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7 2003 6 26 ( ) 5 5.1 F K 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) ( qk p k p ) k q k q i p j q i p j δ ik δ jk δ ij 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 (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 ( 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 ( 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 [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 w, l (44) ij ij ij u, i J ij ([v, w]), j v, i J ij ([u, w]), j ij J ij J (u, i (v, k w, l ), j v, i (u, k w, l ), j ) J ij J (u, i v, k v, i u, k )w, lj + ij J ij J (u, i v, kj v, i u, kj )w, l (45) ([v, w]), j J (v, k w, l ), j J (v, kj w, l +v, k w, lj ) (45) u, i v, k v, i u, k i k J ij J i, k ij 1 2 (J ijj J kj J il )(u, i v, k v, i u, k )w, lj J ij J J kj J il l j w, lj l, j l, j (45) J ij J (u, i v, kj v, i u, kj )w, l ij i j J ij J ij J (u, i v, kj +v, j u, ki )w, l ij ij J ij J (u, i v, j ), k w, l J [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 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 ( Qi P j Q ) i P j δ ij (47) q k p k p k q k ( Pi P j P ) i P j 0 q k p k p k q k ( Pi Q j P ) i Q j δ ij q k p k p k q k 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 dp i H (51) p i H q i (41) dq i dp i [q i,h] (52) [p i,h] q p q i,p i F (q, p) (51) df (q, p) ( F dq i q i i + F ) dp i p i ( F H F ) H q i i p i p i q i [F (q, p),h] (52) df (q, p) [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) [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) 1 (F (ˆq, ˆp)H(ˆq, ˆp) H(ˆq, ˆp)F (ˆq, ˆp)) (62) i (61) (62) 12

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