j=1 N j=1 (4.4) ( ) d qi = d ( ) qi dt Q k dt Q k L Q k = ( ) d L dt Q = k = N i=1 N i=1 L Q k = (4.8) (4.10) ( ) { d L dt Q L N = k Q k = i=1 N i=1 d dt { d dt N i=1 d dt d dt N i=1 L q i q i Q = k 2 q i Q j Q k Q j (4.4) 2 q i Q k Q j Q j (4.5) N i=1 ( ) L qi N + q i Q k i=1 ( ) L qi N + q i Q k L q i + q i Q k N i=1 = q i Q k (4.6) L q i (4.7) q i Q k i=1 ( ) L qi N + q i Q k i=1 ( ) L L } qi q i q i Q k ( ) L d qi q i dt Q k L q i (4.8) q i Q k (4.9) L q i (4.10) q i Q k L q i q i Q k } { N i=1 L q i + q i Q k 60 N i=1 L q i q i Q k }

4.1 d dt ( L q k ) L q k = 0 (k = 1, 2,..., N) (4.11) ( ) d L dt Q L = 0 k Q k (k = 1, 2,..., N) (4.12) Q 1 = Q 1 (q 1 (t), q 2 (t),, q N (t), t) (4.13) Q 2 = Q 2 (q 1 (t), q 2 (t),, q N (t), t) (4.14). Q N = Q N (q 1 (t), q 2 (t),, q N (t), t) (4.15) y=1 y=-1 x-y 1: y = 1 2: y = 1 m 1 2 1 2 k l 0 A 1 x x 1 2 x x 2 L(x 1, x 2, ẋ 1, ẋ 2 ) = m 2 (ẋ2 1 + ẋ 2 k { 2) (x1 x 2 ) 2 + 4 } (4.16) 2 61

4 ( ) d L dt ẋ 1 ( d L dt ẋ 2 ( L x 1 ( L ) ) = 0 (4.17) ) = 0 (4.18) x 2 mẍ 1 + k(x 1 x 2 ) = 0 (4.19) mẍ 2 + k(x 2 x 1 ) = 0 (4.20) B (r, s) (x 1, x 2 ) r = x 1 + x 2 2 s = x 1 x 2 2 r x (4.21) (4.22) x 1 = r + s (4.23) x 2 = r s (4.24) ẋ 1 = ṙ + ṡ (4.25) ẋ 2 = ṙ ṡ (4.26) L(r, s, ṙ, ṡ) = m ( ṙ 2 + ṡ 2) 2k ( 1 + s 2) (4.27) (4.21) (4.22) (4.17), (4.18) ( ) ( ) d L L = 0 (4.28) dt ṙ r ( ) ( ) d L L = 0 (4.29) dt ṡ s r = 0 (4.30) m s + 2ks = 0 (4.31) 62

4.2 (4.30) r(t) = c 0 + c 1 t (4.32) c 0, c 1 (4.30) (4.28) L r = 0 (4.33) 4.2 4.2.1 N L(q 1, q 2,..., q N, q 1, q,..., q N ) p i = L q i (4.34) q i L(q 1, q 2,..., q N, q 1, q,..., q N ) q k q k (4.27) r q k p k = L q k (4.35) d dt ( L q k ) L q k = 0 (4.36) L q k = 0 (4.37) dp k dt = 0 (4.38) 63

4 U 2 {r, ϕ} r v r = ṙ ϕ v ϕ = r ϕ v 2 = v 2 r + v 2 ϕ = ṙ 2 + r ϕ 2 (4.39) L(r, ϕ, ṙ, ϕ) = m 2 ( ṙ 2 + r 2 ϕ2 ) U(r) (4.40) ϕ ϕ 4.2.2 p ϕ = L ϕ = mr2 ϕ (4.41) L(q 1, q 2,..., q N, q 1, q 2,..., q N ) p j q j N h = p j q j L(q 1, q 2,..., q N, q 1, q 2,..., q N ) (4.42) j=1 dh = d N p dt dt j q j L(q 1, q 2,..., q N, q 1, q 2,..., q N ) j=1 N N N = ṗ j q j + p j q j L N L q j + q j 2 4 q j q j = = j=1 j=1 j=1 N N L ṗ j q j q j q j N j=1 { d dt j=1 ( L q j ) L q j j=1 } q j j=1 = 0 L(q 1, q 2,..., q N, q 1, q 2,..., q N ) = K U (4.43) L(q, q, t) = 1 2 q2 + 1 2 q2 + cos t 64

4.2 U q i K = N j=1 h m 2 q j q j (4.44) p i = L q i = K q i = m q i (4.45) h N N N h = m q j q j (K U) = m q j q j j=1 j=1 j=1 m 2 q j q j + U = K + U (4.46) dh/dt = 0 4.2.3 L L t t t t = t + t 0 t 0 (4.47) L x L 3 x 4.1 p.61x y x (4.16) x 1 x 2 2 3 2 x = (x, y, z) (4.48) X = (X, Y, Z) (4.49) 65

4 (x, X) = (x, y, z, X, Y, Z), L(x, X, ẋ, Ẋ) (4.50) δr (x, X ) = (x, y, z, X, Y, Z ) x x = (x + δr x, y + δr y, z + δr z ) (4.51) X X = (X + δr x, Y + δr y, Z + δr z ) (4.52) ẋ x = ẋ (4.53) Ẋ X = Ẋ (4.54) δl = L(x, ẋ, X, Ẋ ) L(x, ẋ, X, Ẋ) (4.55) = L x δr x + L y δr y + L z δr z + L X δr x + L Y δr y + L Z δr z (4.56) x X p P δl = ṗ x δr x + ṗ y δr y + ṗ z δr z + P x δr x + P y δr y + P z δr z (4.57) = d dt (p x + P x ) δr x + d dt (p y + P y ) δr y + d dt (p z + P z ) δr z (4.58) = d (p + P ) δr (4.59) dt δr δl = 0 d (p + P ) = 0 (4.60) dt N (q 1, q 2,..., q N ) L(q 1, q 2,..., q N, q 1, q 2,..., q N ) (4.61) K : (X 1, X 2, X 3 ) q i = q i (X 1, X 2, X 3 ) (4.62) K δr K : (X 1, X 2, X 3) X α X α + δr α (α = 1, 2, 3) (4.63) 66

4.2 K q i q i = q i (X 1, X 2, X 3) = q i (X 1, X 2, X 3 ) + 3 α=1 q i (X 1, X 2, X 3 ) X α δr α (4.64) δq i = 3 α=1 q i X α δr α (4.65) δ q i = 3 α=1 q i X α δr α = 3 α=1 ( ) d qi δr α (4.66) dt X α δl(q 1, q 2,..., q N, q 1, q 2,..., q N ) = (4.65) (4.66) δl = = = N j=1 N L q j 3 α=1 ṗ j j=1 α=1 3 N α=1 j=1 3 d dt q j X α δr α + q j X α δr α + ( p j q j X α N j=1 N N j=1 L q j 3 p j j=1 α=1 L q j δq j + 3 α=1 d dt N j=1 L q j δ q j (4.67) ( ) d qj δr α (4.68) dt X α ( qj X α ) δr α (4.69) ) δr α (4.70) p j = L/ q j δr α L δl = 0 N j=1 ( ) d q j p j = 0 (α = 1, 2, 3) (4.71) dt X α N N Y α = Yα j q j = p j (α = 1, 2, 3) (4.72) X α j=1 j=1 67

4 K ω ϵ K (X 1, X 2, X 3 ) 3 X α X α = X α + ϵ 3 β=1 γ=1 3 ϵ αβγ ω β X γ (4.73) q i q i = q i + δq i (4.74) δq i = = ϵ = ϵ 3 α=1 3 q i X α δx α (q i ) (4.75) 3 3 α=1 β=1 γ=1 3 3 3 α=1 β=1 γ=1 q i X α ϵ αβγ ω β X γ (q i ) (4.76) ω β ϵ βγα X γ (q i ) q i X α (4.77) d 3 3 N q j ϵ βγα X γ (q j ) p j = 0 (β = 1, 2, 3) (4.78) dt X α γ=1 α=1 j=1 Y j (4.72) N X(q j ) Y j = 0 (4.79) j=1 4.3 L K U L = K U ϵ αβγ 68

4.3 1 q L(q, q) q W (q) L dw (q) (q, q) = L(q, q) + (4.80) dt dw (q) dt = dw (q) dq q (4.81) dw (q)/dq { ( )} d dw ( ) dw = d ( ) dw (q) d2 W (q) dt q dt q dt dt dq dq 2 q (4.82) = d2 W (q) dq 2 q d2 W (q) dq 2 q (4.83) = 0 (4.84) q W (q) dw (q)/dt 1 L(q, q) W (q) = sin (q 3 ) dw dt = d sin (q3 ) dt = 3q 2 q cos (q 3 ) (4.85) (q 1, q 2,..., q N ) W (q 1, q 2,..., q N ) dw (q 1, q 2,..., q N ) dt = N j=1 W dq N j q j dt = W q j (4.86) q j L j=1 L (q 1,..., q N, q 1,..., q N ) = L(q 1,..., q N, q 1,..., q N ) + dw (q 1,..., q N ) dt (4.87) d dt ( ) L = d ( L q k dt q k L q k ) + d dt = L + W (4.88) q k q k ( ) W (q1,..., q N ) = d ( ) L + 2 W q j q k dt q k q k q j (4.89) 69

4 (4.87) L q k = L + 2 W q j (4.90) q k q k q j L ( ) d L L = 0 (4.91) dt q k q k { d dt ( L q k ) } { } + 2 W L q j + 2 W q j = 0 (4.92) q k q j q k q k q j d dt ( L q k ) L q k = 0 (4.93) L {q 1, q 2,..., q N } L W (q 1, q 2,..., q N ) L = L + dw dt (4.94) L L L 4.4 +e E(x) U ψ E U U(x) = eψ(x) (4.95) E(x) = ψ(x) (4.96) m L(x 1, x 2, x 3, ẋ 1, ẋ 2, ẋ 3 ) = 3 j=1 m 2 ẋjẋ j eψ(x) (4.97) L(x, ẋ) = m 2 ẋ2 eψ(x) (4.98) dw (q)/dt L(q, q) = m 2 q2 L (q, q) = e m q L 70

4.4 B E(x, t) B(x, t) +e m L(x, ẋ, t) = m ẋ ẋ + ea(x, t) ẋ eψ(x, t) (4.99) 2 A ψ (4.102) (4.103) A α = A α (x 1, x 2, x 3, t) (α = 1, 2, 3) (4.100) ψ = ψ(x 1, x 2, x 3, t) (4.101) E = A ψ t (4.102) B = A (4.103) A A = A + Λ (4.104) ψ ψ = ψ Λ t (4.105) A ψ Λ(x, t) E = A t ψ = A t ψ = E (4.106) B = A = A + Λ = A = B (4.107) (4.106) (4.107) (4.99) L L = m 2 ẋ ẋ + ea ẋ eψ (4.108) = m ( 2 ẋ ẋ + e (A + Λ) ẋ e ψ Λ ) (4.109) t = m ( 2 ẋ ẋ + ea ẋ eψ + e Λ ẋ + Λ ) (4.110) t Λ ẋ + Λ t = 3 j=1 Λ ẋ j + Λ dλ(x, t) = x j t dt (4.111) 71

4 L = L + d(eλ) dt (4.112) (4.94) W = eλ 4.5 4.5.1 z B y x A = (A x, A y, A z ) = (0, x, 0) (4.113) B = A = (0, 0, 1) (4.114) z m e (4.99) L(x, ẋ) = m 2 (ẋ2 + ẏ 2 + ż 2) + exẏ (4.115) (4.115) 72

4.5 y z p y = L ẏ = mẏ + ex = mẏ + ea y (4.116) p z = L ż = mż (4.117) (4.117) z (4.116) A y = 0 y 4.5.2 2 z Q ψ(r) = Q r (4.118) e r U U(r) = eψ = eq/r e (4.119) {r, ϕ, z} A A = (A r, A ϕ, A z ) = (0, rb 0 /2, 0) B 0 (4.120) 73

4 ( A) r = 1 r A z ϕ A ϕ z ( A) ϕ = A r z A z r ( A) z = 1 r r (ra ϕ) 1 r A r ϕ (4.121) (4.122) (4.123) B = A z B = (B r, B ϕ, B z ) = (0, 0, B 0 ) (4.124) (4.120) z (4.113) A x-y z B 0 B 0 Q z B -Q E? ẋ = (v r, v ϕ, v z ) = (ṙ, r ϕ, v z ) (4.125) (4.120) L(r, ϕ) = m 2 ( ) ṙ 2 + r 2 ϕ2 + eb 0 eq 2 r2 ϕ + r (4.126) 74

4.5 (a) mr 2 ϕ + eb 0 2 r2 B 0 = 0 mr 2 ϕ (b) ṙ = v r (4.127) v r = rvϕ 2 + eb 0 m rv ϕ eq 1 m r 2 (4.128) ϕ = v ϕ (4.129) v ϕ = 2 v r r v ϕ eb 0 v r m r (4.130) 4 = Processing charged_planet.pde web page 75

4 (a) ϕ ϕ (b) r p ϕ = L ϕ = mr2 ϕ + eb 0 2 r2 = mr 2 ϕ + eraϕ (4.131) {m r} d dt L ṙ = mṙ ( ) L = m r ṙ L r = mr ϕ 2 + eb 0 r ϕ Qe r 2 { mr ϕ 2 + eb 0 r ϕ Qe } r 2 = 0 r r ϕ 2 eb 0 m r ϕ + eq m 1 r 2 = 0 ϕ d dt ( ) L ϕ = d dt L ϕ = eb 0 mr2 ϕ + 2 r2 ( ) mr 2 eb 0 ϕ + 2 r2 = 2mrṙ ϕ + mr 2 ϕ + eb0 rṙ = 0 ϕ + 2ṙ r ϕ + eb 0 ṙ m r = 0 r ϕ r = r ϕ 2 + eb 0 m r ϕ eq m ϕ = 2ṙ r ϕ eb 0 ṙ m r v r = ṙ v ϕ = ϕ 1 r 2 (4.132) (4.133) (4.127) (4.130) 76

4.5 4.5.3 (4.99) mẍ = e {E + ẋ B} (4.134) (4.99) x 1 L ẋ 1 = mẋ 1 + e A 1 ( ) d L = mẍ 1 + e dt ẋ 1 L x 1 = e 3 j=1 3 j=1 A 1 ẋ j + e A x j t A j ẋ j e ψ x 1 x 1 x 1 3 mẍ A 1 1 + e ẋ j + e A x j t e j=1 3 j=1 A j ẋ j e ψ x 1 x 1 = 0 j 1 3 j = 1 A 1 / x j A j / x 1 { mẍ 1 + e A 1 ẋ 2 + e A 1 ẋ 3 + e A x 2 x 3 t } { e A 2 ẋ 2 + e A 3 ẋ 3 e ψ } = 0 x 1 x 1 x 1 { ( A2 mẍ 1 = e ẋ 2 A ) ( 1 A1 ẋ 3 A )} 3 e A x 1 x 2 x 3 x 1 t e ψ x 1 ( A) 3 = A 2 x 1 A 1 x 2 ( A) 2 = A 1 x 3 A 3 x 1 mẍ 1 = e {ẋ 2 ( A) 3 ẋ 3 ( A) 2 } e ( A t + ψ ) x 1 77

4 (4.102) (4.103) mẍ 1 = e (ẋ 2 B 3 ẋ 3 B 2 ) + e E 1 mẍ 1 = e (ẋ B) 1 + e E 1 x 2 x 3 78

Chapter 5 5.1 1 ( ) d L L dt q q = 0 (5.1) t 2 4 = q = F (q, q) (5.2) q = v (5.3) v = F (q, v) (5.4) 2 2 L(r, ϕ, ṙ, ϕ) = m ( ) ṙ 2 + r 2 ϕ2 + eb 0 eq 2 2 r2 ϕ + r (5.5) 2 2 r = r ϕ 2 + eb 0 m r ϕ eq m ϕ = 2ṙ r ϕ eb 0 ṙ m r 1 r 2 (5.6) (5.7) v r = ṙ (5.8) 79

5 v ϕ = ϕ (5.9) ṙ = v r (5.10) v r = rvϕ 2 + eb 0 m rv ϕ eq 1 m r 2 (5.11) ϕ = v ϕ (5.12) v ϕ = 2 v r r v ϕ eb 0 v r m r (5.13) 4 = N L(q 1,..., q N, q 1,..., q N ) q i p i = L q i p i = L (q 1,..., q N, q 1,..., q N ) q i (5.14) ṗ i = L (q 1,..., q N, q 1,..., q N ) q i (5.15) 2N q i q i = F q (q 1,..., q N, p 1,..., p N ), (5.16) ṗ i = F p (q 1,..., q N, p 1,..., p N ) (5.17) (5.16) (5.17) p i = L(, q i, ) q i (5.18) q i = H(, p i, ) p i (5.19) H 80

5.2 5.2 f(x) β x x f(x) x β β = df(x) dx (5.20) x x β x β (5.21) β (5.20) x = dg(β) dβ (5.22) g(β) x β = f + g (5.23) g(β) = x β f(x) (5.24) g(β) = x(β) β f(x(β)) (5.25) dg dβ = x + β dx dβ df dx dx dβ = x (5.26) (5.20) (5.24) x f g (Legendre) g (5.23) f 0 (x) = x 2 g 0 (β) = β2 4 (5.27) 81

5 f 0 (x) x f 1 (x) = (x 1) 2 g 1 (β) = β2 4 + β (5.28) f 0 (x) f 2 (x) = c x 2 g 2 (β) = β2 4c (5.29) L(q, q) q L( q) p = L( q) q (5.30) q = (?) p (5.31) L( q) q H(p) = p q L( q) (5.32) q = H(p) p (5.33) H(p) p = q + p q p L q q p q = q + p p p q = q (5.34) p L H L H 82

5.3 f(x) β x x β f(x) g(β) df dx = β dg dβ = x... q p L(, q) H(, p) L q = p H p = q 5.3 L(q, q) q 1 H(q, p) = p q L(q, q) (5.35) Hamiltonian q p q p q q p q = H (5.36) p 83

5 H q H q = {p q L(q, q(q, p)} (5.37) q q(q, p) = p L q q L q (5.38) q q ( ) L q = p (5.39) = L q (5.40) = ṗ (5.41) ( ) d L = dp dt q dt = L q 5.4 5.4.1 1 (5.42) (5.42) H q = ṗ (5.43) H(q, p) 1 H(q, p) q = (5.44) p H(q, p) ṗ = (5.45) q L(q, q) H(q, p) H(q, p) = p q(q, p) L(q, q(q, p)) (5.46) H L p q p = L(q, q) q 84

5.4 5.4.2 1 N N q 1,, q N N p 1,, p N 2N H(q 1, q 2,..., q N, p 1, p 2,..., p N ) L(q 1,, q N, q 1,, q N ) N N H(q 1,, q N, p 1,, p N ) = p i q i L(q 1,, q N, q 1,, q N ) (5.47) i=1 N q i = H p i (1 i N) (5.48) ṗ i = H q i (1 i N) (5.49) p i q i p i = L q i (5.50) 85

5 5.4.3 p q L(q 1, q 2,..., q N, q 1, q 2,..., q N ) H(q 1, q 2,..., q N, p 1, p 2,..., p N ) (5.51) 2N (q 1 (t), q 2 (t),..., q N (t), p 1 (t), p 2 (t),..., p N (t)) N t (q 1 (t), q 2 (t),..., q N (t) ( q 1 (t), q 2 (t),..., q N (t) (q 1, q 2,..., q N, p 1, p 2,..., p N ) 2N (5.48), (5.49) 2N (q 1 (t), q 2 (t),..., q N (t), p 1 (t), p 2 (t),..., p N (t)) 86

5.4 5.4.4 : q=0 q k m 1 q = 0 q L p L(q, q) = K U = m 2 q2 k 2 q2 p = L q = m q ( H(q, p) = p q L = p2 p 2 m 2m k ) 2 q2 = p2 2m + k 2 q2 H(q, p) = p2 2m + k 2 q2 (5.52) q = H p = p (5.53) m ṗ = H = kq (5.54) q (5.53), (5.54) (5.3) (5.4) (5.52) q q H = m 2 q2 + k 2 q2 = K + U (5.55) 87

5 5.5 (5.52) U(q) K q f(q) U(q, q) = U(q), K(q, q) = f(q) q 2 (5.56) L L = K U = f(q) q 2 U(q) (5.57) U q K p = K q (5.58) p = K q = 2f(q) q (5.59) H = p q L (5.60) = 2f(q) q q ( f(q) q 2 U(q) ) (5.61) = f(q) q 2 + U(q) (5.62) = K + U (5.63) H = += ( ) (5.64) H q p 88

5.6 5.6 p H(q,p) = ( H q, H ) p 接線方向 ( dq dt, dp ) dt = ( ) H p, H q q H= 一定の曲線 ( 曲面 ) 上を動く 1 H(q, p) 2 (q, p) (q 0, p 0 ) ( ) q-p H(q, p) (gradient) ( ) H/ q H = (5.65) H/ p H ( ) ( ) q H/ p = (5.66) ṗ H/ q ( ) q u ṗ (5.67) H u H = H H p q H H q p = 0 (5.68) H(q, p) 89

5 5.7 p 体積は変わらない q N 2N u 1 q 1 u 2 q 2. u =. u N u N+1 u N+2. u 2N = q Ṅ p 1 ṗ 2. ṗ N (5.69) 2N u = N j=1 u j q j + N j=1 u N+j p j (5.70) = q j + ṗ j q j p j (5.71) = H H q j p j p j q j (5.72) = 0 (5.73) 90

5.8 5.8 5.8.1 1 z (1,0,1) y θ x p.33 m 1 x 2 + y 2 = 1 (x, y, z) = (1, 0, 1) k 0 (a) x q U K (b) q p (c) q p H(q, p) (d) (e) q 1 (a) (x, y, z) = (cos q, sin q, 0) l l 2 = (cos q 1) 2 + sin 2 q + 1 = 3 2 cos q U = k 2 l2 = k (3 2 cos q) 2 91

5 K = m 2 q2 (b) (5.56) p = K q = m q (c) (d) H = K + U H(q, p) = p2 2m + k (3 2 cos q) 2 q = H p = p m (5.74) ṗ = H q = k sin q (5.75) (d) q 1 sin q q (5.75) ṗ = k q (5.74) q = ω 2 q ω 2 = k/m q(t) = c 1 cos (ωt + c 2 ) c 1 c 2 5.8.2 2 92

5.8! g φ 2 m " 2 m g ( x y q (a) U K (a) q p (b) p q H(q, p) (c) (d) q 1 (a) (x, y) (x, y) = (sin q, cos q) (ẋ, ẏ) = (cos q q, sin q q) K K = m 2 (ẋ2 + ẏ 2 ) = m 2 q2 U U = mgy = mg cos q 93

5 (b) p = K q = m q (c) H = K + U = p2 + mg cos q 2m (d) q = H p = p m ṗ = H = mg sin q q (e) q 1 sin q q q = p m ṗ = mgq 2 q = gq c 1 c 2 q(t) = c 1 e g t + c 2 e g t c 2 = 0 q(t) = c 1 e g t g 94

5.8 5.8.3 3 q 1 1 x y 1 m k 0 x q (1-1) H(q, p) (1-2) (1-3) (1-1) q m q 2 /2 U(q) = k 2 { (1 + cos q) 2 + (1 + sin q) 2} q p = K q = m q H(q, p) = K + U = p2 + k (cos q + sin q) 2m H(q, p) = p2 2m + ( 2k cos q π ) 4 (5.76) 95

5 (1-2) q = p (5.77) m ṗ = k (sin q cos q) (5.78) (1-3) q = 5π/4 l g l q m g l m l q p = ml 2 q H(q, p) = p2 mg cos q (5.79) 2ml2 2 (5.76) (5.79) (5.79) (5.76) l 1 k 2k q q + 5π/4 x 5π/4 2k 5.8.4 4 L(q, q) = 1 2 (q + q)2 (a) H(q, p) (b) (c) q(0) = p(0) = 1 q(t) p(t) 96

5.8 (a) q L(q, q) = 1 (q + q)2 2 p = L q = q + q H L (b) H(q, p) = p q L(q, q) = p (p q) 1 2 p2 = p2 2 pq q = H p ṗ = H q H(q, p) (c) (5.81) q = p q (5.80) ṗ = p (5.81) p(t) = c e t c p(0) = 1 c = 1 p(t) = e t p(t) (5.80) q(0) = 1 (5.80) q(t) = c 1 e t + c 2 e t (5.82) c 1 + c 2 = 1 (5.83) q(0) = p(0) q(0) = 1 1 = 0 97

5 (5.82) q(0) = c 1 c 2 (5.83) (5.84) c 1 c 2 = 0 (5.84) c 1 = c 2 = 1 2 q(t) = 1 2 et + 1 2 e t = cosh t q(t) = cosh t p(t) = e t 5.8.5 5 q 1 q 2 1 2 m 1 2 k 0 98

5.8 x q 1 q 2 l l = 5 4 cos (q 1 q 2 ) (5.85) U = k 2 {5 4 cos (q 1 q 2 )} L = K U = m 2 q2 1 + m 2 (2 q 2) 2 k 2 l2 (q 1, q 2 ) l (5.85) p 1 = L q 1 = m q 1 (5.86) p 2 = L q 2 = 4m q 2 (5.87) H(q 1, q 2, p 1, p 2 ) = p 1 q 1 +p 2 q 2 L = = p2 1 2m + p2 2 8m +k 2 {5 4 cos (q 1 q 2 )} H = K + U 5k/2 H(q 1, q 2, p 1, p 2 ) = p2 1 2m + p2 2 8m 2k cos (q 1 q 2 ) (5.88) 4 q 1 = H p 1 = p 1 m q 2 = H p 2 = p 2 4m (5.89) (5.90) ṗ 1 = H q 1 = 2k sin (q 1 q 2 ) (5.91) ṗ 2 = H q 2 = 2k sin (q 1 q 2 ) (5.92) 99

Chapter 6 6.1 (q 1,..., q N ) (Q 1,..., Q N ) Q i = Q i (q 1,..., q N ) (i = 1, 2,... N) (6.1) ( ) d L L = 0 dt q i q i ( ) d L dt Q L = 0 i Q i 4.1 p.59 = H(q, p) = q4 p 2 + 1 2 q (6.2) q = H p = q4 p (6.3) ṗ = H q = 2q3 p 2 + 1 q 2 (6.4) 101

6 4 (6.4) 2 q q p q(t) = 1 1 t2 2 q(t) t = 2 = q 1 t (q, p) (Q, P ) N H(q 1,..., q N, p 1,..., p N ) (q 1,..., q N, p 1,..., p N ) (Q 1,..., Q N, P 1,..., P N ) (6.1) q q p Q i = Q i (q 1,..., q N, p 1,..., p N ) (6.5) P i = P i (q 1,..., q N, p 1,..., p N ) (6.6) 102

6.2 k Q k P k 2 (6.5) (6.6) q i = H (q 1,..., q N, p 1,..., p N ) p i (6.7) ṗ i = H (q 1,..., q N, p 1,..., p N ) q i (6.8) Q i = H P i (Q 1,..., Q N, P 1,..., P N ) (6.9) P i = H (Q 1,..., Q N, P 1,..., P N ) (6.10) Q i 6.2 (6.5) (6.6) (6.9) q i = q i (Q 1,..., Q N, P 1,..., P N ) (6.11) p i = p i (Q 1,..., Q N, P 1,..., P N ) (6.12) H P i = N j=1 = H q j q j P i + N j=1 ṗ j q j P i + N j=1 N j=1 H p j p j P i q j p j P i (6.7) (6.8) (6.9) Q i = N j=1 ṗ j Q i p j + N j=1 q j Q i q j 2 (6.10) 103

6 Q i q j = p j P i (6.13) Q i p j = q j P i (6.14) P i p j = q j Q i (6.15) P i q j = p j Q i (6.16) (6.5) (6.6) (6.13) (6.16) 1 1 Q q = p P, (6.17) Q p = q P, (6.18) P p = q Q, (6.19) P q = p Q (6.20) 6.3 (6.1) 1 q q Q ϕ q = ϕ(q) (6.21) q = ϕ (Q) Q (6.22) q Q ϕ d 1 dq ϕ 1 (q) = ϕ (ϕ 1 (q)) Q = ϕ 1 (q) (6.23) (6.24) 104

6.3 Q = q ϕ (ϕ 1 (q)) (6.25) (6.23) (6.21) 0 = Q p = q P (6.18) (6.20) (6.26) P = L(q, q) Q (6.27) = L q L q + q Q q Q (6.28) (6.21) p p = L q (6.29) (6.22) P = p ϕ (Q) (6.30) = p ϕ (ϕ 1 (q)) (6.31) (6.30) (6.31) p = P ϕ (Q) (6.32) P q = p d { ϕ (ϕ 1 (q)) } (6.33) dq = p ϕ (ϕ 1 (q)) d dq ( ϕ 1 (q) ) (6.34) = p ϕ (ϕ 1 (q)) 1 ϕ (ϕ 1 (q)) (6.35) (6.24) P Q (6.32) (6.23) P q = P ϕ (Q) {ϕ (Q)} 2 (6.36) (6.32) p Q = ( ) P Q ϕ = P ϕ (Q) (Q) {ϕ (Q)} 2 (6.37) 105

6 (6.37) (6.36) p Q = P q (6.38) (6.20) (6.17) (6.23) (6.24) (6.32) Q q = 1 ϕ (ϕ 1 (q)) = 1 ϕ (Q) p P = 1 ϕ (Q) Q q = p P (6.19) (6.31) (6.39) (6.40) (6.41) (6.23) (6.21) P p = ϕ (ϕ 1 (q)) (6.42) P p = ϕ (Q) (6.43) q Q = ϕ (Q) (6.44) P p = q Q 6.4 (6.45) (6.13) (6.16) q-p q -p (q 1, q 2,..., p N ) (q 1, q 2,..., p N ) q -p q -p (q 1, q 2,..., p N) (q 1, q 2,..., p N ) 106

6.4 q-p q -p (q 1, q 2,..., p N ) (q 1, q 2,..., p N) (6.13) q i q j = = N k=1 N k=1 = p j p i N q i q k q k + q j ( ) ( p k pj p i k=1 p k q i p k p k q j ) + N ( q k p k=1 i ) ( ) p j q k (6.46) q i q j = p j, p i q i = q j p j p, i p i p j = q j, q i p i q j = p j q i (6.47) q-p q -p (6.2) H(q, p) = q4 p 2 + 1 2 q (6.48) (q, p) (q, p) (Q, P ) Q = Q(q, p) = q 2 p (6.49) P = P (q, p) = 1/q (6.50) (6.49) q = q(q, P ) = 1/P (6.51) p = p(q, P ) = QP 2 (6.52) Q p = q2 107

6 (6.49) q P = 1/P 2 = q 2 Q p = q P (6.17) (6.18) (6.49) (6.50) (Q, P ) H(Q, P ) = q4 (Q, P )p 2 (Q, P ) 2 + 1 q(q, P ) = Q2 2 + P (6.53) Q = H P = 1 P = H Q = Q 6.5 (6.13) (6.16) (6.49) (6.50) q Q W (q, Q) (6.54) 2 P = W (q, Q) (6.55) Q p = W q (q, p) (Q, P ) (6.54) W (q, P ) (6.56) 108

6.5 W (p, Q) W (p, P ) (6.55) (6.56) q p 2 P (q, p) = W (q, Q(q, p)) Q p = W (q, Q(q, p)) q 2 q p 4 Q q P q Q p P p = 1 2 W q Q P q = 2 W q Q 2 W Q Q 2 q P p = 2 W Q Q 2 p 0 = 2 W q 2 1 = 2 W Q q Q p + 2 W Q q Q q ( ) ( ) 2 W 2 W Q 2 q 2 2 W q 1 2 ( 2 W q Q ) 2 2 W Q 2 Q P (6.55) (6.56) P = W (q(q, P ), Q) Q p(q, P ) = W (q(q, P ), Q) q (6.57) Q P q Q p Q q P p P = 1 2 W q Q 2 W Q 2 ( ) ( ) 2 W 2 W Q 2 q 2 1 ( ) 2 2 W q Q 2 W q 2 (6.58) 109

6 (6.57) (6.58) (6.17) (6.20) N W (q 1,..., q N, Q 1,..., Q N ) (6.59) P i = W Q p i = W q (6.60) (6.61) q 1 q 2 1 2 2 m 1 2 k 0 x q 1 q 2 p 1 = m q 1 p 2 = 4m q 2 H(q 1, q 2, p 1, p 2 ) = p2 1 2m + p2 2 8m 2k cos (q 1 q 2 ) (6.62) 110

6.5 (q 1, q 2, p 1, p 2 ) W (q 1, q 2, Q 1, Q 2 ) = (q 2 q 1 )Q 1 (q 1 + q 2 )Q 2 (6.63) P 1 = W = q 1 q 2 Q 1 (6.64) P 2 = W = q 1 + q 2 Q 2 (6.65) p 1 = W = (Q 1 + Q 2 ) p 1 (6.66) p 2 = W = Q 1 Q 2 p 2 (6.67) Q 1 = Q 1 (q, p) = 1 2 (p 1 p 2 ) (6.68) Q 2 = Q 2 (q, p) = 1 2 (p 1 + p 2 ) (6.69) P 1 = P 1 (q, p) = q 1 q 2 (6.70) P 2 = P 2 (q, p) = q 1 + q 2 (6.71) q 1 = q 1 (Q, P ) = 1 2 (P 1 + P 2 ) (6.72) q 2 = q 2 (Q, P ) = 1 2 (P 1 P 2 ) (6.73) p 1 = p 1 (Q, P ) = (Q 1 + Q 2 ) (6.74) p 2 = p 2 (Q, P ) = Q 1 Q 2 (6.75) (Q 1, Q 2, P 1, P 2 ) (6.62) H(q(Q, P ), p(q, P )) = = 5 8m (Q2 1 + Q 2 2) + 1 2m Q 1Q 2 2k cos P 1 H(Q, P ) = 5 8m (Q2 1 + Q 2 2) + 1 2m Q 1Q 2 2k cos P 1 (6.76) Q 1 = 2k sin P 1 (6.77) Q 2 = 0 (6.78) P 1 = 5 4m Q 1 1 2m Q 2 (6.79) P 2 = 5 4m Q 2 1 2m Q 1 (6.80) 111

6 (6.78) Q 2 = c ( ) (6.81) Q 1 = 2k sin P 1 (6.82) P 1 = 5 4m Q 1 c 2m (6.83) P 2 = 5c 4m 1 2m Q 1 (6.84) 1 c = 0 Q 1 = 2k sin P 1 (6.85) P 1 = 5 4m Q 1 (6.86) P 2 = 1 2m Q 1 (6.87) (6.85) (6.86) P 1 P 1 = 0 c = Q 2 = 0 (6.69) p 1 + p 2 = 0 q 1 q 2 p 1 + p 2 Q 2 = 0 Q 2 p 1 + p 2 (6.81) p 1 + p 2 2 (6.62) p 1 + p 2 (6.76) P 2 Q 2 Q 2 = 0 112

6.6 6.6 N 2N r r = q 1 q 2.. q N p 1 p 2. p N, r i = { q i (1 i N) p i N (N < i 2N) H ṙ i = 2N j=1 J ij H r j (6.88) J J = ( 0 ) 1 1 0 (6.89) 0 1 N N (6.88) J J T J J 1 J ( ) J T 0 1 = (6.90) 1 0 ( ) JJ T = J T 1 0 J = 1 = (6.91) 0 1 J T = J = J 1 (6.92) J 2 = 1 (6.93) J = 1 (6.94) r R { Q i (1 i N) R i = P i N (N < i 2N) 113

6 Ṙ i = (6.95) Ṙ i = = = 2N m=1 2N 2N 2N j=1 J ij H R j (6.95) R i r m ṙ m (6.96) m=1 k=1 2N 2N 2N m=1 k=1 j=1 (6.95) R i H J mk (6.88) (6.97) r m r k R i r m J mk R j r k H R j (6.98) MJM T = J (6.99) M M T M ij = R i r j (6.100) M T ij = R j r i (6.101) M (6.93) (6.99) M T JM = J (6.102) (6.99) (6.102) 6.7 {f, g} q,p = N ( f j=1 q j g g ) f p j q j p j (6.103) 114

6.7 f g J {f, g} q,p = 2N 2N i=1 j=1 f g J ij (6.104) r i r j q,p {f, g} = N ( f j=1 q j g g ) f p j q j p j (6.105) (6.99) 2N 2N 0 1 1 M J 4N 2 4 {Q i, Q j } = 0 (6.106) {P i, P j } = 0 (6.107) {Q i, P j } = δ ij (6.108) {P i, Q j } = δ ij (6.109) (6.103) {g, f} = {f, g} (6.110) c, c 1,c 2 {f, f} = 0 (6.111) {c 1 f 1 + c 2 f 2, g} = c 1 {f 1, g} + c 2 {f 2, g} (6.112) {f, c 1 g 1 + c 2 g 2 } = c 1 {f, g 1 } + c 2 {f, g 2 } (6.113) {c, g} = 0 (6.114) {f, c} = 0 (6.115) q i / q j = δ ij {q i, f} = f p i (6.116) {p i, f} = f q i (6.117) 115

6 { q i, f } { + p j, f } q j p i (6.116) (6.117) = 0 (6.118) {q i, q j } = 0 (6.119) {q i, p j } = δ ij (6.120) {p i, p j } = 0 (6.121) 3 f, g, h {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 (6.122) 1 2N (6.104) (6.122) = f ( ) g h J ij J lm r i r j r l r m + g ( ) h f J lj J mi r l r j r m r i + h ( ) f g J mj J il r m r j r i r l = f g 2 h (J ij J lm + J lj J mi ) r i r l r j r m }{{} (a) (a) = + f r i + g r l h r m h r m 2 h J ij J lm + 2 h J lm J ji r j r m r m r j = 2 h J ij J lm 2 h J lm J ij r j r m r m r j = 0 2 g r j r l (J ij J lm + J mj J il ) 2 f r j r i (J lj J mi + J mj J il ) (6.123) 2 j m (6.92) 116

6.8 (6.123) 2 (6.122) (6.109) (6.110) (q 1,..., q N, p 1,..., P N ) (Q 1,..., Q N, P 1,..., P N ) {Q i, Q j } = 0 (6.124) {P i, P j } = 0 (6.125) {Q i, P j } = δ ij (6.126) 3 Q i =Q i (q 1, q 2,..., q N, p 1,..., p N ), (6.127) P i =P i (q 1, q 2,..., q N, p 1,..., p N ) (6.128) q i =q i (Q 1, Q 2,..., Q N, P 1,..., P N ), (6.129) p i =p i (Q 1, Q 2,..., Q N, P 1,..., P N ) (6.130) 6.8 f df N dt = ( f q j + f ) ṗ j q j p j = j=1 N j=1 f ( f H f ) H q j p j p j q j = {f, H} (6.131) df dt = f + {f, H} (6.132) t (6.131) f H dh dt = {H, H} = 0 (6.111) (6.133) 117

6 f g {f, g} (6.103) f = ġ = 0 d { } dt {f, g} = f, g + {f, ġ} (6.134) d {f, g} = {0, g} + {f, 0} = 0 (6.135) dt (6.122) (6.131) f = q i f = p i q i = {q i, H} (6.136) ṗ i = {p i, H} (6.137) 2 6.9 (6.131) (q 1,... p N ) df/dt (Q 1,... P N ) {f(q, p), H(q, p)} q,p = N j=1 ( f q j H p j H q j ) g p j (6.138) {f(q, P ), H(Q, P )} Q,P = N j=1 ( f Q j H P j H Q j ) g P j (6.139) {f, H} q,p = {f, H} Q,P (6.140) H f g {f, g} q,p = {f, g} Q,P (6.141) 118

6.10 (6.104) {f, g} q,p = = = = 2N 2N f g J ij r i r j i=1 j=1 2N 2N 2N 2N i=1 j=1 I=1 J=1 2N 2N 2N 2N i=1 j=1 I=1 J=1 2N 2N I=1 J=1 ( ) ( ) f R I g R J J ij R I r i R J r j f R I f R I J IJ g J ( ) RI R J g J ij r i r j R J (6.99) = {f, g} Q,P (6.142) 6.10 p t = 0 t = T q t = 0 r = (q 1 (0), q 2 (0),..., p N (0)) t = T R = (q 1 (T ), q 2 (T ),..., p N (T )) r = (q 1, q 2,..., p N ) = (q 1 (0), q 2 (0),..., p N (0)) (6.143) R = (Q 1, Q 2,..., Q N ) = (q 1 (T ), q 2 (T ),..., p N (T )) (6.144) 119

6 r R r R (6.124) (6.126) {q i (t), p j (t)} {Q i, P j } = {q i (T ), p(t )} (6.145) = {q i, p j } t=0 + T d 1 dt {q i, p j } + T 2 d 2 t=0 2! dt 2 {q i, p j } (6.146) t=0 + T 3 d 3 3! dt 3 {q i, p j } + (6.147) t=0 (6.131) f = {q i, p j } 2 (6.120) d dt {q i(t), p j (t)} = {{q i (t), p j (t)}, H} t=0 = {{q i, p j }, H} = {δ ij, H} = 0 t=0 (6.114) (6.148) d 2 dt 2 {q i, p j } = {{{q i, p j }, H}, H} = {0, H} = 0 (6.149) t=0 d 3 dt 3 {q i, p j } = {{{{q i, p j }, H}, H}, H} = 0 (6.150) t=0 {Q i, P j } = {q i (0), p j (0)} = {q i, p j } = δ ij (6.151) {Q i, Q j } = 0 (6.152) {P i, P j } = 0 (6.153) r R 6.11 ( qi p i ) ( Qi P i ) = ( qi p i ) ( δqi + δp i ) (6.154) 2 120

6.11 ϵ G(q 1, q 2,, q N, p 1, p 2,... p N ) δq i = ϵ {q i, G} = ϵ G p i (6.155) δp i = ϵ {p i, G} = ϵ G q i (6.156) G (6.155), (6.156) (6.124) (6.126) (6.124) {Q i, Q j } = {q i + δq i, q j + δq j } (6.157) = {q i, q j } + {q i, δq j } + {δq i, q j } + {δq i, δq j } (6.158) = 0 + {q i, δq j } + {δq i, q j } + 0 (6.159) (6.119) 2 (6.155) {q i, δq j } + {δq i, q j } = ϵ {q i, {q j, G}} + ϵ {{q i, G}, q j } (6.160) = ϵ [{q i, {q j, G}} + {q j, {G, q i }}] (6.161) = ϵ {G, {q i, q j }} (6.122) (6.162) = 0 (6.119) (6.163) {Q i, Q j } = 0 (6.164) (6.125) (6.126) G f(q 1, q 2,..., p N ) δf = ϵ {f, G} (6.165) δf = f(q 1 + δq 1, q 2 + δq 2,..., p N + δp N ) f(q 1, q 2,... p N ) (6.166) N ( f = δq j + f ) δp j q j p j (6.167) = ϵ j=1 N j=1 ( f G f ) G q j p j p j q j (6.168) = ϵ {f, G} (6.169) G k G = p k (6.170) 121

6 q i q i + δq i = q i + ϵ G = q i + ϵδ ik p i (6.171) p i p i + δp i = p i (6.172) k q k ϵ 6.12 H G (6.165) δh = ϵ {H, G} = 0 (6.173) (6.131) dg dt = {G, H} = {H, G} (6.174) dg dt = 0 (6.175) G G H q k (6.171) (6.172) H (6.170) ṗ k = 0 (6.176) N (X 1, X 2, X 3 ) X α ϵ ϵ 6.3 122

6.12 G α (q 1, q 2,..., p N ) = N j=1 p j q i X α (6.177) δq i = ϵ {q i, G} = ϵ q i X α (6.178) (4.65) p.67 ω = (ω 1, ω 2, ω 3 ) ω ϵ G(ω) = 3 3 3 N α=1 β=1 γ=1 j=1 ω β ϵ βγα X γ (q j )p j q j X α (6.179) ω (4.77) δq i = ϵ {q i, G} = ϵ (4.72) 3 3 3 α=1 β=1 γ=1 ω β ϵ βγα X γ (q i ) q i X α (6.180) Y j α = p j q j X α (6.181) N G(ω) = ω X(q j ) Y j (6.182) j=1 ω G(ω) (6.182) N X(q j ) y j (6.183) j=1 p i (6.156) H 123

6 6.13 q i = H p i (6.184) ṗ i = H q i (6.185) 1 t = t n r = (q 1, q 2,..., p N ) = (q n 1, q n 2,..., p n N ) t 1 Q i = q i (t + t) = q n+1 i P i = p i (t + t) = p n+1 i = q n i + t H p i (r n ) (6.186) = p n i t H q i (r n ) (6.187) t = t n+1 = t n + t R = (Q 1, Q 2,..., P N ) = (q1 n+1, q2 n+1,..., p n+1 N ) r R { {Q i, P j } = q i + t H, p j t H } p i q j { = {q i, p j } t q i, H } { } { H H + t, p j t 2, H } q j p i p i q j 2 3 (6.118) { H {Q i, P j } = δ ij t 2, H } p i q j (6.188) δ ij 2 O( t 2 ) t = 0 t = T T/ t O( t 2 ) (T/ t) = O( t) 1 6.14 (q, p) (q, p) (Q, P ) = (q cos ϕ + p sin ϕ, q sin ϕ + p cos ϕ) ϕ 124

6.14 (6.17) (6.20) (6.124) (6.126) 1 (i) Q q = cos ϕ (ii) p P = cos ϕ (Q, P ) = (q cos ϕ + p sin ϕ, q sin ϕ + p cos ϕ) (6.189) (q, p) = (Q cos ϕ P sin ϕ, Q sin ϕ + P cos ϕ) (6.190) (6.17) (6.18) (6.20) 2 (6.111) {Q, Q} = {P, P } = 0 (6.191) {Q, P } = Q P q p P Q q p (6.192) = (cos ϕ) (cos ϕ) ( sin ϕ) (sin ϕ) (6.193) = 1 (6.194) N = 1 (6.124) (6.126) 125

Chapter 7 L(q 1, q 2,, q N, q 1, q 2,, q N ) d dt ( L q i ) L q i = 0 (i = 1, 2,, N) 7.1 V 1 V 2 T y = y(x) V 1 = V 2 T y(x) V 1 V 2 127

7 y x 2 x, y (0, 0) (a, A) y = y(x) +y g y(x) m (a, A) T T y=y(x) a y y = y(x) s m 2 ṡ2 mgy(s) = 0 (7.1) 128

7.1 ṡ = 2gy(s) (7.2) s t t = s ṡ (7.3) = s 2gy (7.4) 1 + (y ) = 2 x 2gy (7.5) T = 1 2g a 0 1 + (y ) 2 y dx (7.6) T y(x) y(0) y(a) T (y, y ) = a 0 1 + (y ) 2 y dx (7.7) y(x) 2 3 y = y(x) (0 x a) x S[y] y y y=y(x) y=y(x) a a 129

7 x x S = 2πy x 2 + y 2 = 2πy 1 + (y ) 2 x S = 2π a 0 y(x) 1 + (y (x)) 2 dx (7.8) y(0) y(a) S(y, y ) y(x) 2π S(y, y ) = a 0 y(x) 1 + (y (x)) 2 dx (7.9) y = y(x) y(x) x I(y) y(x) (7.7) (7.9) y = y(x) y = y(x) x = x 0 y(x 0 + ϵ) = y(x 0 ) + ϵ y(x 0 ) + O(ϵ 2 ) (7.10) y(x 0 ) = y (x 0 ) (7.11) y(x) x 0 x 0 y y y0(x) y0(x)+εη(x) I(y, y ) y 0 (x) y 0 (x) 130

7.2 I y(x) = y 0 (x) + ϵη(x) ϵ η(x) η(a) = η(b) = 0 (7.12) y (x) = y 0(x) + ϵη (x) I(y 0 + ϵη, y 0 + ϵη ) = I(y 0, y 0) + ϵ δi(y 0, y 0) + O(ϵ 2 ) (7.13) δi I I y 0 (x) y 0 (x) δi 7.2 L(q, q) 1 S = tb t a L(q, q) dt (7.14) q = q(t) N q i = q i (t) S = tb t a L(q 1,, q N, q 1,, q N, t) dt (7.15) 131

7 q.. q q0 q = q0 t - g t 2 / 2.. q = q0 - g t t q q = 0 t 1 q = 0 t q(t) q(t) t-q t1 ( m δ 0 2 q2 mgq) dt = 0 (7.16) q(t) q(t) t- q q(t) δq(t) = ϵη(t) (7.17) q(t) q (t) = q(t) + ϵη(t) (7.18) q(t) δ q(t) = ϵ η(t) (7.19) 132

7.2 q k = q(t k ), t k = t a + k t (7.20) t a t b K t = (t b t a )/K S = tb t a L(q, q) dt = S k = tk+1 K 1 k=0 S k (7.21) t k L(q, q) dt (7.22) q k = qk+1 q k t (7.23) S k = L(q k, qk+1 q k ) t + O( t 2 ) (7.24) t q = q(t) k q k S S q k = 0 (7.25) q k S k S k 1 S k q k = L q (qk, qk+1 q k ) t L t q (qk, qk+1 q k ) (7.26) t S k 1 q k = L q (qk 1, qk q k 1 ) (7.27) t (7.25) L q (qk, qk+1 q k ) 1 ( L t t q (qk, qk+1 q k ) L t q (qk 1, qk q k 1 ) ) = 0 t (7.28) t = 0 L q d ( ) L = 0 (7.29) dt q 133

7 7.3 S = t1 t 0 L(q, q) dt (7.30) H(q, p) q p L = p q H (7.31) q(t) p(t) Ŝ = t1 t 0 {p q H(q, p)} dt (7.32) 2 N 2N (q(t), p(t)) Ŝ q p q = q0 t - g t 2 / 2 p0 t q p trajectory 134

7.3 Î q p (7.21) (7.32) Ŝ k = Ŝ = tk+1 Ŝk K 1 k=0 Ŝ k (7.33) t k {p q H(q, p)} dt (7.34) Ŝ k = ( p k q k H(q k, p k ) ) t (7.35) q k = qk+1 q k t (7.36) Ŝ k = p k ( q k+1 q k) H(q k, p k ) t (7.37) k Ŝ q k = 0 (7.38) Ŝ p k = 0 (7.39) q p p0 t 135

7 q k S k S k 1 (7.38) Ŝ q k = Ŝk q k + Ŝk 1 q k = p k H q (qk, p k ) t + p k 1 H q (qk 1, p k 1 ) t = 0 (7.40) t = 0 dp dt = H q p k Ŝk (7.39) (7.41) t = 0 Ŝ p k = ( q k+1 q k) H p (qk, p k ) t = 0 (7.42) dq dt = H p (7.43) (7.41) (7.39) 7.4 (7.34) t = t k t k+1 q p q = q k+1 p = p k S k = p k ( q k+1 q k) H(q k+1, p k ) t (7.44) (7.38) (7.39) S q k = S k q k + S k 1 q k = p k + p k 1 H q (qk, p k 1 ) t = 0 (7.45) S p k = S k p k = ( q k+1 q k) H p (qk+1, p k ) t = 0 (7.46) 136

7.5 = (7.45) k k + 1 p k+1 + p k H q (qk+1, p k ) t = 0 (7.47) (7.48) (7.49) q k+1 = q k + t H p (qk+1, p k ) (7.48) p k+1 = p k t H q (qk+1, p k ) (7.49) H(q, p) = K(p) + U(q) (7.50) q k+1 = q k + t K p (pk ) (7.51) p k+1 = p k t U q (qk+1 ) (7.52) 7.5 = x y(x) I(y, y, x) = y = y(x) y a = y(a), y b = y(b) (7.53) b a L(y(x), y (x), x) dx (7.54) δi = 0 (7.55) y(x) y(x) y(x) + ϵ η(x) (7.56) I I(y + ϵη, y + ϵη, x) = b a L(y(x) + ϵη(x), y (x) + ϵη (x), x) dx (7.57) 137

7 ϵ L(y + ϵη, y + ϵη, x) = L(y, y, x) + L L ϵη(x) + y y ϵη (x) + O(ϵ 2 ) (7.58) I(y + ϵη, y + ϵη, x) = b a + ϵ L(y, y, x) dx b a ( η(x) L y dx + η (x) L ) y dx + O(ϵ 2 ) (7.59) = I(y, y ) + ϵ δi(y, y ) + O(ϵ 2 ) (7.60) δi = δ = b a b (7.53) b a η (x) a L(y, y, x) dx (7.61) ( η(x) L y + η (x) L y ( ) [ ( )] b L L y dx = η(x) y a δi = δ b a L(y, y, x) dx = ) dx (7.62) (7.63) η(a) = η(b) = 0 (7.64) b a { d dx η(x) δi = 0 d dx b a η(x) d dx ( ) L y dx (7.65) ( ) L y + L } η(x) dx = 0 (7.66) y ( ) L y L y = 0 (7.67) = N y i (x) (i = 1,..., N) y i (a) y i (b) (i = 1,..., N) I(y 1, y 2,, y N, y 1, y 2,, y N) = b a L(y 1, y 2,, y N, y 1, y 2,, y N, x) dx 138

7.5 = y i (x) (i = 1,..., N) N = ( ) d L L = 0 (i = 1,, N) dx y i y i p.129 y = y(x) x S S(y, y ) = a 0 y(x) 1 + y (x) 2 dx (7.68) L(y, y ) = y 1 + y 2 (7.69) = (7.67) L y = 1 + y 2 (7.70) L y = yy 1 + y 2 ( ) ( ) d L dx y = d yy dx 1 + y 2 (7.71) (7.72) = y 2 + yy 1 + y 2 (7.73) (7.67) yy = 1 + y 2 (7.74) y = cosh(x) tb δs = δ L(q, q), dt = 0 (7.75) t a δq(t a ) = δq(t b ) = 0 (7.76) = L q d ( ) L = 0 (7.77) dt q 139

7 q(t), p(t) tb δs = δ (p q H(q, p)) dt = 0 (7.78) t a δq(t a ) = δq(t b ) = 0 (7.79) = dq dt = H p dp dt = H q (7.80) (7.81) S ṗ p 140

Chapter 8 8.1 1 K U H(q, p) = K(p) + U(q) (8.1) K U p q t r = (q, p) R = (Q, P ) = x y 1 m k 0 x 141

8 q 1 1 q H(q, p) = p2 + k (cos q + sin q) (8.2) 2m (8.1) K(p) = p2 2m (8.3) U(q) = k (cos q + sin q) (8.4) 8.2 1 1 (8.2) 1 dq dt = p m (8.5) dp = k(sin q cos q) dt (8.6) Listing 8.1: one ball on ring 1stEuler.cpp 1 2 void e u l e r 1 s t ( s t r u c t p a r t i c l e p a r t i c l e, double dt ) 3 { 4 // Hamiltonian 142