Noether symmetry and non-noether conserved quantity of the relativistic holonomic nonconservative systems in general Lie transformations

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1 Vol 16 No 11, November 2007 c 2007 Chin. Phy. Soc /2007/1611/ Chinee Phyic and IOP Publihing Ltd Noether ymmetry and non-noether conerved quantity of the relativitic holonomic nonconervative ytem in general Lie tranformation Luo Shao-Kai ab a Intitute of Mathematical Mechanic and Mathematical Phyic, Zhejiang Sci-Tech Univerity, Hangzhou , China b Key Laboratory of Advanced Textile Material and Manufacturing Technology Zhejiang Sci-tech Univerity, Minitry of Education, Hangzhou , China Received 8 November 2005; revied manucript received 23 May 2007 For a relativitic holonomic nonconervative ytem, by uing the Noether ymmetry, a new non-noether conerved quantity i given under general infiniteimal tranformation of group. On the bai of the theory of invariance of differential equation of motion under general infiniteimal tranformation, we contruct the relativitic Noether ymmetry, Lie ymmetry and the condition under which the Noether ymmetry i a Lie ymmetry under general infiniteimal tranformation. By uing the Noether ymmetry, a new relativitic non-noether conerved quantity i given which only depend on the variable t, q and q. An example i given to illutrate the application of the reult. Keyword: relativity, holonomic nonconervative ytem, Noether ymmetry, non-noether conerved quantity PACC: 0320, 0330 Relativitic analytical mechanic i an important apect of the modern development in the field of theoretical phyic. Since 1987, we have contructed the theory of analytical mechanic for the relativitic ytem, and given it baic theoretical frame. [1 8] The ymmetry and the conerved quantity for a dynamical ytem play an important role in the field of mathematic, mechanic and phyic. There are three main ymmetry method ued to eek the conerved quantitie of the dynamical ytem: Noether ymmetry, [9 14] Lie ymmetry [15 21] and Mei form invariance. [22 29] A i well known, the Noether ymmetry can be led to the conerved quantity directly, and the conerved quantity i called Noether. In general, the Lie ymmetry doe not alway imply the conerved quantity. If and only if the infiniteimal tranformation of Lie ymmetry atify the tructure equation i.e. Noether identity, the Lie ymmetry can be led to a Noether conerved quantity. In 1992, Hojman [30] gave the direct method of finding conerved quantity by uing Lie ymmetry. Thi direct method i generalized by González-Gacón [31] and Lutzky. [32,33] Pillay and Leach [34] demontrated that the invariant derived uing Hojman conervation law i trivial for all Noether ymmetrie. Therefore, uing Hojman method one can find the non-trivial conerved quantity which i led by the non-noether ymmetry and the conerved quantity i called non-noether. In recent year, the Lie ymmetrical Hojman conerved quantitie of ome dynamical ytem have been tudied by uing thi direct method in Ref.[11, 19, 35 40], and a erie of important reult have been obtained. Recently, Ref.[8, 39] gave the Hojman conerved quantity of the relativitic Hamiltonian ytem under the pecial infiniteimal tranformation. But, the direct method of finding conerved quantity preented above i only utilized under the pecial infiniteimal tranformation of group in which the time i invariant. Recently, by uing the Lie ymmetry, Ref.[41, 42] gave directly the non-noether conerved quantitie of the claical dynamical ytem under general infiniteimal tranformation. In thi paper, by uing the Noether ymmetry, we obtain directly the non-noether conerved quantity of the relativitic holonomic nonconervative ytem under general infiniteimal tranformation in which the time i variant. At firt, the relativitic Noether ymmetry, Lie ymmetry and Noether conerved quantity Project upported by the National Natural Science Foundation of China Grant No and mmmplk@163.com

2 No. 11 Noether ymmetry and non-noether conerved quantity of the relativitic holonomic 3183 under general infiniteimal tranformation are given. Secondly, the condition under which the relativitic Noether ymmetry i a Lie ymmetry i obtained. Finally, a et of relativitic non-noether conerved quantity of the ytem i given, which only depend on the variable t, q, and q. An example i given to illutrate the application of the reult. Here we tudy a relativitic holonomic nonconervative ytem compoed of N particle, and it configuration i determined by the n generalized coordinate q = 1,,n. It motion can be decribed by the differential equation a follow: [1] E L = Q, = 1,,n, E = d, dt q L t, q, q = T V, V = V q, N T = m oi c ṙ 2i /c2, i=1 m oi = m i 1 ṙ 2 i /c2, i = 1,,N. 1 Here L = L t, q, q i the relativitic Lagrangian, = Q t, q, q are the nonpotential generalized Q force, and the repeated ubcript repreent the ummation. In general, it i uppoed that ytem 1 i noningular, i.e. 2 L det 0. 2 q k From Eq.1, we can find all generalized acceleration a q = h t, q, q, = 1,,n. 3 We introduce the general infiniteimal tranformation for t and q, t = t + εξ 0 t, q, q, q t = q t + εξ t, q, q, 4 where ε i an infiniteimal parameter, and ξ 0 and ξ are infiniteimal generator. Equation 4 i a oneparameter Lie group of tranformation. If the generator ξ 0 and ξ of infiniteimal tranformation 4 atify the relativitic Noether identity a follow: L t ξ 0 + L ξ + L L q ξ 0 + L ξ + Q ξ q ξ 0 + q q ĠN = 0, 5 or the following relativitic Killing equation have olution: L t ξ 0 + L ξ + L L ξ0 q q t + ξ 0 q k + L q k ξ t + ξ q k + Q ξ q ξ 0 q k = G N G N q, t q L L ξ0 q + L ξ = G N, k = 1,,n, 6 q k q k q k then the ymmetry i relativitic Noether ymmetry. Taking the infiniteimal generator vector and it extenion X 0 = ξ 0 t + ξ, 7 q X 1 = X 0 + ξ q ξ0, 8 X 2 = X 1 + ξ 2 q ξ0 q ξ0 q, 9 the invariance of Eq.1 under the infiniteimal tranformation 4 can lead the atifaction of the relativitic Lie ymmetrical determining equation X 2 {E L } X 1 Q = By uing the relativitic Noether theorem, from the Noether ymmetry or the Lie ymmetry, we can obtain the relativitic Noether conerved quantity a I N = L ξ 0 + L ξ q ξ 0 + G N = cont. 11 We now tudy the relation between the relativitic Noether ymmetry and Lie ymmetry, and obtain the condition under which the relativitic Noether ymmetry i a Lie ymmetry.

3 3184 Luo Shao-Kai Vol.16 For { } X 2 {E L } = E X 1 L + L ξ0 d2 dt 2 E Ġ N = d2 dt 2 G N, [ L q k L q k q k ξ0 ] + E ξ k q k ξ 0 E k L ξk ξ 0 d + q k dt E kl, 12 we have } [ X 2 {E L } X 1 Q = E {X 1 L + L ξ0 + ĠN d2 L dt 2 q k L ξ0 + G ] N q k q k ξk ξ 0 +E ξ k q k ξ 0 Q k + q k Q k X 1 Q. 14 Adding and ubtracting a function E {ξ k q k ξ 0 Q k }, we obtain X 2 {E L } X 1 Q = E {X 1 L + L ξ0 + ξ k q k ξ 0 Q k + ĠN [ d2 L dt 2 q k L ξ0 + G N q k q k ξ k q k ξ 0 Q q k } ] Qk + Q d q q k dt 13 ξ k q k ξ0 q k ξ If the ymmetry i Noether, then Eq.5 and 6 hold. Subtituting Eq.5 and 6 into Eq.15, we obtain X 2 {E L } X 1 Qk Q = Q d Q ξ k q k ξ 0 Q k q q k dt q k q ξ k q k ξ0 q k ξ Hence, we have the following theorem. Theorem 1 For the relativitic holonomic nonconervative ytem 1, if the relativitic Noether ymmetrical generator ξ 0 and ξ atify the following condition: Qk q Q q k d dt Q ξ k q k ξ 0 Q k q k q ξ k q k ξ0 q k ξ 0 = 0, 17 then the generator ξ 0 and ξ certainly are Lie ymmetric under the infiniteimal tranformation 4; otherwie, it i uncertain. We now obtain directly the relativitic non- Noether conerved quantitie from the Noether ymmetry of the relativitic holonomic nonconervative ytem, which only depend on the variable t, q and q. Theorem 2 For the relativitic holonomic nonconervative ytem 1, if the relativitic Noether ymmetrical generator ξ 0 and ξ atify the identity 17 and there i a function µ = µt, q, q uch that h + 1 µ µ t + q µ µ + h = 0, q = 1,, n, 18 then the ytem poee the conerved quantity a follow: I L = 1 µ t µξ µξ + 1 [µ µ q µ q ξ q ξ0 ] ξ 0 = cont. 19 In fact, if the Lie ymmetrical generator ξ 0 and ξ of infiniteimal tranformation 4 atify the condition 18, then ytem 1 poee the conerved quantity 19. So we can obtain directly Theorem 2 from Theorem 1. From Theorem 2, we have the following corollarie. Corollary 1 For the relativitic Lagrangian y-

4 No. 11 Noether ymmetry and non-noether conerved quantity of the relativitic holonomic 3185 tem 1 atifying Q 0, if the relativitic Noether ymmetrical generator ξ 0 and ξ atify the identity 17 and there i a function µ = µt, q, q uch that condition 18 hold, then the relativitic ytem poee the conerved quantity 19. If we only introduce the pecial infiniteimal tranformation for q, then we have t = 0 and ξ 0 = 0, Eq.4, 7 9, 17 and 19 become, repectively, t = t, q t = q t + εξ t, q, q, 20 X 0 = ξ, X 1 = X 0 + q ξ, X 2 = X 1 + ξ, 21 q Qk Q d Q ξ k Q k ξ k q q k dt q k = 0, 22 I L = 1 µξ + 1 µ µ q µ q ξ = cont. 23 Corollary 2 For the relativitic holonomic nonconervative ytem 1, if the relativitic Noether ymmetrical generator ξ of infiniteimal tranformation 20 atify the identity 22, and there i a function µ = µt, q, q uch that condition 18 hold, then the relativitic ytem poee the conerved quantity 23. In the claical approximation ṙ i c, we have m i = m oi. For T, chooing the former two item of the power-erie expreion of 1 ṙ 2 i / c 2 1/2 with repect to ṙ 2 i / c 2, we can obtain T 1 2 N m i ṙi 2 = T, L T V = L. 24 i=1 Therefore, from theorem 2, we have alo the following corollary. Corollary 3 For the claical holonomic nonconervative ytem 1 atifying ṙ i c, if the Noether ymmetrical generator ξ 0 and ξ of infiniteimal tranformation 4 atify the identity 17, and there i a function µ = µt, q, q uch that condition 18 hold, then the ytem poee the claical conerved quantity 19. The above corollarie are alo new reult. A an example, uing the Noether ymmetry, we tudy the Lie ymmetrical non-noether conerved quantitie of a relativitic holonomic nonconervative ytem, of which the Lagrangian and the nonpotential generalized force are, repectively, L = m 0 c q 2 /c 2 V, Q = k 1 1 q 2 /c 2 1/2, 25 where V = cont and k 1 = cont. The equation of motion i q = k 1 m 0 1 q 2 /c 2 = h. 26 The Noether identity 5 give ξ 0 = 1, ξ = q 2, G N = [2m 0 +k 1 q 2 k 1 q] 1 q 2 /c 2 1/2 dt. 27 The generator 27 atifie the condition 17. The condition 18 give 2k 1 q m 0 c 2 + d lnµ = dt And the above equation ha the following olution 2k1 q µ = exp m 0 c According to theorem 2, Eq.27 and 29 give I L = 21 + k 1 q 2 = cont. 30 m 0 c2 Uing the identity 11, we know that the conerved quantity 30 i non-noether. Reference [1] Luo S K 1987 Teach. Mather. Commun in [2] Luo S K 1990 Proc. ICDVC Beijing: Peking Univerity Pre p645 [3] Luo S K 1992 Acta Math. Sci in [4] Luo S K 1996 Appl. Math. Mech [5] Fang J H 2004 Commun. Theor. Phy [6] Qiao Y F 2004 Chin. Phy [7] Zhang Y and Ge W K 2005 Acta. Phy. Sin in [8] Luo S K 2003 Chin. Phy. Lett

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