ADVANCES IN MECHANICS Jan. 25, Newton ( ) ,., Newton. , Euler, d Alembert. Lagrange,, , Hamilton ( )

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1 39 1 Vol. 39 No ADVANCES IN MECHANICS Jan. 25, 2009 *, Noether, Lie,, Lagrange,,.,,, Newton ( ), 3,., Newton d Alembert ( ), Newton, d Alembert Lagrange ( ), Euler, d Alembert,, Lagrange,, 2 Lagran , Hamilton ( ), Hamilton, Hamilton, 3 Hamilton Hertz,, 4. Hamilton, Birkhoff, Hamilton Birkhoff, [1] Santilli II, Hamilton Birkhoff, Birkhoff, Birkhoff Hamilton [2]. Birkhoff , Pauli, Martin Hamilton Hamilton. Hamilton Hamilton,, [3]. Hamilton 5.,, 1.,. [4], Hamilton,,. 1, 1.1 Birkhoff ; 1.2 Birkhoff Hamilton ; 1.3 Birkhoff? : , : ( , ) meifx@bit.edu.cn

2 Newton 3 :,. 3. Lagrange Hamilton Lagrange Hamilton,.,,.,., Newton,,. [5 7],,,, ( ),. Birkhoff,, Birkhoff, Birkhoff, ,, Birkhoff 3. 3,,,?, Newton, Lagrange Hamilton,, Noether [8], Noether. Hamilton. Noether [9]. [10], Noether, Lie,, Lagrange,. 2 Noether. Noether Hamilton. Noether ;, Noether. Noether.,. Noether, Birkhoff.

3 1 : 39 Lagrange Noether E s (L) = 0, s = 1,, n (1) E s = d dt (2) q s L ξ 0 + X (1) (L) + ĠN = 0 (3) X (1) = ξ 0 t + ξ s + q s ( ξs q s ξ0 ) (4) ξ 0, ξ s, Noether I N = Lξ 0 + L (ξ s q s ξ 0 ) + G N = (5) Noether ξ 0, ξ s G N = G N (t, q, q), (5). Noether. Lagrange, Noether, (5) [11,12]., Noether, [13 15], (5). [2,9,16 18]. Birkhoff Noether Noether?., Lagrange, Lagrange [19,20], Hamilton [6,9,11,14,21]. Noether, Noether, Noether t, q, q 2.3 Noether Noether. Noether?, Noether : Noether Noether,.,, Noether, Killing. Noether, [19,20,22 25], [23,26]. 3 Lie., Lie Lie, [27 29] Lutzky Lie [30]. Prince [31] Kepler Lie. [32] Lie. [33 39] Birkhoff Lie., Lie Noether Noether. Lie. Hojman1992 [40], Lagrange Hamilton. Hojman [41 47]. Lagrange (1), Lie q s = F s (t, q, q) (6) ξ s q s ξ0 2 ξ 0 F s = X (1) (F s ) (7), (7) d d dt dt ξ s = F s q k ξ k + F s q k d dt ξ k (8) d dt = t + q s + F s (9) q s Hojman, µ = µ (t, q, q) I H = 1 (µξ s ) + 1 µ q s µ F s + d ln µ = 0 (10) dt. ( µ d ) dt ξ s = (11) Lie (8) (11)., Hojman Noether [48].

4 Hojman, Lagrange Hamilton.,. Noether Lie Hojman. Lie. 3.1 [48]? 3.2 Hojman Noether, Noether., Noether, Noether? 4, Lagrange, Hamilton,,, Birkhoff,. L = L Lagrange (1), Lagrange ) (t, q, dq dt = L (t, q, q)+εx (1) (L)+O ( ε 2) (12) (12) (1), ε 2, G F } E s {X (1) (L) = 0 (13) = G F (t, q, q) X (1) (L) d dt ξ 0 + X { } (1) X(1) (L) + d dt G F = 0 (14) X (1) = ξ 0 t + ξ s + q s I F = X (1) (L) ξ 0 + X (1) (L) ( ) d dt ξ d s q s dt ξ 0 (15) (ξ s q s ξ 0 )+G F = (16),, (16)., 2000 [49]., Lie, Noether, [50 78]. Mei, [50 57,59 63,68,74,76,77].. (13), (14) (16). Noether Lie (16). Noe- ther Noether, Lie Hojman [78] (16) Noether, Noether. Noether, Noether? 4.2 [19,20] Noether? 3, 2, NS Noether, LS Lie, FI. 5 Lagrange 2 NS, LS, FI [2], Lagrange L = 1 2 ( q 2 q 2) (17) L 1 = 1 6 q3 cos t q q2 sin t q 2 q cos t (18) L 2 = 2 q q arctan q q ln ( q 2 + q 2) (19) q + q = 0 (20) (17) d L dt q L q = ( q + q) SA (21)

5 1 : 41 d L dt q L q = [I (t, q, q) ( q + q) SA ] SA (22) I I 1 = q cos t + q sin t = c 1 (23) I 2 = ( q 2 + q 2) 1 = c2 (24), (18) (17), (19) (17) Lagrange, (23), (24) Currie Saletan Lagrange, [79] Hojman Harleston [80]. [9] Lagrange,. [81] Lagrange. Birkhoff, Birkhoff [82]. Lagrange Noether, Lie,, Lagrange,., Birkhoff, Birkhoff Birkhoff,. 5.1 Lagrange. Lagrange Lagrange, Lagrange? 5.2 Birkhoff,? 5.3 Lagrange Noether, Noether, Noether, Noether? 6 [18] Birkhoff, Lie.. Birkhoff Ω µν ȧ ν B a µ R µ = 0 (25) t Ω µν = R ν a µ R µ a ν (26) Birkhoff, B = B (t, a) Birkhoff, R µ (t, a) Birkhoff. t = t + εξ 0 (t), a µ = a µ + εξ µ (t, a) (27) F µ = Ω µν ȧ ν B a µ R µ t (28) X (1) F µ = δ ν µf ν, µ, ν = 1,, 2n (29), [18] det ( δ ν µ) 0 (30) (27) Birkhoff Lie, δµ ρ = (SΩ µν ) Ω νρ ξ ν + Ω µν a l Ω lρ δµ ρ ξ 0 t µ, ν, l, ρ = 1,, 2n (31) S = (ξ µ ȧ µ ξ 0 ) a µ (32) Birkhoff Lie ξ 0 = ξ 0 (t), ξ µ = ξ µ (t, a), (31) δ ρ µ., Birkhoff Noether Lie [78],, Noether, (31) δ ρ µ, Noether Noether., Noether. 7,,.., 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 3.1, 3.2, 4.1, 4.2, 5.1, 5.2, 5.3, 6.1, 6.2.

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