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- ゆずさ かりこめ
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3 n A 79 A A A A
4 0 II 3
5 1 1.1 m d2 r dt 2 = F (1.1) 1 ( r, dr ) (1.2) dt 2 (1.1) p (1.1) p mv (1.3) dp dt = F (1.4) v p (r, p) 1 (1.1) 2 4
6 1.1.2 F (r) 1. 2 A B F (r) F dr (1.5) A B A B 2. rot F = 0 (1.6) 3. F (r) U(r) F = grad U(r) (1.7) F (1.1) dr/dt [ d 1 dt 2 m m d2 r dt 2 = U(r) m d2 r dr = ( U(r)) dt 2 dr dt ( ) dr 2 + U(r)] = 0 dt dt E 1 2 m ( dr dt ) 2 + U(r) (1.8) E 3 5
7 1.1.4 L L r p (1.9) (1.9) r L O (1.4) dl dt = d (r p) dt = dr dt p + r dp dt = r F N (1.10) N F r (1.11) r F = 0 dl dt = 0 (1.12) 1.2 A B A B = F r = r U(r) x m d2 x = kx (1.13) dt2 6
8 x(t) = A cos (ω 0 t + θ) ; ω 0 k m (1.14) A θ (1.13 e λt λ λ 2 + ω 2 0 = 0 λ = ±iω 0 (1.15) λ (1.13) x(t) = C 1 e iω 0t + C 2 e iω 0t (1.16) C 1 C 2 C 1 C 2 (1.16) x(t) (1.13) x C 1 C 2 x (1.13) x c (t) Re[x c (t)] Im[x c (t)] C = Ae iθ x c (t) = Ce iωt (1.14) 1.4 (1.16) x(t) C 1 C 2 t x(t) = x (t) 1.5 (1.14) (1.16) C 1 C x c (t) (1.13) Re[x c (t)] Im[x c (t)] (1.13) 1.7 C = Ae iθ x c (t) = Ce iωt (1.14) 7
9 0 2i ω = ω = 0 C 10 C 5 4i 6i 8i 0 0 π/2 θ 10i π ω 0 ω 0 1.5ω 0 2ω 0 ω 1.1: ω C C θ ω ω 0 γ =1, 0.5, 0.2, 0.1 mω 0 ω 1 0 F/(ω2 0m) m (1.13) m d2 x dt 2 dx = kx γ + F cos ωt (1.17) dt (1.17) m d2 x dt 2 + γ dx dt + kx = F eiωt (1.18) (1.17) (1.18) (1.18) x(t) = Ce iωt (1.19) (1.18) F C = m ( ) ω 2 + iγω/m + ω0 2 = F ω0 2 ω 2 iγω/m ( m ω 2 0 ω 2) 2 ( ) 2 (1.20) + γω/m ω 0 (1.14) C C θ C = F m 1 (ω 2 0 ω 2) 2 ( ) (1.21) 2 + γω/m θ = Arg C atan2 ( γω/m, ω 2 0 ω 2) (1.22) 8
10 (1.17) (1.19) x(t) = Re [ Ce iωt] = C cos(ωt + θ) (1.23) C C θ 1.1 C ω γ γ ω ω 0 ω > 0 C θ < ω ω 0 θ π 1.9 (1.18) (1.19) i m i r i i F i j i F ij i m i d 2 r i dt 2 = j i F ij + F i (1.24) d 2 r i m i dt = ( ) F 2 ij + F i (1.25) i i i j i F ji = F ij (1.26) P T F T (1.25) dp T = F T (1.27) dt 4 θ π < θ π 9
11 i L i r i p i ( ) dl i = r i F ij + F i dt j i L T i L i dl T dt = i r i F ij + i j i r i F i (1.28) F ij ( r i r j ) (1.29) dl T dt = i r i F i = N T (1.30) N T 1.11 (1.26) (1.25) 1.12 (1.29) (1.28) R R 1 m i r i ; M i M i m i (1.31) r i r i r i R (1.32) P T = i m i dr i dt = M dr dt (1.33) M 10
12 L T L T = r i p i = ( ri + R ) ( d ri m i dt + dr ) dt i i = R P T + ( ) d r i r i m i L CM + L (1.34) dt i L CM L K K = i 1 2 m i = 1 2 M ( dr dt ( ) 2 dri dt ) 2 + i 1 2 m i ( ) 2 d ri K CM + K (1.35) dt K CM K 1.13 (1.34) 1.14 (1.35) 1.4 (1.27) (1.30) (1.33), (1.34), (1.35) 11
13 1.2: dp T dt = 0, dl T dt = 0 F T = 0, N T = O P O P 1.2 PP O O O O P P PP = PP + P P = OO + P P P P O P O O P v P ω v P = ω OP 12
14 O v P = V O + ω OP (1.36) 1.15 O ω O O ω O (1.36) P v P ω O = ω O V O = V O + ω O OO OP = OO + O P 13
15 2 2.1 f(x) x f (x) = 0 (2.1) x (2.1) x x x + δx f f(x + δx) f(x) + f (x)δx + (2.2) x δx δx f(x + δx) f(x) f (x)δx δf(x) (2.3) δf f f (2.1) f(x) x x f x δf(x) = 0 (2.4) 14
16 r i i F i N {δr i } {F i } δw δw F i δr i = 0 (2.5) i {δr i } (2.5) δw {δr i } 1 F i = 0; (i = 1, 2,, N) (2.6) (2.6) (2.5) (2.6) {δr i } δw (2.5) (2.5) δw {δr i } 2 15
17 (i) (ii) () {δr i } {δ r i } 3 (2.5) {δr i } δw {δ r i } δ W δ W i F i δ r i = 0 (2.7) i F i S i K i F i = S i + K i (2.8) (2.7) δ W = i ( Si + K i ) δ r i (2.9) S i δ r i = 0 (2.10) i δ W = i K i δ r i = 0 (2.11) 3 δ r δ W δr δw {δ r i } {δ r i } 16
18 {δ r i } δ W (2.10) (2.11) {S i } {K i } {δ r i } (2.10) {S i } {δ r i } (2.11) {K i } {δ r i } {δ r i } {δ r i } δ W {δ r i } (2.11) {K i } S i = K i (2.11) (2.10) F i = K i + S i = 0 {δ r i } {K i } δ W = i K i δ r i = 0 (2.12) {K i } 17
19 {K i } 4 K i = i U(r 1, r 2,, r N ) δ W = i δ r i i U = δ U (2.13) 2.1 δr A δr = 0 A = {r i (t)} r(t) t {F i ( r(t), t ) } 5 {r i (t)} t δr i (t) ( ) ( ) d 2 r i (t) F i r(t), t mi δr dt 2 i (t) = 0 (2.14) i F i (t) m d2 r i (t) dt 2 = 0 (2.15) (2.15) (2.14) t (2.14) (2.15) ( 4 i i,, x i y i z i 5 r (r 1, r 2,, r N ) ) 18
20 δ r i (t) i F i S i K i (2.14) ( ) S i (t) + K i (t) m d2 r i (t) δ r dt 2 i (t) = 0 (2.16) i (2.10) ( ) K i (t) m d2 r i (t) δ r dt 2 i (t) = 0 (2.17) i 19
21 3 3.1 t 2 {r i ( )} {r i (t 2 )} {r i (t)} I t2 ( ) T U dt (3.1) T U {r i } { r i } L ( r, ṙ, t ) T ( ṙ ) U ( r, t ) (3.2) I I r(t) t [, t 2 ] {r i (t)} < t < t 2 {δr i (t)} ( t2 ( ) ) δi = δ T U dt = 0 (3.3) 20
22 δr i ( ) = δr i (t 2 ) = 0 (3.4) t 2 {r i (t)} t (, t 2 ) δr i (t) (2.14) t2 ( ) ( ) d 2 r i (t) F i r(t), t mi δr dt 2 i (t)dt = 0. (3.5) F i U i F i = i U(r 1, r 2,, t) (3.6) 1 (3.5) t2 F i δr i (t)dt = i t2 t2 = = δ δr i (t) i Udt i t ( 1 t2 δu ( r 1 (t), r 2 (t),, t ) dt U ( r 1 (t), r 2 (t),, t ) ) dt (3.7) δu r i (t) r i (t) + δr i (t) U (3.5) t2 ( i [ ) m i r i δr i (t) dt = i ( t2 = δ i m i ṙ i δr i ] t2 ) 1 2 m iṙ i ṙ i dt δ t2 + t 1 ( t2 m i ṙ i δṙ i dt i ) T dt (3.8) (3.4) T T i 1 2 m iṙ i ṙ i (3.9) 1 U 21
23 (3.7) (3.8) (3.5) ( t2 ( δ T ( ṙ(t) ) U ( r(t), t )) ) dt = 0 (3.10) 2 {r i (t)} (3.1) δr i (t) (3.5) (2.15) (2.14) 3.1 f(t) δx(t) t2 f(t)δx(t)dt = 0 f(t) = 0 ( t t 2 ) f(t) f(t) = (3.1) I r(t) I x ẋ t L(x, ẋ, t) t [, t 2 ] x(t) I[x] = t2 L ( x(t), ẋ(t), t ) dt (3.11) I[x] t x x(t) t t 2 x(t) 3 x( ) = x 1, x(t 2 ) = x 2 (3.12) I x(t) 2 (r 1, r 2, ) r 3 t x ẋ L t [, t 2 ] x(t) x(t) ẋ(t) 22
24 x(t) x(t) + δx(t) I δi δx(t) δi x(t) x(t) δx(t) (3.12) δx( ) = δx(t 2 ) = 0 (3.13) I δi δx(t) δi = I[x + δx] I[x] = = = t2 t2 [ L L δx(t) + x [ L ẋ δx(t) ] t2 t2 + ẋ δẋ(t) [ ] L(x + δx, ẋ + δẋ, t) L(x, ẋ, t) dt ] dt ( L x d dt ) L δx(t) dt (3.14) ẋ 1 (3.13) δx(t) δi = 0 d L dt ẋ L x = 0 (3.15) x(t) (3.15) (3.14) (3.11) I[x] (3.15) L (3.2) 3.3 L L(x, ẋ, t) = 1 2 m ẋ2 U(x) (3.15) A (3.11) I t2 t2 ( ) L L L δi = δl(x, ẋ, t) dt = δx + δẋ + x ẋ t δt dt 23
25 4 4.1 (1.1) (3.1) (3.10) (3.15) q 1 = q 1 (x, y, z, t), q 2 = q 2 (x, y, z, t), q 3 = q 3 (x, y, z, t) (4.1) L(q, q, t) = T (q, q, t) U(q, t) (4.2) 1 δi = 0 (3.15) d L L = 0 (4.3) dt q i q i (4.1) q i Q i U q i (4.4) δq i δw δw = Q i δq i (4.5) i (2.5) δw = j F j δr j = i j F j r j q i δq i F j r j = q i j j r j q i j U = U q i = Q i (4.5) (4.1) t 1 T ṙ r T q t 24
26 4.2 (r, θ) (x, y) x = r cos θ, y = r sin θ ẋ = ṙ cos θ r θ sin θ, ẏ = ṙ sin θ + r θ cos θ L = 1 2 m(ẋ2 + ẏ 2 ) U(x, y) L = 1 2 m (ṙ 2 + ( r θ ) 2 ) U(r, θ) U(r, θ) (r, θ) d L dt ṙ L r = 0 m r = mr θ 2 U r d L L dt θ θ = 0 d ( ) mr 2 θ = U dt θ (4.6) (4.7) (4.6) (4.7) mr 2 θ r (4.7) mr 2 θ = l (4.8) 1 (4.6) m r = ( ) 1 l 2 r 2 mr + U(r) 2 (4.9) 4.1 (4.7) mr 2 θ 4.2 (r, θ, ϕ) 25
27 4.3 (x, y) θ (X, Y ) (X, Y ) ω θ = ωt x y (e x, e y ) X Y (e X, e Y ) ( ) ( ) ( ) ( ) e X cos θ, sin θ e x = e Y sin θ, cos θ ˆR(θ) e x (4.10) ˆR(θ) 2 (x, y) (X, Y ) r r = xe x + ye y = Xe X + Y e Y (4.10) (e X, e Y ) (e x, e y ) x = X cos ωt Y sin ωt (4.11) y = X sin ωt + Y cos ωt θ = ωt T T = 1 2 m( ẋ 2 + ẏ 2) e y e y = 1 2 m [Ẋ2 + Ẏ 2 + 2ω ( ẊY + Ẏ X) + ω 2( X 2 + Y 2)] = 1 2 m [ (Ẋ ωy ) 2 + (Ẏ + ωx ) 2 ] (4.12) mẍ = 2mωẎ + mω2 X U X mÿ = 2mωẊ + mω2 Y U Y (4.13) (4.14) 4.3 (4.10) ˆR(θ) ˆR(θ) 1 = ˆR( θ) ˆR(θ) (4.11) 4.4 (4.12) (4.13) (4.14) 2 ˆR t = ˆR 1 26
28 4.4 f l (q 1, q 2,, q n, t) = 0; l = 1, 2,, m (4.15) 3 (4.15) f(q 1, q 2,, q n, t) > 0 (4.16) n a k (q, t) dq k + a 0 (q, t)dt = 0 (4.17) k=1 a θ ϕ (x, y) dx a cos θ dϕ = 0, dy a sin θ dϕ = 0 θ, ϕ 4 (4.15) n m n m (4.3) ω L 3 t 4 27
29 J[y] b a G(y, y, x)dx = s (4.18) s I[y] b a L(y, y, x)dx (4.19) y(x) y(x) y(a) = y a, y(b) = y b (4.20) (4.18) y(x) δy(x) 5 y(x) (4.18) y s (x) δy(x) (4.18) δ y s (x) { J[y s ] = s δ J[y J[y s + δ s ] J[y s + δ y s ] J[y s ] = 0. (4.21) y s ] = s I[y s ] δ I[y s ] = δ ( b a ) L(y s, y s, x)dx = 0 (4.22) y s (x) δ (4.18) (4.21) δ y s (x) 5 (4.20) 28
30 λ Ĩ[y] I[y] + λj[y] = b a ( L(y, y, x) + λg(y, y, x) ) dx Ĩ[y] δy(x) [ b δĩ[y] = δ ( L(y, y, x) + λg(y, y, x) ) ] dx = 0 (4.23) a y(x; λ) λ y(x; λ) (4.18) s λ λ λ s y s (x) y s (x) = y ( x; λ s ) (4.24) (4.24) (4.18) (4.22) (4.18) λ s (4.22) y ( x; λ s ) δy(x) (4.23) δ y s (x) δ Ĩ[y s ] = δ I[y s ] + λ δ J[y s ] = 0 (4.25) y(x; λ s ) (4.18) s δ y s (x) (4.21) δ J[y s ] = 0 y(x; λ s ) (4.22) (4.23) y(x) L(y, y, x) + λg(y, y, x) ( d dx y ) ( L(y, y, x) + λg(y, y, x)) = 0 (4.26) y 4.7 J l [y] b a G l (y, y, x)dx = s l ; l = 1, 2,, m 29
31 (±a, 0) l y = y(x) y(x) y(x) L[y] 0 = y(±a) (4.27) a a U[y] = a a 1 + y 2 dx = l (4.28) g y ρ 1 + y 2 dx (4.29) ρ g (4.27) (4.28) (4.29) U y(x) y(x) y(x) + δ y(x) (4.28) δ y U[y] δ U = 0 y(x) gρλ I[y] U[y] + gρλ L[y] δy y(x; λ) y(x; λ) (4.28) λ 4.1: 30
32 J[y] δy(x) (4.26) ( ) d (y + λ)y 1 + y dx 2 = 0 (4.30) 1 + y 2 y(x; λ) = C cosh x + D C λ (4.31) 6 C D cosh C, D λ (4.27) D = 0, 0 = C cosh a C λ (4.28) l = 2C sinh a C C λ a, l 4.8 (4.31) (4.30) (4.17) a k (q, t)dq k + a 0 (q, t)dt = 0 (4.32) k q k (t) q k (t) = dq k /dt a k (q, t) q k (t) + a 0 (q, t) = 0 (4.33) k f(q 1, q 2,, q n, t) = 0 (4.34) f dq 1 + f dq 2 + f dq n + f dt = 0 (4.35) q 1 q 2 q n t 6 II p.18 31
33 (4.32) (4.32) (4.32) n a k (q, t) δ q k (t) = 0 (4.36) k=1 δ q k (t) ( t2 ) δ I[q] = δ L(q, q, t) dt = 0 (4.37) δ q k (t) (4.36) (4.32) dt = 0 dq k δ q k (t) (4.36) (4.37) ( t2 ) t2 n ( L δ L dt = d ) L δ q k (t) dt = 0 (4.38) q k dt q k k=1 δq k (t) (3.15) (4.36) (4.36) λ(t) δq k (t) ( t2 n ) δĩ[q] δi[q] + λ(t) a k (q, t)δq k (t) dt = 0 (4.39) k=1 (4.36) λ(t) t2 n ( L d ) L + λ(t)a k (q, t) δq k (t) dt = 0 (4.40) q k dt q k k=1 7 δ 32
34 δq k (t) q k (t) L d L + λ(t)a k (q, t) = 0 ; k = 1, 2,, n (4.41) q k dt q k t λ(t) q k (t; λ) t λ(t) q k (t; λ) (4.33) (4.34) λ(t) λ s (t) q k (t; λ) q k (t; λ s ) 4.9 (4.39) (4.40) q k (t; λ s ) q k (t; λ s ) δq k (t) (4.39) δ δ ( t2 n ) δ Ĩ[q] = δ I[q] + λ(t) a k (q, t)δ q k (t) dt = 0 (4.42) k=1 δ q k (t) (4.36) a k (q, t) q k (t; λ s ) (4.42) 2 q k (t) (4.37) (4.41) d L L = λ(t)a k (q, t); k = 1, 2,, n (4.43) dt q k q k λ(t)a k (q, t) Q k ; k = 1, 2,, n (4.44) q k 4.10 (4.32) m a lk (q, t)dq k + a l (q, t)dt = 0; l = 1, 2,, m k 4.11 I
35 4.2: l θ L 4.2 (x, y) L = 1 2 m(ẋ2 + ẏ 2 ) + mgx (4.45) ( t2 ) [ t2 ( L δ L dt = d ) ] L δ x i (t) dt = 0 (4.46) x i dt ẋ i i=1,2 x 1 = x, x 2 = y l x 2 + y 2 = l 2 (4.47) δ x i (t) xδ x(t) + yδ y(t) = 0 x, y λ(t) δx i (t) [ t2 ( L d ) ] L + λ(t)x i δx i (t) dt = 0 (4.48) x i dt ẋ i i=1,2 (4.47) λ(t) 34
36 (4.48) L d L + λ(t)x i = 0; i = 1, 2 x i dt ẋ i (4.45) mẍ = mg + λ(t)x mÿ = λ(t)y t λ(t) λ(t) x = l cos θ, y = l sin θ mẍ = mg + λ(t) l cos θ mÿ = λ(t) l sin θ λ T T (t) = λ(t)l 35
37 5 5.1 L d L L = 0 (5.1) dt q i q i L T U (5.2) q i p i (5.1) p i L q i (5.3) dp i dt = L q i (5.4) L q i L/ q i = 0 (5.4) dp i dt = 0 p i = const. (5.5) q i L 1 q i q i α q i + α 1 36
38 1 L = 1 2 mẋ2 U(x) x p x = mẋ 2 L = 1 2 m (ṙ 2 + ( r θ ) 2 ) U(r, θ) r p r = mṙ θ p θ = mr 2 θ 5.1: 2 q i q i q i p i x x x p x θ 5.1 x p x θ ϕ z 5.2 L t H i p i q i L (5.6) H 3 dh dt = 0. (5.7) dh dt = i = i = i ( ṗ i q i + p i q i L q i L ) q i q i q i ( ( ) d L ṗ i q i + p i q i q i L ) q i dt q i q i ) (ṗ i q i + p i q i ṗ i q i p i q i = p (5.3) q q p (5.6) q 7 37
39 (5.6) H U q i T q i H = T + U (5.8) T q i a ij T = 1 a ij q i q j ; a ij = a ji (5.9) 2 i,j a ij q U q i p i = L = ( ) T U = a ij q j (5.10) q i q i j p i q i = i i,j (5.6) a ij q i q j = 2T H = 2T (T U) = T + U. 5.2 f(x) 5x x 1 x 2 + 3x 2 2 f(x) = 1 a ij x i x j a ij 2 i,j 5.3 (5.9) a ij a ij a ij T = 1 a ij q i q j 2 ij 5.4 (5.10) 1/2 5.5 L (5.3) p i (5.6) H A H
40 5.1 ( ) q i q i (t) q i (t, α) (5.11) L α Q = q i p i (5.12) α α=0 i p i q i q i (t, α) t α α = 0 q i q i (t, α) α=0 = q i (t). (5.13) L α d ( L ( q i (t, α), q i (t, α), t )) = 0 dα i q i α = α i ( L q i q i α + L ) q i = 0 q i α d dt q i(t, α) = d q i (t, α) dt α ( L q i q i α + L q i d dt ) q i = 0 α α 0 L L = d q i q i dt L (α 0) q i [( ) d L qi + L ( d qi )] = 0 dt q i α α=0 q i dt α α=0 i ( ) ( ) d L q i = d q i p i = 0 dt q i α α=0 dt α α=0 i i 39
41 5.2 ( ) I = t2 L(q, q, t)dt (5.14) t 2 t i t i t i + α q i (t) q i ( t) q i (t) = q i ( t α) H H = i L q i q i L I = = = t 2 t 1 t2 +α +α t2 +α +α L ( q( t), q( t), t ) d t L ( q( t α), q( t α), t ) d t L ( q(t α), q(t α), t ) dt α t t I α di = 0 α = 0 dα 0 = di dα = L ( ) ( ) q 2, q 2, t 2 L q1, q 1, α=0 t2 ( L q i + L ) q i dt q i i q i t2 dl t2 = dt dt [( ) d L q i + L ] d q i dt dt q i q i dt = t2 d dt [ L i i L q i q i t 2 ] dt L i L q i q i = H 40
42 5.6 (5.12) 5.7 (x, y) ( x(α), ȳ(α) ) x(α) = x cos α y sin α, ȳ(α) = x sin α + y cos α (x, y) ( x(α), ȳ(α) ) α Q Q 41
43 6 6.1 q q = q 0 U(q) du(q) d 2 U(q) dq = 0, q=q0 dq 2 > 0 q=q0 U(q) = U(q 0 ) + du dq (q q 0 ) + 1 d 2 U q0 2 dq 2 (q q 0 ) 2 + q0 1 2 kx2 + const. (6.1) k d2 U dq 2 > 0 (6.2) q0 x q q 0 (6.3) T = 1 2 a(q) q2 1 2 a(q 0) q 2 = 1 2 mẋ2 (6.4) m a(q 0 ) > 0 m > 0 L = 1 2 mẋ2 1 2 kx2, (6.5) mẍ = kx, ẍ = ω 2 0x (6.6) ω 0 42 k m. (6.7)
44 (6.6) x = A cos ω 0 t + B sin ω 0 t = C cos ( ω 0 t + δ ) = Re [ De iω 0t ] (6.8) A, B, C, δ D 6.1 (6.8) (C, δ) (A, B) (C, δ) D 6.2 (6.6) x = e iωt ω = ±ω 0 (6.6) x = C 1 e iω 0t + C 2 e iω 0t C 1 C 2 x C 1 C 2 (6.8) V (x, t) U(x) x = 0 V (x, t) V (x, t) = V (0, t) + x x + x=0 xf (t) + const. (6.9) V (0, t) const. V (x, t) F (t) x (6.10) x=0 L = 1 2 mẋ2 1 2 kx2 + xf (t) (6.11) mẍ + kx = F (t) (6.12) F (t) = f cos ωt = Re [ fe iωt] (6.13) (6.12) = + (6.14) 43
45 (6.8) (6.12) (6.12) x = Ee iωt ( ) mω 2 + mω0 2 Ee iωt = fe iωt, E = f m ( ω 2 0 ω 2) for ω ω 0 (6.12) [ ] x = Re De iω0t f + m ( ω0 2 ω 2)eiωt (6.15) ω = ω 0 [( ) ] f x = Re D i t e iω 0t (6.16) 2mω 0 2 t (6.12) 6.3 ω = ω 0 (6.16) (6.12) 6.2 q = (q 1, q 2,, q n ) q = q 0 U q i = 0 (6.17) q=q0 q 0 U(q) = U(q 0 ) U 2 q i q j (q i q 0,i )(q j q 0,j ) + q0 = 1 2 C ijx i x j + const. (6.18) 1 x q q 0 (6.19) 1 n n C ij x i x j = C ij x i x j i=1 j=1 44
46 C ij C ij 2 U q i q j (6.20) q0 T = 1 2 B ij(q) q i q j 1 2 B ij(q 0 )ẋ i ẋ j (6.21) L = 1 2 B ijẋ i ẋ j 1 2 C ijx i x j = 1 2ẋt ˆBẋ 1 2 xt Ĉx (6.22) ˆB (B ij ) Ĉ (C ij) B ij = B ji, C ij = C ji (6.23) 6.4 C ij C ij x i C ij x i x j = x t Ĉx > B ij C ij (6.22) x i p i p i = L ẋ i = B ij ẋ j (6.24) B ij ẍ j + C ij x j = 0 (i = 1, 2,, n) (6.25) x j = a j e iωt (6.26) a j ω a j 2 (6.25) ω 2 B ij a j + C ij a j = 0 (i = 1, 2,, n) (6.27) ω 2 ˆBa = Ĉa (6.28) 2 45
47 (6.28) ω 2 a ω 2 a ˆB Ĉ (6.28) ( ) ω 2 ˆB Ĉ a = 0 a = 0 ( ω 2 ˆB Ĉ ) det ω 2 ˆB Ĉ = 0 (6.29) 3 ω 2 (6.29) n n ω 2 n n ω 2 α (α = 1, 2,, n) ω 2 α ω 2 α (6.28) aα ˆB Ĉ (6.28) ωα 2 aα (1) ω 2 α (2) a α a β a α ˆB a β = 0 (6.30) 4 ˆB Ĉ (3) ω 2 α a α (6.30) a α ˆB a β = δ α,β (6.31) Â Â ( a 1, a 2,, a n) A iα a α i (6.32) Â ˆBÂ = Î (6.33) Î 3 a = ( ω 2 ˆB Ĉ ) 1 0 = 0 a = 0 4 a a a t 46
48 6.6 (6.25) (6.26) x j ω x j 6.7  x = 0 x = 0 det  = (1), (2), (3) (6.28) (ω 2 α, a α ) (6.26) x α j (t) = a α j e iω αt x α (t) = a α e iω αt (6.34) C α n x(t) = C α x α (t) = α=1 n C α a α e iωαt = α=1 n a α C α e iω αt α=1 (6.35) Q α (t) C α e iω αt (6.36) (6.35) x(t) = n a α Q α (t) (6.37) α=1 Q α (6.37) (6.32)  x =  Q x j = a α j Q α (6.38) x j Q α Q α (6.22) L = 1 2ẋ ˆB ẋ 1 2 x Ĉ x (6.39) 47
49 T T = 1 2ẋ 1 ) ˆB ẋ = (Â Q ˆB (Â Q) 2 = 1 2 Q Â ˆB Â Q = 1 2 Q Î Q = α 1 2 Q 2 α (6.40) (6.33) Q α U U = 1 2 x Ĉ x = 1 ) ) (Â Q Ĉ (Â Q 2 = 1 2 Q Â Ĉ Â Q (6.41) Â Ĉ Â = Â Ĉ ( a 1, a 2,, a n) = ( Â Ĉ a 1, Ĉ a2,, Ĉ an) a 1 a 2 ( = ω 2 1 ˆB a 1, ω 2 ˆB 2 a 2,, ω 2 ˆB n a n) = a n ω ω ωn 2 ˆΩ 2 d (6.42) U = 1 2 Q ˆΩ2 d Q = α 1 2 ω2 αq 2 α (6.43) Q (6.40) (6.43) L = n α=1 1 ( ) Q 2 α ω 2 2 αq 2 α (6.44) n P α = L Q α = Q α (6.45) Q α = ω 2 αq α (6.46)
50 6.9 ω 2 ω ±ω α (6.34) (6.35) +ω α 6.10 (6.33) (6.42) 49
51 L(q, q, t) = T U (7.1) ( t2 ) δ L(q, q, t)dt = 0 (7.2) d L dt q L q = 0 (7.3) (7.3) q p L q ṗ = L q (7.4) (7.5) L (q, q, t) q (7.4) p (q, p, t) L (5.6) H H p q L(q, q, t) (7.6) L dl = L q H L L dq + d q + q t dt = ṗ dq + p d q + L t dh = p d q + q dp dl = ṗ dq + q dp L t 50 dt (7.7) dt (7.8)
52 (q, p, t) H (7.4) q q = q(q, p, t) (7.9) (7.6) H (q, p, t) H(q, p, t) 1 H(q, p, t) dh = H q (7.8) dq + H p dp + H t dt q = H p, ṗ = H q, (7.10) H t = L t q q p (7.10) (7.4) L t H t H dh dt = H p ṗ + H q = q ṗ ṗ q = 0. q 7.1 (7.8) 7.2 H (7.6) (7.9) H/ q H/ p (7.10) 7.3 A H p = ( ) p q L(q, q, t) = ( ) p q ( ) L(q, q, t) = q 0 p p p 7.4 L = (1/2)mẋ 2 U(x) H 7.5 H L = 1 2 m [ (Ẋ ωy ) 2 + (Ẏ + ωx ) 2 ] U(X, Y ) H H 1 (5.8) H 51
53 7.2 (3.3) (7.10) t [, t 2 ] ( q(t), p(t) ) ( t2 ( ) ) δ p q H(q, p, t) dt = 0 (7.11) q(t) p(t) (7.11) F (q, q, p, t) d F dt q = F q d F dt ṗ = F p ṗ = H q 0 = q H p (7.12) (7.10) (7.2) L (7.6) 2 q(t) ( q(t), p(t) ) q(t) q(t) p L/ q q(t) p(t) q p (7.12) q( ) q(t 2 ) q(t) δq (q( ), p( )) (q(t 2 ), p(t 2 )) 2 52
54 q δq p δp (7.11) ṗ (7.12) q p ( δq(t1 ), δp( ) ) = ( δq(t 2 ), δp(t 2 ) ) = 0 (7.13) 3 (3.1) (7.11) 7.7 (7.11) ṗ q H(q, p, t) (7.10) 7.3 q i p i u(q, p) v(q, p) {u, v} ( u v u ) v (7.14) q i i p i p i q i 3 Classical Mechanics third ed. (Goldstein, Poole, and Safko, 2002 ) 8.5 (p.353) 53
55 {, } dq i = H = {q i, H} (7.15) dt p i dp i dt = H = {p i, H} (7.16) q i (q, p, t) F (q, p, t) df dt = ( F q i + F ) ṗ i + F q i i p i t = ( F H F ) H + F q i i p i p i q i t = {F, H} + F (7.17) t t F {F, H} = 0 (7.18) 7.8 {q i, q j } = {p i, p j } = 0, {q i, p j } = δ i,j 7.9 H = 1 ( ) p 2 2m x + p 2 y + p 2 z + U(r); r = x2 + y 2 + z 2 l r p (7.3) q q = q(q, t) Q = Q(q, t) (7.19) Q Q (7.3) Q 3.1 (7.19) 54
56 (7.3) Q (7.10) (7.10) (7.19) Q = Q(q, p, t), P = P (q, p, t) (7.20) (7.20) (7.10) K(Q, P, t) (7.10) Q = K P, P = K Q (7.21) K(Q, P, t) (7.10) (7.21) (q, p) (Q, P ) (7.10) (7.11) (7.11) (Q, P ) p q H(q, p, t) = P Q K(Q, P, t) (7.22) ( t2 ( ) ) δ P Q K(Q, P, t) dt = 0 (7.23) (7.21) 55
57 (7.22) 4 (7.22) t p q H(q, p, t) = P Q K(Q, P, t) + d F (q, p, Q, P, t) (7.24) dt (Q, P ) (7.21) F (q, p, Q, P, t) ( t2 δ ( P Q K(Q, P, t) + df dt df/dt t2 ) ) dt = 0 (7.25) df [ ] dt dt = t2 F (q, p, Q, P, t) (7.26) (q, p, Q, P ) t = t 2 t (, t 2 ) (7.25) [ ] t2 P δq + δf + ( t2 ( P + K Q ) ( δq + Q K ) ) δp dt = 0 (7.27) P (7.13) (q, p) ( δq(t1 ), δp ( ) ) = ( δq(t 2 ), δp (t 2 ) ) = 0 (7.28) (7.27) [ [ ] t2 P δq + δf = P δq(t) + F F δq(t) + q p δp(t) + F F δq(t) + δp (t) Q P (7.27) (7.21) ] t2 = 0 (q, p) (7.24) (Q, P ) df = p dq P dq + (K H)dt (7.29) F 4n (q, p) (Q, P ) 2n 5 F 4 (7.22) q Q q Q (7.19) 5 (q, p) (Q, P ) n (q i, p i ), (Q i, P i ) (i = 1, 2,, n) 56
58 1. F q, Q t F = W 1 (q, Q, t) 2. F = P Q + W 2 (q, P, t) 3. F = pq + W 3 (p, Q, t) 4. F = pq P Q + W 4 (p, P, t) 1. F = W 1 (q, Q, t) (7.29) F (q, Q, t) F (7.29) F = W 1 (q, Q, t) (7.30) dw 1 (q, Q, t) = p dq P dq + (K H)dt (7.31) p = W 1(q, Q, t) q P = W 1(q, Q, t) Q K = H + W 1(q, Q, t) t (7.32) (q, p) (Q, P ) K F W 1 2. F = P Q + W 2 (q, P, t) F df = P dq Q dp + dw 2 (q, P, t) (7.29) dw 2 (q, P, t) = p dq + Q dp + (K H)dt (7.33) p = W 2(q, P, t) q Q = W 2(q, P, t) P K = H + W 2(q, P, t) t (7.34) 57
59 3. F = pq + W 3 (p, Q, t) F df = p dq + q dp + dw 3 (p, Q, t) (7.29) dw 3 (p, Q, t) = q dp P dq + (K H)dt (7.35) q = W 3(p, Q, t) p P = W 3(p, Q, t) Q K = H + W 3(p, Q, t) t (7.36) 4. F = pq P Q + W 4 (p, P, t) F df = p dq + q dp P dq Q dp + dw 4 (p, P, t) (7.29) dw 4 (p, P, t) = q dp + Q dp + (K H)dt (7.37) q = W 4(p, P, t) p Q = W 4(p, P, t) P K = H + W 4(p, P, t) t (7.38) W i (i = 1 4) 7.10 (7.25) (7.27) (7.27) 7.11 W 1 = qq (Q, P ) 7.12 Q i = f i (q, t) (i = 1,, n) W 2 (q, P ) W 2 (q, P ) (7.20) Q p (7.22) 6 58
60 7.4.2 (7.34) W 2 (q, P ) = qp (7.39) p = W 2 q = P, Q = W 2 P = q, K = H (7.40) Q = q + δq, P = p + δp (7.41) (7.39) W 2 (q, P ) = qp + ϵg(q, P ) (7.42) (7.34) p = P + ϵ G(q, P ) q Q = q + ϵ G(q, P ) P G(q, p) P = p ϵ q G(q, p) Q = q + ϵ p (7.43) ϵ P p G(q, p) (7.43) G G(q, p) δp = ϵ q G(q, p) δq = +ϵ p = ϵ{p, G} = ϵ{q, G} (7.44) G = H H ϵ = t (7.43) P Q = p t H q = q + t H p = p + ṗ t = q + q t (7.45) 59
61 H 7.14 (7.39) W i 7.15 G(q, p) (1) G = p x, (2) G = l z xp y yp x ( z ) 7.16 (7.44) G p δq q δq L 5.3 Q G 7.17 G(q, p) (7.43) g(q, p) δg g(q, P ) g(q, p) δg = ϵ{g, G} g = p q (7.44) 7.18 H G n Ω J n dq 1 dq n dp 1 dp n (7.46) Ω 2n (q, p) j(q, p) 2n j(q, p) = 0 60
62 j(q, p) (q, p) j(q, p) ( q, ṗ) j = i = i ( qi + ṗ ) i q i p i ( H ) H = 0 q i p i p i q i (7.14) (q, p) {u, v} q,p i ( u v u ) v q i p i p i q i (7.47) u, v, w (q, p) 1. {u, v} = {v, u} (7.48) 2. c 1, c 2 {c 1 u + c 2 v, w} = c 1 {u, w} + c 2 {v, w} {w, c 1 u + c 2 v} = c 1 {w, u} + c 2 {w, v} (7.49) 3. {u v, w} = u{v, w} + {u, w}v {w, u v} = u{w, v} + {w, u}v (7.50) {q i, u} = u p i, {p i, u} = u q i (7.51) {u, {v, w}} + {w, {u, v}} + {v, {w, u}} = 0 (7.52)
63 7.5.1 (7.52) F (q, p, t) G(q, p, t) {F, G} (i) F, G t (7.52) {H, {F, G}} + {F, {G, H}} + {G, {H, F }} = 0 F G (7.18) {F, G} {F, H} = {G, H} = 0 {H, {F, G}} = 0 (ii) F, G t {F, G} (7.17) d dt {F, G} = {{F, G}, H} + {F, G} t { } { F = {{G, H}, F } {{H, F }, G} + t, G + F, G } t { = F, {G, H} + G } { + {F, H} + F } t t, G { = F, dg } { } df + dt dt, G = (7.47) {q i, q j } q,p = {p i, p j } q,p = 0, {q i, p j } q,p = δ i,j (7.53) (q i, p i ) (i = 1, 2,, n) Q i = Q i (q, p), P i = P i (q, p), (i = 1, 2,, n) (7.54) (Q, P ) (Q, P ) {Q i, Q j } q,p = {P i, P j } q,p = 0, {Q i, P j } q,p = δ i,j (7.55) 62
64 n = 1 (7.55) {Q, Q} q,p = {P, P } q,p = 0 (7.47) {Q, P } q,p = 1 (7.56) (Q, P ) W 1 (q, Q) (7.54) p = W 1(q, Q) q ϕ(q, Q), P = W 1(q, Q) Q ψ(q, Q) (7.57) P Q (q, p) P p = ψ Q Q p, P q = ψ q + ψ Q Q q (7.56) {Q, P } q,p = Q q = Q q = ψ Q = Q p P p Q P p q ψ Q Q p Q ( ψ p q + ψ Q ( Q Q q p Q Q p q ψ q ) Q q ) Q ψ p q (7.58) Q = Q(q, p) = Q ( q, ϕ(q, Q) ) 1 = Q Q = Q ϕ p q Q (7.57) ϕ Q = 2 W 1 Q q = 2 W 1 q Q = ψ q (7.59) 1 = Q ϕ p q Q = Q ψ p q q (7.59) (7.60) (7.61) (7.58) (7.56) 63
65 (Q, P ) (7.54) p = W 1(q, Q), P = W 1(q, Q) q Q (7.62) W 1 (q, Q) (7.54) p = ϕ(q, Q), P = ψ(q, Q) (7.63) ϕ Q = ψ q (7.64) ϕ(q, Q)dq ψ(q, Q)dQ = dw 1 (q, Q) W 1 (q, Q) (7.56) (7.58) 1 = {Q, P } q,p = Q ψ (7.65) p q (7.54) (7.63) Q(q, p) = Q(q, ϕ(q, Q)) 1 = Q Q = Q p ϕ Q (7.65) (7.64) (7.66) (q, p) (u, v) (q, p) (Q, P ) {u, v} q,p = {u, v} Q,P (7.67) {u, v} q,p = u q k = u q k = v Q l v u v p k p k q ( k v Q l + v ) P l Q l p k P l p k ( u Q l u Q l q k p k p k q k u ( v p k ) + v P l Q l + v ) P l Q l q k P l q k ( u P l u ) P l q k p k p k q k = v Q l {u, Q l } q,p + v P l {u, P l } q,p. (7.68) 64
66 u = Q s (7.55) {Q s, v} q,p = v Q l {Q s, Q l } q,p + v P l {Q s, P l } q,p u = P s = v P l δ s,l = v P s (7.69) {P s, v} q,p = v Q s (7.70) v u (7.68) {u, v} q,p = + v Q l ( {Q l, u} q,p ) + v P l ( {P l, u} q,p ) = v u + v u Q l P l P l Q l = {v, u} Q,P = {u, v} Q,P {u, v} 65
67 8 8.1 q c L p c q c H ṗ c = H q c = 0 ϕ z p ϕ (Q, P ) K P P i P i = K(P ) = 0 P i = α i (8.1) Q i Q i P i (8.1) Q i = K(P ) P i v i (α) Q i = v i (α)t + β i (8.2) β i α i K = 0 K Q P K = 0 66
68 P i = 0 Q i = 0 P i = α i Q i = β i (8.3) K = (q i, p i ), H(q, p, t) (8.4) W (q, P, t) W 2 (q, P, t) p i = W (q, P, t) q i, Q i = W (q, P, t) P i (8.5) (Q, P ) K = H(q, p, t) + W (q, P, t) t = 0 (8.6) W K (8.6) (8.5) ( H q, W ) q, t + W = 0 (8.7) t W (q, P, t) n (8.7) n q t n + 1 n + 1 (8.7) W W q t (8.7) P 67
69 8.2.1 (8.7) n + 1 α i (i = 1,, n + 1) W (q 1,, q n, α 1,, α n, t) + α n+1 (8.8) α n+1 W P i (q, p) (8.5) P i = α i ; n = 1,, n (8.9) p i = W (q 1,, q n, α 1,, α n, t) q i (8.10) Q i = W = W (q 1,, q n, α 1,, α n, t) = β i P i α i (8.11) (8.11) Q i β i (8.10) (8.11) (q, p) 2n (α, β) t = 0 (q, p) (8.10) (8.11) (8.7) q t H H t H (8.7) ( W t + H q, W ) = 0 (8.12) q q t W (q, t) = S(q) + Θ(t) (8.13) (8.13) (8.12) ( H q, S ) = E (8.14) q Θ = E Θ = Et + (8.15) t 68
70 E S 1 (8.14) E n W n 1 α i (n = 2,, n) W S(q 1,, q n, α 2,, α n ; E) + α 1 (8.16) W = Et + S(q 1,, q n, α 2,, α n ; E) + α 1 (8.17) P i W E P 1 P 1 = E, P i = α i (i = 2,, n) (8.18) (8.5) (8.6) p i = W q i Q i = W P i K = H + W t = S q i (8.19) { Q1 = t + S E = β 1 Q i = S α i = β i (i = 2,, n) (8.20) = 0 E = H(q, p) (8.21) 2n E, α, β (8.21) E (8.20) β 1 (8.20) (i = 2,, n) t q q n (8.12) q n W (q) = S(q 1,, q n 1 ) + Φ(q n ) Et (8.22) t ( ) S S Φ H q 1,, q n 1,,,, = E (8.23) q 1 q n 1 q n 1 S W 69
71 q n Φ q i ( H q 1,, q n 1, S S ),,, L = E (8.24) q 1 q n 1 Φ(q n ) q n = L Φ(q n ) = Lq n + (8.25) L q n p n W (q, α, t) (q, p) H(q, p, t) ( H q, W ) q, t + W = 0 (8.26) t W (q, α, t) p i = W (q, α, t) q i, β i = W (q, α, t) α i (i = 1, 2,, n) (8.27) (q, p) q i = H p i, ṗ i = H q i (i = 1, 2,, n) (8.28) α i (8.26) W (q, α, t) β i n = 1 2 q (8.27) 2 t α β q(t) t 0 = q 2 W q α + 2 W (8.29) t α ( ) 2 1 W 2 W q = q α t α (8.30) (8.26) α W α H(q, p, t) p = W/ q α 2 W α q 2 i H(q, p, t) p + 2 W p= W/ q α t = 0 (8.31) 70
72 (8.30) (8.31) 2 W/ t α ( ) 2 1 W 2 W H(q, p, t) q = q α α q p = H (8.32) p= W/ q p 3 (8.28) p (8.27) t ṗ = q 2 W q W t q (8.33) (8.26) q H q + 2 W H q 2 p + 2 W p= W/ q q t = 0 (8.34) 2 W/ q t ( ṗ = q H ) 2 W p q 2 H q = H q (8.35) (8.32) (8.28) n n = H = 1 2m p2 x (8.7) W t + 1 ( ) 2 W = 0 2m x t W (x, t) = Θ(t) + S(x) Θ(t) + 1 ( 2 S (x)) = 0 2m 3 W/ q = p 71
73 1 t 2 x t x E E 1 ( 2 S (x)) = E 2m S(x) = ± 2mE x + Θ(t) = E Θ(t) = Et + W (x, t) = Et ± 2mE x + E P (8.19) (8.21) p x = W x = ± 2mE Q = W P = W E = t ± m 2E x = β ( ) K = H + W t = H E = 0 H = E E 2 2E ( ) x = ± t + β m y H = 1 2m p2 y + mgy ( ) 2 1 W + mgy + W 2m y t = 0 t W (y, t) = S(y) + Θ(t) 72
74 E ( ) 2 1 S + mgy = E 2m y Θ t = E Θ(t) = Et + const. S(y) = 2 1 2m (E mgy) 3/2 + const. 3 mg W (y, t) = 2 1 2m (E mgy) 3/2 Et + const. 3 mg E P p y = W y = ± 2m ( E mgy ) Q = W P = W 2 E == 1 E mgy t = β g m β Q 1 p y y 1 2m p2 y + mgy = E 2 y = 1 2 g(t + β)2 + E mg x H = 1 ( ) p 2 2m x + p 2 y + mgy W ( ( W ) 2 ( ) ) 2 1 W + + mgy + W = 0 (8.36) 2m x y t H t x x t W = Et + αx + S(y) 73
75 E α (8.36) ( ( ) ) 2 1 S α mgy E = 0 2m y ( ) S y = ± 2m E α2 2m mgy W = Et + αx ± dy 2m ( ) E α2 2m mgy (8.37) E α P 1 = E, P 2 = α β 1 = Q 1 = W P 1 = W E = t ± β 2 = Q 2 = W = W P 2 α = x ± m dy ( ) (8.38) 2m E α2 2m mgy α dy ( ) (8.39) 2m E α2 2m mgy β i y t x y y = h 1 2 g( t + β 1 ) 2 y = h 1 2 g ( m α ( x β2 ) ) 2 h 1 ) (E α2 2m 2m β i α p x H = 1 2m p kq2 = 1 ( p 2 + m 2 ω 2 q 2) ; ω k/m 2m 74
76 [ ( W ) ] m 2 ω 2 q 2 + W 2m q t = 0 W = S(q) + Θ(t) 1 2m [ ( S ) ] 2 + m 2 ω 2 q 2 = E (8.40) q Θ(t) = Et K E K = H + W E = 0 H = E (8.40) S S 2mE q = ± 2mE m 2 ω 2 q 2 S(q) = ± m2 ω 2 q 2 dq p = W q = ± 2mE m 2 ω 2 q 2 1 ) (p 2 + m 2 ω 2 q 2 = E 2m Q = W P = S E t = ± m 2mE m2 ω 2 q 2 dq t = β 2 q = ± 1 2E ω m sin( ω(t + β) ) 1 p = ± 2mE cos ( ω(t + β) ) ± β 4 ( ) H = 1 ( ) p 2 r + p2 θ + V (r) (8.41) 2m r 2 4 β β + π/ω 75
77 HJ ( ( W ) m r r 2 ( W θ ) 2 ) + V (r) + W t = 0 (8.42) θ t W W = Et + lθ + S(r) (8.43) E l HJ 1 2m ( ( S ) ) 2 + l2 + V (r) = E (8.44) r r 2 S(r) S(r) = ± dr 2m ( E V (r) ) l 2 /r 2 (8.45) E l P 1 = E, P 2 = l Q 1 = W P 1 = W E, Q 2 = W = W P 2 θ Q i = β i β 1 = W E, β 2 = W θ β 1 = t ± m dr 2m ( E V (r) ) (8.46) l 2 /r 2 β 2 = θ ± l/r 2 dr 2m ( E V (r) ) (8.47) l 2 /r 2 r t r θ 76
78 : V (r) = k r (8.47) 1 θ β 2 = du (2m/l )( 2 E + ku ) ; u 1 u 2 r ( l 2 = ± cos 1 / ( mk r ) ) l2 E/(mk 2 ) 1 r = mk ( 1 ± ϵ cos ( ) ) θ θ l 2 0 ; ϵ 1 + 2l2 E mk 2 (8.48) θ 0 β 2 ϵ > 1 (E > 0) ϵ = 1 (E = 0) 0 < ϵ < 1 ( mk 2 /2l 2 < E < 0 ) ϵ = 0 ( E = mk 2 /2l 2) 8.4 n n (8.27) 2 t β j 0 = q j 2 W q j α i + 2 W t α i (8.49) δ ji = q l β j (8.50) 2 W q l α i (8.50) A ij q j β i, B ij 2 W q i α j (8.51) Â = ˆB 1 Â ˆB = ˆB Â = Î (8.52) 77
79 q l β i 2 W q l α j = 2 W q i α l q j β l = δ ij (8.53) (8.49) Â ( ) 2 W 0 = q j + 2 W qk q j α i t α i β i = q j δ jk + 2 W q k t α i β i q k = 2 W t α i q k β i (8.54) (8.26) α i q k = q k β i 2 W H + 2 W α i q j p j α i t = 0 (8.55) 2 W α i q j H H = δ kj = H (8.56) p j p j p k (8.53) p (8.27) t ṗ l = q i 2 W + 2 W (8.57) q i q l t q l (8.26) q l H q l + 2 W q l q i H p i + 2 W q l t = 0 (8.58) ( ) 2 W H ṗ l = q i + 2 W H q i q l q l q l q i p i ( = q i H ) 2 W H = H (8.59) p i q i q l q l q l 78
80 A A.1 m r = qe + qṙ B (A.1) E = grad Φ A t B = rot A (A.2) (A.3) Φ A (A.2) B A.1.1 (A.1) L = 1 2 m ṙ2 + q ( ṙ A Φ ) (A.4) (A.4) d L = d dt ẋ j dt (m ẋ j + qa j ) = m ẍ j + q ( L A i = q ẋ i Φ ) x j x j x j ( Aj ẋ i + A ) j x i t 79
81 ( ( Ai mẍ j = q ẋ i A ) ) j Φ A j x j x i x j t (A.5) (A.2) (A.3) (A.1) 3.2 I[r] = t2 L dt (A.6) p = L ṙ = m ṙ + qa mṙ (A.7) H = 1 ( ) 2 p qa + Φ (A.8) 2m A.1 (A.5) (A.1) A.2 H (A.8) A.1.2 Λ { Φ A Φ = Φ Λ t A = A + grad Λ (A.9) (A.2) (A.3) L L = 1 2 m ṙ2 + q ( ṙ A Φ ) ( = L + q ṙ grad Λ + Λ ) t = L + q dλ dt (A.10) (A.10) 80
82 E B dλ/dt t2 t2 ( δ L dt = δ L + q dλ ) t2 dt = δ L dt (A.11) dt (A.10) Φ A L Φ = Φ Γ t A = A + grad Γ L = L + q dγ dt (A.12) (A.10) L Φ A L L = 1 2 m ṙ2 + q ( ṙ A Φ ) (A.13) A.2 F r F r = k v (A.14) k F r U(r) m r = U + F r (A.15) (A.14) R R 1 2 k v2 = 1 2 k ṙ2 F r = R ṙ 81 (A.16) (A.17)
83 (A.15) d L L = R, dt ẋ i x i ẋ i ( r (x1, x 2, x 3 ) ) (A.18) L L = 1 2 m ṙ2 U(r) (A.19) (A.16) R F r W dw = F r dr = F r v dt = kv 2 dt = 2R dt 2R q j = q j (x, t), x i = x i (q, t) (A.20) (A.18) d L L = Q j dt q j q j (A.21) Q j (A.14) Q j = F r,i x i q j = R ẋ i x i q j (A.22) ẋ i = x i q j q j + x i t ẋ i q j = x i q j Q j = R ẋ i ẋ i q j = R q j (A.21) q d L L = R dt q j q j q j (A.23) (A.18) A.3 H = p ṙ L L dh dt = 2R R ṙ 2 82
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1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :
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x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
ver F = i f i 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 i δx
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
) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
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chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
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dynamics-solution2.dvi
1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
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I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
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0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
master.dvi
4 Maxwell- Boltzmann N 1 4.1 T R R 5 R (Heat Reservor) S E R 20 E 4.2 E E R E t = E + E R E R Ω R (E R ) S R (E R ) Ω R (E R ) = exp[s R (E R )/k] E, E E, E E t E E t E exps R (E t E) exp S R (E t E )
Morse ( ) 2014
Morse ( ) 2014 1 1 Morse 1 1.1 Morse................................ 1 1.2 Morse.............................. 7 2 12 2.1....................... 12 2.2.................. 13 2.3 Smale..............................
2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)
D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y
AC Modeling and Control of AC Motors Seiji Kondo, Member 1. q q (1) PM (a) N d q Dept. of E&E, Nagaoka Unive
AC Moeling an Control of AC Motors Seiji Kono, Member 1. (1) PM 33 54 64. 1 11 1(a) N 94 188 163 1 Dept. of E&E, Nagaoka University of Technology 163 1, Kamitomioka-cho, Nagaoka, Niigata 94 188 (a) 巻数
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DVIOUT
A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
x x x 2, A 4 2 Ax.4 A A A A λ λ 4 λ 2 A λe λ λ2 5λ + 6 0,...λ 2, λ 2 3 E 0 E 0 p p Ap λp λ 2 p 4 2 p p 2 p { 4p 2 2p p + 2 p, p 2 λ {
K E N Z OU 2008 8. 4x 2x 2 2 2 x + x 2. x 2 2x 2, 2 2 d 2 x 2 2.2 2 3x 2... d 2 x 2 5 + 6x 0 2 2 d 2 x 2 + P t + P 2tx Qx x x, x 2 2 2 x 2 P 2 tx P tx 2 + Qx x, x 2. d x 4 2 x 2 x x 2.3 x x x 2, A 4 2