722 Chen Xiang-wei et al. Vol. 9 r i and _r i are repectively the poition vector and the velocity vector of the i-th particle and R i = dm i dt u i; (

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1 Volume 9, Number 10 October, /2000/09(10)/ CHINESE PHYSICS cfl 2000 Chin. Phy. Soc. PERTURBATION TO THE SYMMETRIES AND ADIABATIC INVARIANTS OF HOLONOMIC VARIABLE MASS SYSTEMS * Chen Xiang-wei(ΞΠ) a)b) and Mei Feng-xiang( Λ) a) a) Department of Applied Mechanic, Beijing Intitute of Technology, Beijing , China b) Department of Phyic, Shangqiu Teacher College, Shangqiu , China (Received 20 March 2000; revied manucript received 7 June 2000) The perturbation problem of ymmetrie for the holonomic variable ma ytem under mall excitation i dicued. The concept of high-order adiabatic invariant i preented, and the form of adiabatic invariant and the condition for their exitence are given. Then the correponding invere problem i tudied. Finally an example i preented to illutrate thee reult. Keyword analytical mechanic, variable ma, ymmetry, perturbation, adiabatic invariant PACC 0320 I. INTRODUCTION More and more attention ha been paid to the reearch of ymmetrie and conerved quantitie of mechanical ytem and many reult have prung up during recent year. Thee reult turn the knowledge of people to the eence of mechanical ytem. [1;2] In 1917, Burger firt propoed adiabatic invariant which pointed to a pecial kind of Hamilton ytem. [3] The adiabatic invariant mean that they are almot not changed when the parameter varie very lowly and they play a very important role in the reearch on quai-integrability of a mechanical ytem. The rigorou mathematical meaning of the adiabatic invariant i a ubtle work, o far, it ha not been olved yet. [4] In fact, that the parameter varie very lowly, i equal to the action of a mall perturbation, o we would make reearche by the latter model in thi paper. Due to the development of the pace and other technologie, the tudy on the dynamic of variable ma ytem become more and more important. The differential equation of motion, [57] the Hamilton' principle, [7;8] the Noether' theory, [9] the Lie ymmetrie and conerved quantitie of the variable ma ytem, [10] etc., have been obtained. But the problem of adiabatic invariant of the ytem ha not been dealt with. The tudy of adiabatic invariant ha become a popular ubject in mechanic, [11] atomic and molecular phyic, [12;13] etc. In Ref.[14, 15] adiabatic invariant of general mechanical ytem were tudied and ome important reult obtained. In thi paper, we dicu the adiabatic invariant of the holonomic variable ma ytem from the viewpoint of ymmetrie and conerved quantitie. Firtly, we propoe the concept of higher order adiabatic invariant. Then, we prove the condition for their exitence and give their form. Finally we preent the relationhip between the ymmetrie of the mechanical ytem and the adiabatic invariant. II. EQUATIONS OF MOTION OF THE SYS- TEMS Let u conider a mechanical ytem of N particle. At time t, the ma of the i-th particle i m i (i = 1;;N); at time t dt, the ma of the corpucle abandoned from (or merging into) the particle i dm i. Let the poition of the ytem be determined by the n generalized coordinate q ( = 1; ; n) and let [5] mi = mi(t; q; _q); (1) the equation of motion can be written in the form [6] d dt = Q P ( = 1; ; n); (2) L i Lagrangian of the ytem, Q are generalized nonpotential force, P are generalized revere thrut, and we have [5;6] P = NX i=1 n (R i _m i _r i ) r 1 i 2 _r m i i _r i o ; (3) d dt 1 2 _r i _r i m i Λ Project upported by the National Natural Science Foundation of China (Grant No ), and by the Doctoral Program Foundation of Intitution of Higher Education of China and by the Natural Science Foundation of Henan Province, China.

2 722 Chen Xiang-wei et al. Vol. 9 r i and _r i are repectively the poition vector and the velocity vector of the i-th particle and R i = dm i dt u i; (4) u i i the corpucle' velocity relative to the i-th particle. III. INFINITESIMAL TRANSFORMATIONS AND CONSERVED QUANTITIES Introducing the infiniteimal tranformation t Λ = t t; q Λ = q q ( = 1;;n) (5) or their expanded formula t Λ =t "fi 0 (t; q; _q); q Λ = q "ο 0 (t; q; _q) ( = 1; ; n) (6) and taking the generator vector of the infiniteimal tranformation X (0) = fi 0 t and it firt extended generator X (1) = X (0) ο 0 ; (7) ( ο _ 0 _q _fi 0 ) ; (8) we have Propoition 1. If there exit a gauge function G 0 = G 0 (t; q; _q) atifying the tructure equation L _fi 0 X (1) (L) (Q P )(ο 0 _q fi 0 ) _ G 0 = 0; (9) then the holonomic variable ma ytem ha the following conerved quantity I = Lfi 0 Proof di dt = _ Lfi 0 L _fi 0 = (ο 0 _q fi 0 )G 0 = cont (10) (ο 0 _q fi 0 ) d dt L _fi 0 X (1) (L) ( _ ο 0 q fi 0 _q _fi 0 ) d (ο 0 _q fi 0 ) dt (Q P )(ο 0 _q fi 0 ) Q P = 0 IV. PERTURBATION TO THE SYMME- TRIES AND ADIABATIC INVARIANTS Firtly we give the concept of high-order adiabatic invariant. Definition If I z (t; q; _q;") i a phyical quantity including " in which the highet power i z in a mechanical ytem, and it derivative with repect to time t i in direct proportion to " z1, then I z i called a z-th order adiabatic invariant of the mechanical ytem. Suppoe the holonomic variable ma ytem correponding to Eq.(2) i perturbed by a mall quantity "W, then the equation of motion of the ytem become d = Q P "W ( = 1;;n); (11) dt due to the action of "W, the primary ymmetrie and conerved quantitie of the ytem may vary. Suppoe the variation i a mall perturbation baed on the ymmetrical tranformation of the ytem without perturbation, and fi(t; q; _q) and ο (t; q; _q) expre the generator of time and pace repectively after being perturbed, then fi = fi 0 "fi 1 " 2 fi 2 ; ο = ο 0 "ο 1 " 2 ο 2 ; (12) and they atify L _fi X (1)0 (L) " X (1)0 = X (0)0 (Q P )(ο _q fi) W (ο _q fi) _G = 0; (13) = fi t ( ο q _fi) ο ( ο q _fi) ; with G in Eq.(13) being a gauge function. Let X (1)Λ = fi k G = G 0 "G 1 " 2 G 2 ; (14) t ο k ( ο _ k _q _fi k ) (15) Subtituting Eq.(12) and (14) into Eq.(13), we obtain L _fi k X (1)Λ (L) (Q P )(ο k _q fi k ) W (ο k1 _q fi k1 ) _G k = 0 (k = 0; 1; 2; ) (16)

3 No. 10 Perturbation to the Symmetrie and Adiabatic Invariant of When k = 0, the condition W =0 hold, and we may prove that, for the differential equation (11) of the mechanical ytem, the following quantity # I z = " "Lfi k k ( ο _ k _q _q fi k )G k (17) i a z-th order adiabatic invariant of the mechanical ytem, i.e., i in direct proportion to " z1. In dt fact dt = k " Lfi _ k L _fi k = ( _ ο k q fi k _q _fi k ) (ο k _q fi k ) d L _fi k X (1) Λ (L) dt (Q P )(ο k _q fi k ) W (ο k1» X n " k _q _fi k1 ) Λ ( d Q P )(ο k dt q _q fi k ) W (ο k1 _q _fi k1 ) Uing Eq.(11), we have» X n dt = " k "W (ο k _q fi k ) W (ο k1 _q _fi k1 ) Expanding the above formula and making ummation, we have dt = "z1 W (ο z _q fi z ) (18) So we have Propoition 2. For a holonomic variable ma ytem perturbed by a mall quantity "W, if the generator fi k (t; q; _q) and ο k (t; q; _q) under infiniteimal tranformation atify Eq.(16), and G k (t; q; _q) i the gauge function, then # I z = " "Lfi k k (ο k _q _q fi k )G k i a z-th order adiabatic invariant of the mechanical ytem. V. INVERSE PROBLEM Suppoe that the ytem ha a firt order adiabatic invariant a follow I 1 = 0 (t; q; _q)" 1 (t; q; _q) (19) Differentiating thi with repect to t, we have di 1 dt = 0 t 0 _q 0 q "» 1 t 1 _q 1 q (20) Multiplying Eq.(11) by ο _q fi, and taking ummation for, we have d (ο _q fi) Q P "W = 0 (21) dt From formula (3) we know that P are linear with repect to q in general cae If we have P = V r = i=1 V r (t; q; _q) q r V (t; q; _q); (22) ρ mi _q r (u i _r i ) r i 1 2 _r 2 m i i _r i m i _r i r i _q r ff q r m i = 0 (i = 1; ; N; = 1; ; n); V r = 0 By uing Eq.(18), and adding Eq.(20) and (21), we have 0 t "» 1 t =" 2 0 _q 0 q (ο _q fi) 1 _q 1 q d Q P "W dt W (ο 1 _q fi 1 ); (23) the expreion of fi and ο are Eq.(12). Firtly, we eek the generator fi 0 and ο 0 of the infiniteimal tranformation without perturbation. Separating the term not containing " in Eq.(23), then

4 724 Chen Xiang-wei et al. Vol. 9 eparating the term containing q r and taking their coefficient a zero, we have 0 2 L V _q r _q r r (ο 0 _q fi 0 ) = 0 (24) Let det(! r ) = det V r 2 L 6= 0; _q r from Eq.(24) we can obtain Further let 0 = Lfi 0 ο 0 _q fi 0 = ~! k! kr = ffi r ~! r 0 _q r ; (25) (ο 0 _q fi 0 )G 0 (26) Thu we can eek the generator fi 0 and ο 0 of the infiniteimal tranformation from Eq.(25) and (26) when the gauge function G 0 i given. Further, eparating the term containing " in Eq.(23), then eparating the term containing q r and taking their coefficient a zero, we have 1 = Lfi 1 ο 1 _q fi 1 = ~! r 1 _q r ; (27) (ο 1 _q fi 1 )G 1 (28) Thu we can eek the generator fi 1 and ο 1 of the infiniteimal tranformation from Eq.(27) and (28) when the gauge function G 1 i given. VI. AN ILLUSTRATIVE EXAMPLE Conider a variable ma particle whoe ma i m = m 0 e fft (m 0 = cont;ff= cont;ff > 0) (29) It Lagrangian i L = 1 2 m(t)( _q2 1 _q 2 2); (30) the generalized nonpotential force are Q 1 = Q 1 (t; q 1 ; _q 1 ; _q 2 ); Q 2 = C 2 _q 2 C 1 q 1 _q 1 ; (31) C 1 and C 2 are contant. The corpucle' abolute velocity i zero, i.e., u = _r = _q 1 i _q 2 j; (32) and the ytem i perturbed by the mall quantitie a follow W 1 = D 1 1b 2 t 2 ; W 2 = D 2bt 1b 2 t 2 ; (33) D 1 ;D 2 and b are contant. Try to tudy the adiabatic invariant of the ytem. From Eq.(3), (4) and (32) we know Equation(11) give (m _q 1 ) = Q 1 D 1" 1b 2 t 2 ; P 1 = P 2 = 0 (34) (m _q 2 ) = C 2 _q 2 C 1 q 1 _q 1 D 2"bt 1b 2 t 2 (35) Firtly, we eek the 0-th order adiabatic invariant, i.e., the exact invariant. The tructure equation (16) give L _fi 0 fi 0 t ο 0 ( ο _ 0 _q _fi 0 ) Q (ο 0 _q fi 0 ) _G 0 = 0 (36) Chooing the infiniteimal generator a then Eq.(36) become o we obtain fi 0 = 0; ο 0 1 = 0; ο 0 2 = 1; (37) Q 2 _ G 0 = 0; (38) G 0 = C 2 q C 1q 2 1 (39) From propoition 1, we can obtain the 0-th order adiabatic invariant I 0 = m(t)_q 2 C 2 q C 1q 2 1 = cont (40) Secondly, we eek the firt order adiabatic invariant. The tructure equation (16) give L _fi 1 fi 1 t ο 1 Q (ο 1 _q fi 1 ) ( ο _ 1 _q _fi 1 ) W (ο 0 _q fi 0 )_G 1 = 0 (41)

5 No. 10 Perturbation to the Symmetrie and Adiabatic Invariant of Chooing the infiniteimal generator a then Eq.(41) become o we obtain fi 1 = 0; ο 1 1 = 0; ο 1 2 = 1; (42) Q 2 W 2 _G 1 = 0; (43) G 1 = C 2 q C 1q 2 1 D 2 2b ln(1 b2 t 2 ) (44) From propoition 2, we can obtain the firt order adiabatic invariant I 1 =m(t)_q 2 C 2 q C 1q 2 1 " 1 2 C 1q 2 1 D 2 2b ln(1 b2 t 2 ) h i m(t)_q 2 C 2 q 2 = cont (45) Further we can obtain higher order adiabatic invariant. REFERENCES [1] Z. P. Li, Acta Phyica Sinica, 30(1981), 1659(in Chinee). [2] Y. Y. Zhao, F. X. Mei, Adv. Mech., 23(1993),360(in Chinee). [3] D. Y. Wu, Claical Dynamic (Science Pre, Beijing, 1983), p.26(in Chinee). [4] V. I. Arnold, Mathematical Method of Claical Mechanic (Springer-Verlag, New York, 1978), p.297. [5] V. S. Novoelov, Vetn LGU, 7(1959),112(in Ruian). [6] L. W. Yang, F. X. Mei, Mechanic of Variable Ma Sytem (Beijing Intitute oftechnology Pre, Beijing, 1989), p.1( in Chinee). [7] F. X. Mei, Foundation ofmechanic ofnonholonomicsytem (Beijing Intitute oftechnology Pre, Beijing, 1985), p. 96(in Chinee). [8] Z. M. Ge, Y. H. Cheng, Appl. Math. Mech., 4(1983), 291. [9] S. K. Luo, Jiangxi Science, 10(1992),131(in Chinee). [10] F. X. Mei, Appl. Math. Mech., 20(1999),629. [11] J. S. Hu, SIAM J. Appl. Math., 59(1999),777. [12] L. Wang, J. Kevorkian, Phy. Plama, 3(1996),1162. [13] V. N. Otrovky, N. V. Prudov, J. Phyic, B20(1995), [14] Y. Y. Zhao, F. X. Mei, Acta Mechanica Sinica, 28 (1996),207(in Chinee). [15] Y. Y. Zhao, F. X. Mei, Symmetrie and Invariant of Mechanical Sytem (Science Pre, Beijing, 1999), p.164(in Chinee).