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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 : q i (t) (i 1,,N) 1 1 1

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0 {L(q(t)+δq(t), q(t)+δ q(t),t) L(q(t), q(t),t)} dt i ( L δq i (t)+ L ) δ q i (t) dt (4) q i q i δq i (t) δ q i (t) q i(t) q i (t) d dt (q i(t)+δq i (t)) d dt q i(t) d dt δq i(t). (5) 2

(5) (4) t1 ( L δs δq i (t)+ L ) d t 0 q i i q i dt δq i(t) dt t1 { L δq i (t)+ d ( ) ( L d L δq i (t) t 0 q i i dt q i dt q i [ ] t1 L t1 { L δq i (t) d L q i q i dt q i i t 0 + t 0 i ) } δq i (t) dt } δq i (t)dt (6) (3) t1 { L δs d } L δq i (t)dt. (7) q i dt q i t 0 i δq i (t) δs 0 δq i (t) d L L 0 (i 1,,N) (8) dt q i q i 1.2 L(q(t), q(t),t) r (x, y, z) U(r) m m d2 r U(r) (9) dt2 T (ṙ) 1 2 m ( dr dt ) 2 1 2 m ( ẋ 2 +ẏ 2 +ż 2) (10) 3

(11) d T dt ẋ U x, d T dt ẏ U y, d T dt ż U z (11) q 1 x, q 2 y, q 3 z ( ) d T U (i 1, 2, 3) (12) dt q i q i L(r, ṙ) T (ṙ) U(r) (13) (8) (11) T (ṙ) U(r) (13) (8) T (q i, q i ),U(q i ) L(q, q, t) T (q, q, t) U(q, t) (14) q i 2 [ ] r (x, y) xe x + ye y r, φ x r cos φ, y r sin φ r r(cos φ, sin φ) re r e r e φ φ e r ( sin φ, cos φ) e φ (15) r, φ r ( dr dr r + dφ ) (re r )dre r + rdφ φ φ e r dre r + rdφe φ (16) 2 4

e r dr e φ rdφ v dr dt v ṙe r + r φe φ (17) e r e r e φ e φ 1 e r e φ 0 T (r, ṙ, φ) m 2 v2 m 2 (ṙ2 + r 2 φ2 ) (18) U(r, φ) L(r, φ, ṙ, φ) m 2 (ṙ2 + r 2 φ2 ) U(r, φ) (19) L ṙ mṙ, L 2 mr φ r L φ L mr2 φ, φ U φ r, φ m ( r r φ ) 2 d dt U r ( ) mr 2 φ U φ (20) (21) U(r) φ (21) p φ mr 2 φ (22) 5

2 2003 5 8 ( ) 1.3 q i p i L q i (1) (1) q i (t) q i (t) q i(t)+εf i (q, t) (2) I i L(q, q, t) f i (q, t) (3) q i (t) ε δq i (t) εf i (q, t) δl(q, q, t) L(q(t)+δq(t), q(t)+δ q(t),t) L(q(t), q(t),t) ( L δq i (t)+ L ) δ q i (t) q i i q i δ δl(q, q, t) ( L δq i (t)+ L ) d q i i q i dt δq i(t) ( ) d L δq i (t) + ( L d ) L δq i (t) (4) dt q i i q i i dt q i 1

(2) δl 0 (4) di dt 0, I i L q i f i [ 1] n r i, (i 1,,n) L(ṙ i, r i ) n 1 2 m iṙi 2 U(r i r j ) (5) i>j x, y, z r i r i r i + ɛ (6) P n L ṙ i n m i ṙ i (7) [ 2] L(r, ṙ) 1 2 mṙ2 U(r) (8) U(r) r r n ε r ε εn L ṙ δr ε r (9) (ε r) ε ( r L ) ṙ ε l dl dt 0, l r L ṙ r p (10) 2

[ 3] t ε t 0 q i t t t ε (11) q i(t )q i (t) (12) q i (t) q i (t) q i(t + ε) q i (t)+δq i (t), δq i (t) ε q i (t). (13) δ L(t) L (t) L(t + ε) L(t)+δL(t), δl(t) ε dl(t). (14) dt (13) (4) δl(q, q) L(q (t), q (t)) L(q(t), q(t)) ( ) d L δq i (t) dt q i i (15) (14)(15) de dt 0, E i L q i q i L (16) 1 L(r, ṙ) 1 2 mṙ2 U(r) (17) E 1 2 mṙ2 + U(r) (18) (16) 1 E 3

3 2003 5 15 ( ) 1.4 q i, (i 1,,N) q 1,,q N k f l (q 1,,q N,t)0, (l 1,,k) (1) (7) N δq i (8) (eqn:constraints) Lagrange Multiplier Method [ ] l m r φ (??) r l U(r, φ) mgl(1 cos φ) L(φ, φ) m 2 l2 φ2 mgl(1 cos φ). (2) g l [ ] δq i k A li δq i (t) 0, ( 1 A li f ) l. (3) q i

t λ l (t) t l t1 t 0 k λ l (t)a li (q, t)δq i (t)dt 0. (4) l1 (7) { } t1 L d L k + λ l (t)a li (q, t) δq i (t)dt 0 (5) q i dt q i t 0 l1 δq i (t) (i 1,,N) N k λ l (t) (l 1,,k) i 1,,k L d L + q i dt q i k λ l (t)a li (q, t) 0, (i 1,,k) (6) l1 λ l (t) (l 1,,k) λ l (t) (l 1,,k) (5) { } t1 L d L k + λ l (t)a li (q, t) δq i (t)dt 0 (7) q i dt q i t 0 ik+1 l1 N k δq k+1,,δq N δq i, (i k +1,,N) L d L + q i dt q i k λ l (t)a li (q, t) 0, (i k +1,,N) (8) l1 i 1,,N N d L L dt q i q i k λ l (t)a li (q, t), (i 1,,N) (9) l1 k (1) f l (q 1,,q N,t)0, (l 1,,k) N q i (t) (i 1,,N) k λ l (t) (l 1,,k) (9) k λ l (t)a li (q, t) (10) l1 2

(3) (1) L L + k λ l f l (q, t) (11) l1 N q i (t) (i 1,,N) k λ l (t) (l 1,,k) q i (9) (1) 3

4 2003 5 22 ( ) 2 2.1 Lagrange N N q i (t) (i 1 N) Hamilton q i (t) p i (t) 2N Lagrangian L(q, q, t) q i q i q i p i (Hamiltonian) H(q, p, t) p i q i L(q, q, t) (1) q i p i 2N q i p i q i p i L q i (2) q i Lagrange dl(q, q, t) ( L dq i + L ) d q i + L q i i q i t dt (ṗ i dq i + p i d q i )+ L t dt i dh(q, p, t) i dh(q, p, t) ( dp i q i + p i d q i L dq i L ) d q i L q i q i t dt ( q i dp i ṗ i dq i ) L dt (3) t ( H dq i + H ) dp i + q i p i H(q, p, t) dt (4) t 1

dq i dt dp i dt dh(q, p, t) dt i H(q, p, t), p i H(q, p, t) q i (5) ( q i ṗ i ṗ i q i ) L(q, q,t) t L(q, q,t) t 1 q i (t), p i (t) 2N (q 1,,q N,p 1, p N ) (phase space) N (q 1,,q N ) (configuration space) q i (i 1,,N) q i,p i, (i 1,,N) [ ] (6) L(q, q) 1 2 m q2 1 2 mω2 q 2 (7) p (1) L(q, q) q m q (8) H(q, p) p q L(q, q) p2 2m + 1 2 mω2 q 2 (9) (8) q (5) q(t) p m, (10) ṗ(t) mω 2 q (11) p(t) q(t) m q mω 2 q (12) q(0) q o,p(0) p 0 q(t) q 0 cos (ωt)+ p 0 sin ωt ω (13) p(t) q 0 ω sin (ωt)+p 0 cos (ωt). (14) 1 q i (t), p i (t) L 2

H(q, p) p2 2m + 1 2 mω2 q 2 E ( ). (15) ((q, p) ) q ( 2 2E ) + p2 1. (16) 2mE mω 2 2E q p 2mE mω 2 m 1 ω 1 H(q, p) 1 2 p2 + 1 2 q2 (q, p) [ q ṗ ] [ H(q,p) p H(q,p) q ] [ p q ] (17) p 2 1 q -1 0 1 2-1 -2 t x t + t [ ] [ x q q x + x + p ṗ [ q p ] ] t ( ) x t 0.1 x (q, p) t 0 (1, 0) (0, 1.6) t 0 (q, p) (0, 0) [ ] ( ) V (q) 1 2 mκ2 q 2 m 1 κ 1 H(q, p) 1 2 p2 1 2 q2 (18) 3

[ q ṗ ] [ H(q,p) p H(q,p) q ] [ p q ] (19) V(q) -2-1 1 2 0 q -0.5-1 -1.5-2 ( ) ( q, p) ( q, ṗ) t t 0.1 ( q, p) (q, p) (19) p q q q (20) A, B q(t) Ae t + Be t, p(t) Ae t Be t (21) q(0) q 0, p(0) p 0 q(t) q 0 2 (et +e t )+ p 0 2 (et e t ), (22) p(t) q 0 2 (et e t )+ p 0 2 (et +e t ) (23) q 0,p 0 (18) E 1 2 p2 0 1 2 q2 0 (24) a t 0 (2, 2) E a 0 (22) q(t) p(t) 2e t 4

b t 0 (2, 1.5) E b < 0 p 0 p>0 c t 0 (1.5, 2) E c > 0 q p 0 a (4) (1) (2) d t 0 (1, 0) (22) q(t) cosh t, p(t) sinh t q p e t 0 (0, 1) q(t) sinh t, p(t) cosh t q p 2 p e (4) 1 (1) d -2-1 0 1 2 q (2) -1 a b c -2 5

5 2003 6 4 ( ) 2.2 (q 1,q 2,,q N ) (q 1,,q N,p 1,,p N ) 2N x x 1 x 2N q 1 : q N p 1 : p N [ [ ] dx dt J H, H(q,p) p H(q,p) q q p ], H H x 1 H x N J [ H q 1 : H q N H p 1 : H p N 0 1 N 1 N 0 ] [ H q H p 1 N N N x (q, p) x + x H(x) H(x + x) H(x) x H(x) x H(x) 0 x H(x) H(q, p) H ] (1) J H (1) v dx dt v H(x) ( H H v H(x) J H H H ) H 0 p i q i q i p i 1

v H H 1 Liouville t D x t t + t D x x + ẋ t D D dq 1 dq N dp 1 dp N dq 1 dq N dp 1 dp N (2) D D v(x) J H div v v ( H ) H 0 (3) q i p i p i q i D D Jacobian (x 1,x 2,,x 2N ) (x 1,x 2,,x 2N ) 1+ v t + O( t)2 (4) t 1 t t t dq 1 dq N dp 1 dp N D D D (x 1,x 2,,x 2N ) (x 1,x 2,,x 2N ) dq 1 dq N dp 1 dp N dq 1 dq N dp 1 dp N D 2.3 1 2

S t1 t 0 ( p i q i H(q, p, t))dt (5) 2N C C q, p C S t 0 t t 1 C q i (t), p i (t) q i (t) q i (t) q i(t)+δq i (t) p i (t) p i (t) p i(t)+δp i (t) (6) C C C δs S[C ] S[C] t1 p i q i H(q,p,t))dt t 0 ( t1 t1 t 0 ( p i q i H(q, p, t))dt δ( p i q i H(q, p, t))dt t 0 0 (7) 2 δq i (t 0 )δq i (t 1 )δp i (t 0 )δp i (t 1 )0. (8) (7) q, p (6) δs t1 t 0 ( ) H(q, p, t) H(q, p, t) δp i q i + p i δ q i δq i δp i dt (9) q i p i (6) (9) δ q i (t) q i (t) q i(t) d dt δq i(t) δs (p i (t 1 )δq i (t 1 ) p i (t 0 )δq i (t 0 )) t1 + t 0 {δp i ( q i ) ( H(q, p, t) δq i ṗ i + p i )} H(q, p, t) dt. (10) q i 2 t t 0 t t 1 3

(8) q i,p i S δq i,δp i q i H(q, p, t) p i ṗ i H(q, p, t) q i (i 1, 2,,N). 4

6 2003 6 11 ( ) 4 q i Q i f i (q 1,q 2,,q N,t) (i 1, 2,,N) (1) q i p i (q i,p i ) q i H(q, p, t) H(q, p, t), ṗ i p i q i Q i (q, p, t), P i (q, p, t) Q i K(Q, P, t), K(Q, P, t) P i P i Q i (q, p) (Q, P ) Q i Q i (q, p, t) (2) P i P i (q, p, t) (q, p) (Q, P ) (q, p) (Q, P ) p i q i H(q, p, t) P i Q i K(Q, P, t)+ df dt (3) { t1 N } p i q i H(q, p, t) dt t 0 t1 t 0 { N } P i Q i K(Q, P, t) dt + F (t 1 ) F (t 0 ) F (t 1 ) t 1, q i (t 1 ), p i (t 1 ) F (t 0 ) t 0, q i (t 0 ), p i (t 0 ) { t1 N } { t1 N } δ p i q i H(q, p, t) dt δ P i Q i K(Q, P, t) dt t 0 t 0 (q, p) (Q, P ) 1

(2) (p i dq i P i dq i ) (H(q, p, t) K(Q, P, t))dt df (4) F q i,q i,t F F 1 (q i,q i,t) df (q, Q, t) ( F1 dq i + F ) 1 dq i + F 1 q i Q i t dt (4) dq i,dq i,dt p i F 1(q, Q, t), P i F 1(q, Q, t) (5) q i Q i K(Q, P, t) H(q, p, t)+ F 1(q, Q, t) t 1 F 1 (q, Q) i q iq i q i p i q, Q (5) p i F 1(q, Q, t) Q i, P i F 1(q, Q, t) q i q i Q i Q i p i, P i q i q i p i 1 q i q Q i f i (q 1,,q N,t) q p q i,p i F P i Q i + F 2 (q i,p i,t) (6) 2 df (dp i Q i + P i dq i )+ ( F2 dq i + F ) 2 dp i + F 2 q i P i t dt (4) dq i dq i,dp i,dt p i F 2(q, P, t), Q i F 2(q, P, t) (7) q i P i K(Q, P, t) H(q, p, t)+ F 2(q, P, t) t F 1 F 2 1 2 Q i P i Legendre 2

2 F 2 (q, P) i q ip i q, P (7) p i F 2(q, P, t) P i, Q i F 2(q, P, t) q i q i P i 3 F 2 (q, P, t) N f i(q 1,,q N,t)P i Q i F 2(q, P, t) f i (q 1,,q N,t) (8) P i p i F 2(q, P, t) f j Q j P j P j (9) q i q i q i (9) P i Q j q i P i j1 q i Q k δ jk j1 j1 q j Q i p j (10) (8) p i q i [ ] 1. p i,q i F i p iq i + F 3 (p i,q i,t) q i F 3(p, Q, t), P i F 3(p, Q, t) (11) p i Q i K(Q, P, t) H(q, p, t)+ F 3(p, Q, t) (12) t 2. p i,p i F i (p iq i P i Q i )+F 4 (p i,p i,t) q i F 4(p, P, t), Q i F 4(p, P, t) (13) p i P i K(Q, P, t) H(q, p, t)+ F 4(p, P, t) (14) t 3. m 1 m 2 U(r 1 r 2 ) L m 1 2 ṙ2 1 + m 2 2 ṙ2 2 U(r 1 r 2 ) (15) V r r r V t (16) 3

4.1 ɛ F 2 i q i P i + ɛg(q,p, t) (17) F 2 p i F 2(q, P, t) G(q, P, t) P i + ɛ q i q i Q i F 2(q, P, t) P i q i + ɛ G(q, P, t) P i q i Q i q i, p i P i p i ɛ G(q, p, t) G(q, p, t) q i ɛ, p i ɛ (18) p i q i P p ɛ G(q, p, t) P p G(q, p, t) x ɛj G (19) 1. Q i q i + ɛ, P i p i G(q, p, t) G(q, p, t) Q i q i + ɛ q i + ɛ p i G(q, p, t) P i p i p i ɛ q i G(q, p, t) i p i P total 2. z (θ ε) z X x cos θ y sin θ x ɛy x + ɛ G p x Y x sin θ + y cos θ y + ɛx y + ɛ G p y G(x, y, p x,p y,t) G xp y yp x L z z (20) x, y x, y L x,l y 4

7 2003 6 26 ( ) 5 5.1 F K dt 0 (q 1,,q N,p 1,,p N ) (Q 1,,Q N,P 1,,P N ) Q i Q i (q, p). (1) P i P i (q, p), (2) (p i dq i P i dq i )df. (3) [ ] Q αq + βp, P γq + δp α, β, γ, δ [ ] PdQ pdq (γq + δp)(αdq + βdp) pdq γαqdq + δβpdp + δαpdq + γβqdp pdq d( 1 2 γαq2 + 1 2 δβp2 + γβqp)+(δα γβ 1)pdq δα γβ 1 (3) C (p i dq i P i dq i ) 0 (4) C 1

C (q, p) (1)(2) C (Q, P ) ( Qi (q, p) P i (q, p) dq j + Q ) i(q, p) dp j P i dq i (5) q j p j C i,j1 (q,p) C D C (Q,P) D C Phase Space (1)(2) C p i dq i P i dq i (6) C (u, v) C 0 D 0 ( Av (A u (u, v)du + A v (u, v)dv) C D u A ) u dudv. (7) v C D C (u, v) C (u, v) C 0 C (u, v) D 0 C 2

D (q i,p i ) (u, v) (6) (7) C p i dq i C 0 ( qi p i u du + q ) i v dv (8) A u (u, v) p i q i u, A v(u, v) p i q i v. (9) (7) A v u A u v N ( pi q i u v p ) i q i v u (p i,q i ) p (u, v) i p i u v p i u q i u q i v (p i,q i ) (u, v), (10) q i v p i q i v u. (11) (10) p i dq i C D dp i dq i (12) dp i dq i D D 0 (p i,q i ) dudv (13) (u, v) (1)(2) D D D (Q, P ) (q, p) D (q, p) (u, v) D (Q, P ) (u, v) D D dp i dq i D 0 (P i,q i ) dudv. (14) (u, v) (3) D dp i dq i D dp i dq i (15) D D (1)(2) 3

5.2 (15) (13)(14) (p i,q i ) (u, v) dudv D 0 D 0 (P i,q i ) (u, v) dudv. u v {u, v} q,p ( qi p i u v p i u ) q i v (15) (16) {u, v} q,p {u, v} Q,P (17) q, p u, v q i p j [ ] {q i,p j } q,p {q i,q j } 0, {p i,p j } 0, {q i,p j } δ ij (18) k1 ( qk p k p ) k q k q i p j q i p j δ ik δ jk δ ij k1 2N x 1 q 1 X 1 Q 1 : : [ ] x, X 0 1 N, J 1 N 0 x 2N q N p 1 : p N X 2N 1 N N N x T, X T ( qi p i {u, v} q,p u v p ) i q i x T J x (19) u v u v ( Qi P i {u, v} Q,P u v P ) i Q i X T J X (20) u v u v 4 Q N P 1 : P N

x i X i X i (x 1,,x 2N ) (21) X i u j X i x j x j u j M ij x j u M M ij X i x j (22) X v M x v, T X u x T M T. (23) u (20) (19) M (21) (21) M T JM J. (24) M det J (det M) 2 1 det M ±1 (Liouville ) dq 1 dq N dp 1 dp N ±dq 1 dq N dp 1 dp N. det M 1 (24) M 1 M T J JM 1 5

J J 2 1 (25) JM T M 1 J M MJM T J (26) 5.3 q k,p k (k 1, 2,,N) (Poisson Bracket) ( u v [u, v] v ) u q k p k q k p k k1 (27) [ ] x 1 q 1 X 1 Q 1 : : q N, p 1 Q N P 1 : : (28) x 2N p N X 2N P N [u, v] q,p k1 ( u v v ) u q k p k q k p k 2 i,j1 u v J ij ( u) T J ( v) (29) x i x j i v v x i j v X j X j x i j M ji v X j (30) v M T v (31) ( v) T ( v) T M (32) 6

(31) [u, v] q,p ( u) T J ( v) ( u) T MJM T ( v) ( u) T J ( v) [u, v] Q,P (33) M (26) [q i,p j ] [q i,q j ] [p i,p j ] ( qi p j p ) j q i δ ik δ jk δ ij (34) q k p k q k p k k1 ( qi q j q ) j q i 0 (35) q k p k q k p k ( pi p j p ) j p i 0 (36) q k p k q k p k k1 k1 k1 [q i,q j ] 0 [p i,p j ] 0 (37) [q i,p j ] δ ij [u, v] [v,u] (38) [αu + βv,w]α[u, w]+β[v, w] (α, β ) (39) [uv, w] u[v, w]+[u, w]v (40) [q i,f(q, p)] F, [p i,g(q, p)] G p i q i (41) [u, [v, w]] + [v, [w, u]] + [w, [u, v]] 0 (Jacobi ) (42) (38)(39)(41) (40) (uv) q k u q k v + u v q k, (uv) p k u p k v + u v p k 7

J ij (31) u x i u, i 2 u x i x j u, ij (43) [v, w] 2 k,l1 [u, [v, w]] + [v, [w, u]] [u, [v, w]] [v, [u, w]] v, k J kl w, l (44) ij ij ij u, i J ij ([v, w]), j v, i J ij ([u, w]), j ij J ij J kl (u, i (v, k w, l ), j v, i (u, k w, l ), j ) kl J ij J kl (u, i v, k v, i u, k )w, lj + ij kl J ij J kl (u, i v, kj v, i u, kj )w, l (45) kl ([v, w]), j kl J kl (v, k w, l ), j kl J kl (v, kj w, l +v, k w, lj ) (45) u, i v, k v, i u, k i k J ij J kl i, k ij kl 1 2 (J ijj kl J kj J il )(u, i v, k v, i u, k )w, lj J ij J kl J kj J il l j w, lj l, j l, j (45) J ij J kl (u, i v, kj v, i u, kj )w, l ij i j J ij J ij J kl (u, i v, kj +v, j u, ki )w, l ij kl ij kl J ij J kl (u, i v, j ), k w, l kl kl J kl [u, v], k w, l [[u, v],w] [w, [u, v]] 8

[ 1] L x yp z zp y,l y zp x xp z,l z xp y yp x Poisson Blacket [x, L z ], [y, L z ], [z, L z ] [p x,l z ], [p y,l z ], [p z,l z ] [L x,l y ], [L y,l z ], [L z,l x ] 5.4 (26) (24) q, p Q, P Q Q(q, p), P P (q, p) MJM T J 2N 2N (i, j) 2 k,l1 X i X j J kl x k l J ij (46) i, j 1 N N +1 2N k k 1,,N k N +1,, 2N ( Qi Q j Q ) i Q j 0 q k p k p k q k k1 ( Qi P j Q ) i P j δ ij (47) q k p k p k q k k1 ( Pi P j P ) i P j 0 q k p k p k q k k1 ( Pi Q j P ) i Q j δ ij q k p k p k q k k1 q, p Q, P Q Q(q, p), P P (q, p) [Q i,p j ] 0, (48) [Q i,p j ] δ ij (49) [P i.p j ] 0 (50) (37) 9

[ 2] (N1) q, p Q, P Q Q(q, p), P P (q, p) MJM T J M [ Q q P q Q p P p ] [ J 0 1 1 0 [Q, Q] 0, [P, P] 0, [Q, P ]1 ] 5.5 dq i dt dp i dt H (51) p i H q i (41) dq i dt dp i dt [q i,h] (52) [p i,h] q p q i,p i F (q, p) (51) df (q, p) ( F dq i dt q i i dt + F ) dp i p i dt ( F H F ) H q i i p i p i q i [F (q, p),h] (52) df (q, p) dt [F (q, p),h] (53) 10

q, p F (q, p) F F F (q, p) G(q, p) [F, G] F (q, p), G(q, p) [F, H] 0, [G, H] 0 (42) [H, [F, G]] [G, [F, H]] [F, [G, H]] 0 [F, G] 5.6 G(q, p, t) (41) G(q, p, t) G(q, p, t) q i ɛ, p i ɛ (54) p i q i q i ɛ [q i,g], p i ɛ [p i,g] (55) q i,p i F (q, p) F (q, p) ( ) F F q i + p i q i i p i ɛ ( G F G ) F p i i q i q i p i ɛ [F, G] (56) G(q, p, t) (56) G (53) H(q, p) ɛ [H, G] (57) dg(q, p) dt [H, G] 0 (58) [G(q, p),h] 0 (59) G [ 3]( ) 11

1. (P total i p i) q i ɛ [q i,p total ] p i ɛ [p i,p total ] 2. n (θ ε) x i ɛ [x i, n L] p i ɛ [p i, n L] [ 4]( ) r x 2 + y 2 + z 2 L i (i x, y, z) p 2 p 2 x + p2 y + p2 z L i (i x, y, z) H 1 2m p2 + V (r) L i (i x, y, z) 5.7 A, B Â, ˆB [A, B] 1 i (Â ˆB ˆBÂ). (60) [q i,q j ]0 1 i (ˆq iˆq j ˆq j ˆq i )0, [q i,p j ]δ ij 1 i (ˆq iˆp j ˆp j ˆq i )δ ij, (61) [p i,p j ]0 1 i (ˆp iˆp j ˆp j ˆp i )0. (53) df (ˆq, ˆp) dt 1 (F (ˆq, ˆp)H(ˆq, ˆp) H(ˆq, ˆp)F (ˆq, ˆp)) (62) i (61) (62) 12

8 2003 7 3 ( ) 6 6.1 t 1 q 1 t 2 q 2 S[C] t2 t 1 C L(q(t), q(t))dt (1) C q 1 t 1 q 2 t 2 S(q 1,t 1,q 2,t 2 ) q 1 q 1 q 1 + q 1, t 1 t 1 t 1 + t 1, (2) q 2 q 2 q 2 + q 2, t 2 t 2 t 2 + t 2. (3) t t 2 C C q q+dq q+dq t 1 q 1 q 2 q 1

C C C t q C q q q + q, t t t + t, (4) q(t) q (t ) q(t) δq(t) q (t) q(t) q(t) q (t ) q(t) q (t + t) q(t) q (t)+ t q(t) q(t) δq(t)+ t q(t) (5) t q q S(q 1,t 1,q 2,t 2 ) t2 + t 2 t 1 + t 1 C L(q (t ), q (t ))dt L(q 2, q 2 ) t 2 L(q 1, q 1 ) t 1 + t2 t 1 C t2 t 1 L(q(t), q(t))dt (L(q, q ) L(q, q)) dt t2 [L t] t 2 t 1 + δl(q, q)dt. (6) t 1 d δq(t) δ q(t) dt t2 ( ) S(q 1,t 1,q 2,t 2 ) [L t] t L(q, q) L(q, q) 2 t 1 + δq i + δ q i dt q i q i [L t] t 2 t 1 + t2 + t 1 t 1 [ i i L q i δq i ] t2 t 1 ( L(q, q) d q i dt i ) L(q, q) δq i dt (7) q i (5) S(q 1,t 1,q 2,t 2 ) [ p i q i ( ] t2 p i q i L) t i i t 1 [ ] t2 p i q i H t i i t 1 p 2 i q 2 i H t 2 i p 1 i q 1 i + H t 2 (8) 2

p 2 i S, H(q 2 qi 2 i,p 2 i ) S t p 1 i S, H(q 1 qi 1 i,p1 i ) S t (9) t 1, q 1 t 2,q 2 i t, q i (9) (10) (11) S(q i,t) p i S q i (10) H(q i,p i ) S(q i,t) t (11) H(q i, S ) S(q i,t) q i t (12) S S(q i,t) S(q i,t) (10) p(q,t) S(q,t) (13) S(q i,t) S(q i,t) 3

6.2 S(Q, t 1,q,t) t q, p t 1 Q, P q, Q (q, p) (Q, P ) p i F 1(q, Q, t), q i (i 1,,N) (14) P i F 1(q, Q, t) Q i (15) K(Q, P, t) H(q, p, t)+ F 1(q, P, t) t (16) K F 1 (q, Q, t) Q, P Q i K 0 P i P i K 0 Q i Q, P (14) (16) (10)(11) S(Q, t 1,q,t) t q, p t 1 Q, P [ 1] [ ] t 1 q 1 t 2 q 2 q q 2 q 1 t 2 t 1 S(q 1,t 1,q 2,t 2 ) t2 t 1 m 2 q2 dt m 2 (q 2 q 1 ) 2 t 2 t 1 (17) S(q 1,t 1,q 2,t 2 ) q 2 m q2 q 1 t 2 t 1 p 2, S(q 1,t 1,q 2,t 2 ) t 2 m 2 (9) ( q2 q 1 t 2 t 1 ) 2 p2 2 2m 4

6.3 Q P F 2 (q, P, t) q, P (q, p) (Q, P ) p i F 2(q, P, t), q i (i 1,,N) (18) Q i F 2(q, P, t) P i (19) K(Q, P, t) H(q, p, t)+ F 2(q, P, t) t (20) K F 2 (q, P, t) Q, P Q i K 0 P i P i K 0 Q i ( ) S(q, t) H q i, + q i S(q, t) t 0 (21) α i (i 1,,N) N S (21) S P i (19) β i S(q, t, α) α i β i (22) q i q i 2N α i,β i 2N q i (t) p i (18) S p i S(q, t, α) q i (23) 5

[ 2] [ ] N 1 (21) S(q, t) W (q) αt. (24) α W (q) q t (24) (24) ( ) 2 1 dw (q) α 2m dq (24) W (q) 2mαq (25) S(q, α, t) 2mαq αt (26) S α α P Q Q p S(q, α, t) α S(q, α, t) q m 2α q t β (27) 2mα (27) q α, β q 2α (t + β), m p 2mα. 6.4 1 2m ( S q )2 + mω2 2 q2 + S t 0 (28) 1 2m S(q, t) W (q) αt (29) ( W(q) q ) 2 + mω2 2 q2 α (30) 6

W (q) W (q) 2m α mω2 2 q2 dq. (31) (29)(31) (22) m 2 1 dq t β. (32) α mω2 2 q2 ( ) 1 mω 2 ω sin 1 2α q t β. q q(t) 2α sin ω(t + β) mω2 α, β p(t) (23) p(t) 2m α mω2 2 q2 2mα cos ω(t + β) (33) α (30) β 6.5 r x ) H 1 2m ( p 2 r + p2 φ r 2 + V (r) (34) S(r, φ, t) W (r, φ) α 1 t (35) W (r, φ) ( ( W(r, ) 2 1 φ) + 1 ( ) ) 2 W(r, φ) + V (r) α 2m r r 2 1 (36) φ 7

φ r (38) 1 2m W (r, φ) W 1 (r)+α 2 φ (37) ( (dw1 (r) dr ) 2 + α2 2 r 2 ) + V (r) α 1 (38) (37) W (r, φ, α 1,α 2 ) dr 2m(α 1 V (r)) α2 2 r + α 2φ 2 (39) (22) W(r, φ, α 1,α 2 ) α 1 W(r, φ, α 1,α 2 ) α 2 mdr t + β 1 (40) 2m(α 1 V (r)) α2 2 r 2 α 2 dr r 2 2m(α 1 V (r)) α2 2 r 2 + φ β 2 (41) r, φ 4 r, p r,φ,p φ p r,p φ (23) p r W(r, φ, α 1,α 2 ) r p φ W(r, φ, α 1,α 2 ) φ α 2 2m(α 1 V (r)) α2 2 r 2 (40) (41) 8

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