ver. 1.0 18 6 20 F = f m r = F r = 0 F = 0 X = Y = Z = 0 (1 δr = (δx, δy, δz F δw δw = F δr = Xδx + Y δy + Zδz = 0 (2 δr (2 1 (1 (2 n (X δx + Y δy + Z δz = 0 (3 1
F F = (X, Y, Z δr = (δx, δy, δz S δr δw = S δr = 0 F m r = F F m r = 0 F F = m r F F m r = 0 ( = 1, 2, 3, (4 r (t (4 (F m r δr = 0 (5 {(X m ẍ δx + (Y m ÿ δy + (Z m z δz } = 0 (6 F U(r X = U x, Y = U y, Z = U z δu = {( U x δx + ( U y δy + ( } U δz z (6 m (ẍ δx + ÿ δy + z δz = δu (7 2
Hamlton P 2 (t 2 C δr (t C P 1 ( n r ( = 1, 2,, n, 3n P 1 t 2 P 2 P 1 P 2 C t 2 t δr (δx, δy, δz 1 C C P 1 ( P 2 (t 2 (7 (7 ẍ δx ẍ δx = d (ẋ δx ẋ d δx (8 t C x dx C ( dx d (x + δx = + δ ( dx ( dx (9 + d(δx (10 (9 (10 ( dx δ = d(δx δẋ = d δx (11 2 (11 δẏ = d δy, δż = d δz (12 1 dr δr 2 3
(8 ẍ δx = d (ẋ δx x δẋ = d (ẋ δx 1 2 δ(ẋ2 (13 ÿ δy, z δz (7 d m (ẋ δx + ẏ δy + ż δz = m 2 δ(ẋ2 + ẏ 2 + ż 2 δu (14 K K = m 2 (ẋ2 + ẏ 2 + ż 2 (14 K δk d m (ẋ δx + ẏ δy + ż δz = δk δu t 2 t 2 δx = δy = δz = 0 (δk δu = 0 δ (K U = 0 (15 (15 P 1 ( P 2 (t 2 K U (K U L = K U (16 (Lagrangan δ L = 0 (17 (K U L n L L = { t2 δ L = δl = δ = m 2 (ẋ2 + ẏ 2 + ż 2 U (18 m 2 (ẋ2 + ẏ 2 + ż 2 U { m (ẋ δẋ + ẏ δẏ + ż δż U x δx U y δy U z δz } } (19 4
(19 (11(12 [ ] t2 t2 δ L = m (ẋ δx + ẏ δy + ż δz { m (ẍ δx + ÿ δy + z δz + U δx + U δy + U } δz x y z {( = m ẍ + U ( δx + m ÿ + U ( δy + m z + U } δz (20 x y z δx, δy, δz (20 ( U m d 2 x 2 = U x, m d 2 y 2 = U y, m d 2 z 2 d 2 r m = U ( = 1, 2,, n 2 = U z ( = 1, 2,, n l T σ x x + dx du U (dx2 + (du 2 dx 1 ( 2 u 2 x U = 1 ( 2 u 2 T dx x L { L = K U = 1 2 l δ l 0 L = = l 0 l 0 0 { σ u t δ { σ u t σ ( 2 u T t ( u T u t x δ ( } 2 u dx x u (δu T t x x (δu ( } u dx x } dx = 0 5
1 t 2 x l { l [ δ L = σδu u ] t2 } t2 σδu 2 u 0 0 t t 2 dx { t2 [ T δu u ] l } l T δu 2 u x x 2 dx = 0 t 2 δu x = 0 x = l δu 0 l ( σ 2 u t 2 + T 2 u x 2 δudx = 0 0 δu ( 0 σ 2 u t 2 0 0 = T 2 u x 2 0.1 (x 1, x 2, x 3 (r, θ, φ (q 1, q 2, q 3 x 1 = x 1 (q 1, q 2, q 3, x 2 = x 2 (q 1, q 2, q 3, x 3 = x 3 (q 1, q 2, q 3 (21 q 1 = q 1 (x 1, x 2, x 3, q 2 = q 2 (x 1, x 2, x 3, q 3 = q 3 (x 1, x 2, x 3 (22 (x 1, x 2, x 3 (q 1, q 2, q 3 (q 1, q 2, q 3 3 (q 1, q 2, q 3 n 3n (q 1, q 2,, q 3n h f α = f α (q 1, q 2,, q 3n (α = 1, 2,, h (23 f α = f α (q 1, q 2,, q 3n, t (α = 1, 2,, h (24 f f = 3n h f (q 1, q 2,, q f (23 (24 q α q α (holonomc 6
0.2 f δw F 1, F 2,, F f δw = F 1 δx 1 + F 2 dx 2 + + F f δx h = δx = x q 1 δq 1 + x q 2 δq 2 + + x q f δq f = f F δx (25 =1 f j=1 x q j δq j (25 (25(26 (26 δw = f f =1 j=1 Q j δw = f =1 F x q j δq j (26 F x q j (27 f Q j δq j (28 j=1 Q j q j δw J=N m Q j [ ] [q j ] q j Q j q j Q j 0.3 (q 1, q 2,, q f q p L = L(q 1, q 2,, q f, q 1, q 2,, q f p L q (29 L = m 2 ẋ2 (29 p = mẋ 7
L L = L(q 1, q 2,, q f, q 1, q 2,, q f (30 q 1, q 2,, q f q 1, q 2,, q f δ q q δq δ q L L(q 1 + δq 1,, q f + δq f, q 1 + δ q 1,, q f + δ q f = L(q 1, q 2,, q f, q 1, q 2,, q f + ( L δq + L δ q + (31 q q 2 δl = ( L δq + L δ q q q δ L = δl = (11(12 q ( L δq + L δ q = 0 (32 q q (32 [ ] t2 L δq q t1 + t1 δ q = d δq { L q d ( L q } δq = 0 (33 δq δq { } ( d L L = 0 ( = 1, 2,, f (34 q q (34 (q 1, q 2,, q f (Q 1, Q 2,, Q f ( d L L = 0 ( = 1, 2,, f (35 q q 8
(Q 1, Q 2,, Q f L = L(Q 1, Q 2,, Q f, Q 1, Q 2,, Q f (36 ( d L Q L = 0 Q ( = 1, 2,, f (37 (q 1, q 2,, q f (Q 1, Q 2,, Q f q = q (Q 1, Q 2,, Q f ( = 1, 2,, f (38 q Q 1, Q 2,, Q f q q = q Q 1 + q Q 2 + + q Q 1 Q 2 Q f f f = k=1 q Q k Q k = j=1 Q f q Q j Q j (39 (Q 1, Q 2,, Q f (36 (38 (39 Q 1, Q 2,, Q f Q 1, Q 2,, Q f (39 q Q k = q Q k (40 (40 Q 1, Q 2,, Q f (40 ( d q = Q k j 2 q Q j Q k Q j (41 (39 q / Q j Q 1, Q 2,, Q f (39 q Q k = j 2 q Q k Q j Q j (42 (42 (41 ( d q = d ( q Q k Q = q (43 k Q k (q 1, q 2,, q f (Q 1, Q 2,, Q f L (38 (39 Q 1, Q 2,, Q f Q 1, Q 2,, Q f Q Q q q L L Q k L = Q k ( L q q + L Q k q q Q k (44 9
L Q k (38 q Q Q L Q k = = = ( L q q Q + L k q ( L q q Q k ( L q q Q k q (40 (45 (43 ( d L Q = { ( d L q k q Q + L ( } d q k q Q k = { ( d L q q Q + L } q (46 k q Q k (46 (44 ( d L Q L = { d k Q k ( L q Q k (45 L } q (47 q Q k (47 (q 1, q 2,, q f (35 (Q 1, Q 2,, Q f q = q (Q 1, Q 2,, Q f ( = 1, 2,, f Q Q Q U U = Cm r L m L = K U = 1 2 m(ṙ2 + r 2 θ2 + Cm r L ṙ = mṙ, L θ = mr2 θ, d L ṙ L d L L θ θ = d L r = mr θ 2 Cm r 2 L θ = 0 ( r r = m r θ + Cr 2 = 0 (48 ( mr 2 θ = 0 (49 10
(49 mr 2 θ L(q 1, q 2,, q f, q 1, q 2,, q f q d ( L d q = L q ( L = 0 q p = L/ q θ q n n A Φ m q L L = 1 2 m(ẋ2 + ẏ 2 + ż 2 + q (A ṙ qφ (50 3 U m (50 F = q (E + ṙ B B = A E = Φ A t 11
L L q 1, q 2,, q f q 1, q 2,, q f p (29 p L q q q 1, q 2,, q f, p 1, p 2, p f H = f p q L (51 =1 (Hamltonan q q q H q 1, q 2,, q f, p 1, p 2, p f q 1, q 2,, q f p 1, p 2,, p f p q (51 H L = f p q H { } t2 δ L = δl = ( q δp + p δ q δh ( = q δp + p δ q H δq H δp q p ( 2 δ q = d δq δ L = δl [ ] t2 = p δq + t 1 =1 ( q δp ṗ δq H δq H δp q p t 2 δq = 0 1 {( δ L = q H ( δp ṗ + H } δq = 0 (52 p q δq δp ( dq = H p, dp = H q ( = 1, 2,, f (53 q, p 12
L = K U K q 2 K = a j q q j (a j = a j j K q = 2 j a j q j U q q p = L q = K q = 2 j a j q j H H = p q L = 2 a j q j q K + U = 2K K + U j = K + U (54 H H = H (q (t, p (t t q p dh = ( H dq q + H dp p (53 dh = ( H q H p H p H q = 0 (55 U Hamltonan (54 K U L (x, y, z L = K U = 1 2 m(ẋ2 + ẏ 2 + ż 2 U(x, y, z x, y, z p x, p y, p z p x = L ẋ = mẋ, p y = L ẏ = mẏ, p z = L ż = mż 3 Hamltonan H = 1 ( p 2 2m x + p 2 y + p 2 z + U(x, y, z (56 3 p x = mẋ, p y = mẏ, p z = mż Hamltonan 13
Hamltonan p E p x x, p y y, p z z, E t Hamltonan E H = E Ψ(x, y, x, t ( { 2 2m x 2 + y 2 + } Ψ(x, y, x, t z 2 + U(x, y, z Ψ(x, y, x, t = t (57 { } 2 Ψ(r, t 2m 2 + U(r Ψ(r, t = t Shrödnger equaton (58 m 1 Lagrangan x P L = 1 2 mẋ 1 2 mω 0x 2 p = L ẋ = mẋ Hamltonan ( ( ( 1 1 1 1 H = pẋ L = p m p 2 m m p m p + 1 2 mω2 0x 2 = 1 2m p2 + 1 2 mω2 0x 2 ( x, p ẋ = H p = p m, ṗ = H x = mω2 0x 2 1 p = mẋ 2 A Φ m q L (50 L = 1 2 m(ẋ2 + ẏ 2 + ż 2 + q (A ṙ qφ x, y, z p x = L ẋ = mẋ + qa x, p y = L ẏ = mẏ + qa y, p z = L ż = mż + qa z 14
Hamltonan H = p x ẋ + p y ẏ + p z ż 1 2 m(ẋ2 + ẏ 2 + ż 2 q (A ṙ + qφ 1 = p x m (p 1 x qa x + p y m (p 1 y qa y + p z m (p z qa z m { 1 2 m 2 (p x qa x 2 + 1 m 2 (p y qa y 2 + 1 } m 2 (p z qa z 2 { } 1 q A x m (p 1 x A x + A y m (p 1 y A y + A z m (p x A x + qφ = 1 { (px qa x 2 + (p y qa y 2 + (p z qa z 2} + qφ 2m A Φ m q Hamltonan H = 1 { (px qa x 2 + (p y qa y 2 + (p z qa z 2} + qφ 2m = (p qa 2 + qφ (59 2m =x,y,z Hamltonan H = H H qφ =x,y,z p 2 2m p p qa ( = x, y, z 3 U m Hamltonan (59 Hamltonan F = q (E + ṙ B B = A E = Φ A t 15