696 Fu Jing-Li et al Vol. 12 form in generalized coordinate Q ffiq dt = 0 ( = 1; ;n): (3) For nonholonomic ytem, ffiq are not independent of

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1 Vol 12 No 7, July 2003 cfl 2003 Chin. Phy. Soc /2003/12(07)/ Chinee Phyic and IOP Publihing Ltd Lie ymmetrie and conerved quantitie of controllable nonholonomic dynamical ytem Fu Jing-Li(ΛΠ±) a)b), Chen Li-Qun( Φ) b), Bai Jing-Hua( ΠΞ) c), and Yang Xiao-Dong(ΩΨ ) b) a) Intitute of Mathematical Mechanic and Mathematical Phyic, Shangqiu Teacher College, Shangqiu , China b) Shanghai Intitute of Applied Mathematic and Mechanic, Shanghai Univerity, Shanghai , China c) Department of Mathematic, Kaifeng Univerity, Kaifeng , China (Received 6 September 2002; revied manucript received 12 March 2003) Thi paper concentrate on tudying the Lie ymmetrie and conerved quantitie of controllable nonholonomic dynamical ytem. Baed on the infiniteimal tranformation, we etablih the Lie ymmetric determining equation and retrictive equation and give three definition of Lie ymmetrie before the tructure equation and conerved quantitie of the Lie ymmetrie are obtained. Then we make a tudy of the invere problem. Finally, an example i preented for illutrating the reult. Keyword: control, nonholonomic dynamical ytem, Lie ymmetry, conerved law PACC: Introduction The ymmetric principle i one of the key iue in mathematic and phyic. Reearch on the conerved quantitie of mechanical and phyical ytem not only ha important mathematical ignificance, but alo how profound phyical eence. Two of the modern method of tudying on the conerved quantitie of mechanical ytem and phyical ytem are the Noether method [1] and the Lie method. Li [2] tudied Noether ymmetrie of the claical and the quantum contrained ytem. Mei [3] tudied Noether ymmetrie of Birkhoffian ytem. Noether ymmetrie of nonholonomic ytem have been tudied by many author. [4 9] The approach of Lie ymmetrie wa et up in the 19th century, but it application in mechanic had not tarted until [10] In recent year, many reult have been obtained in the tudie of the Lie method. [11 22] Lately, Mei [23] tudied Noether ymmetrie a well a Lie ymmetrie of a variety of contrained mechanical ytem, and obtained relationhip between the Noether ymmetrie and Lie ymmetrie, and alo between the ymmetrie and the conerved quantitie. Along with the development of cience and technologie, the control ytem theory become more and more important. [24;25] Thi paper tudie the Lie ymmetrie of the controllable nonholonomic ytem, including both direct and invere problem. 2.Equation of motion of the controllable nonholonomic dynamical ytem Conider a ytem compoed of N particle, and the ma of a particle i m i (i = 1; ;N). The configuration of the ytem can be expreed by n generalized coordinate q ( = 1; ;n). We aume that thi ytem i ubjected to m holonomic contraint of the form f ρ (q ;u r ;t) = 0 (ρ = 1; ;m; i = 1; ;N; r = 1; ;p); (1) and g nonholonomic contraint of the form ψ fi (q ; _q ;u r ; _u r ;t) = 0 (fi = 1; ;g); (2) where u r (r = 1; ;p) are controlled parameter. The motion of the mechanical ytem i controlled by uing the controlling parameter u r, which are important for a mechanical ytem. In the cae of ideal contraint, the d'alembert Lagrangian principle of the contrained ytem can be expreed in the following

2 696 Fu Jing-Li et al Vol. 12 form in generalized coordinate Q ffiq dt = 0 ( = 1; ;n): (3) For nonholonomic ytem, ffiq are not independent of each other, and they have to be contrained by the Chetaev condition in Eq.(3), which are expreed a follow: ffiq = 0 ( = 1; ;n; fi = 1; ;g): (4) From Eq.(3) and (4), we can obtain the Routh equation of the controllable nonholonomic ytem with the uual Lagrange multiplier method a follow: = Q + fi : (5) dt Equation (5) are different from the uual equation of non-controllable nonholonomic ytem, becaue there are control parameter and their derivative with repect to time in the oppoing force of contraint. To control the motion of the ytem by controlling the oppoing force of the contraint in the controlled theory, we can control u r (t) for the purpoe of practical realization. We firt obtain the contrained multiplier fi by uing Eq.(2), (4) and (5), and then ubtitute it into Eq.(5) to obtain an explicit form [27] where q l + A 1 l (k; m; )_q k _q ρ =A @B _q k + k ff k _q k + fi _q l [k; m; ] = m + A km : (7) Suppoe that the controllable parameter u r are dependent only on time, i.e., u r = u r (t)(r = 1; ;p), then differentiating Eq.(2) with repect to t l _q l fl _q l fl _u r fl ü r fl _u = 0: (8) We obtain the contrained multiplier fi by ubtituting Eq.(6) into Eq.(8). Subtituting fi into Eq.(5) lead to q = ff (q; _q;u r ; _u r ; ü r ;t): (9) 3. Direct problem of Lie ymmetry in controllable nonholonomic dynamical ytem 3.1.Infiniteimal tranformation and determining equation of Lie ymmetrie in controllable nonholonomic dynamical ytem We firt introduce infiniteimal tranformation in term of time and generalized coordinate ( t ffl = t + "ο 0 (t; q; _q); (10) q(t) ffl = q (t)+"ο (t; q; _q); where " i a mall parameter, and ο 0 and ο are infiniteimal generator. We again introduce the vector of generator X (0), X (1), X (2). [23] Then baed on the invariance of the differential Eq. (9) under the infiniteimal tranformation (10), if and only if X (2) [ q ff (t; q; _q;u r ; _u r ; ü r )]j q =ff = 0; (11) we can have ο _q ο0 2 _ ο 0 ff = X (1) (ff ) ( = 1; ;n): (12) Eq. (12) are called the determining equation of Lie ymmetrie when the generator ο 0 and ο ( = 1; ; n) atify the infiniteimal tranformation (10). 3.2.Three definition of Lie ymmetrie for controllable nonholonomic dynamical ytem Definition 1 Tranformation (10) are called Lie ymmetric tranformation correponding to the controllable holonomic ytem when the infiniteimal generator ο 0 and ο atify the determining Eq. (12). For the nonholonomic ytem, the contrained Eq. (2) alo retain the form of invariance after the infiniteimal tranformation (10), i.e. X (1) (ψ fi (q; _q;u r ; _u r ;t)) = 0 (fi = 1; ;g; r = 1; ;p): (13) Eq. (13) are called retrictive equation of controllable nonholonomic contraint in Lie ymmetrie. Definition 2 Tranformation (10) are called weak Lie ymmetric tranformation correponding to the controllable nonholonomic ytem when the infiniteimal generator ο 0 and ο atify both the determining Eq. (12) and the retrictive Eq. (13). To obtain Eq.(5), we exploit the Chetaev condition (7) which applie retriction to ffiq, ffiq = q _q t = "(ο _q ο 0 ); (14)

3 No. 7 Lie ymmetrie and conerved quantitie of and ubtitute Eq.(14) into Eq.(4), then (ο _q ο 0 ) = 0 (fi = 1; ;g; = 1; ;n): (15) Eq. (15) are called upplementary retrictive equation. Definition 3 Tranformation (10) are called trong Lie ymmetric tranformation correponding to the controllable nonholonomic ytem when the infiniteimal generator ο 0 and ο atify the determining Eq. (12), the retrictive Eq. (13) and the upplementary retrictive Eq. (15). 3.3.Lie ymmetrical tructural equation and conerved quantitie in the controllable nonholonomic dynamical ytem Theorem 1 If the infiniteimal tranformation generator ο 0 and ο atify the determining Eq. (12), and if the function G = G(t; q; _q;u r ; _u r ) atifie the following equation L _ ο 0 + X r _u r ο _u r ü r ο 0 + _G = 0; (16) there will be a conerved quantity correponding to the Lie ymmetrie for the controllable holonomic ytem, i.e. I = Lο 0 (ο _q ο 0 )+G = contant:; (17) where L = T V i the Lagrangian of the ytem. Proof: di dt = Lο _ 0 + L ο _ 0 + (ο _q ο 0 ) _q _ο q ο 0 _q ο0 _ L _ ο ( _ ο _q _ ο0 _u r ο ü r ο 0 Q 00 (ο _q ο 0 _u r =(ο _q ο 0 Q 00 = 0: dt Eq. (16) i called the tructural equation for the controllable holonomic ytem. Theorem 2 If the infiniteimal tranformation generator ο 0 and ο atify both the determining Eq. (12) and retrictive Eq. (13), and if there exit a function G = G(t; q; _q;u r ; _u r ) atifying the tructural Eq. (16), there will be a conerved quantity (17) correponding to the weak Lie ymmetrie for the controllable nonholonomic ytem. Theorem 3 If the infiniteimal tranformation generator ο 0 and ο atify the determining Eq. (12), retrictive Eq. (13) and upplementary retrictive Eq. (15), and if there i a function G = G(t; q; _q;u r ; _u r ) atifying the tructural Eq. (16), then there will be a conerved quantity (17) correponding to the trong Lie ymmetrie for the controllable nonholonomic ytem. 4. The invere problem of the Lie ymmetrie for controllable nonholonomic dynamical ytem Finding Lie ymmetrie from the given conerved quantitie i called the invere problem, which can be olved by the following tep. Firtly, we find out the correponding Lie ymmetrie from the known conerved quantitie. Suppoe that there exit conerved quantitie in the controllable nonholonomic ytem then we have I = I(t; q; _q;u r ; _u r ) = contant:; (18) di dt dt _q r _u _u r ü r = 0: (19) Multiply Eq.(5) by then umming up for lead to ο = ο _q ο 0 ; (20) Q 00 dt = 0; Q 00 = Q + fi : (21) Adding the two ide of Eq.(20) and (21), expanding the reulting equation, and letting the coefficient of the term including q k to be zero, we have where ο = _q k ; (22) h k h kr = ffi r ;h k _q k : (23) Let the conerved quantity in Eq.(17) equal to Eq.(18), i.e., Lο 0 ο + G = I; (24)

4 698 Fu Jing-Li et al Vol. 12 and then we can obtain generator ο 0 and ο from Eq.(22) and (24). ο 0 and ο are Noether ymmetrie of Eq.(5). Secondly, ubtituting generator ο 0 and ο into Eq.(12), (17) and (15), we can determine the Lie ymmetrie. Theorem 4 If the infiniteimal tranformation generator ο 0 and ο determined by Eq.(22) and (24) atify the determining Eq. (12), the ymmetry i called Lie ymmetrie for the controllable holonomic ytem; if generator ο 0 and ο alo atify retrictive Eq. (13), then the ymmetry i called weak Lie ymmetrie for the controllable nonholonomic ytem. Furthermore, if generator ο 0 and ο atify the upplementary retrictive Eq. (15), then the ymmetry i called trong Lie ymmetrie for the controllable nonholonomic ytem. 5. Example Suppoing that the configuration of the ytem can be pecified with the generalized coordinate q 1 and q 2, it Lagrange function i and the contrained equation i L = 1 2 (_q2 1 2) kq 2 ; (25) f = _q 2 u(t)_q 1 = 0; (26) where u(t) i the controllable parameter. The ytem i a holonomic one when the controllable parameter u(t) doe not include time t, and u(t) i neither a linear function with repect to time t. If the momentum along the generalized coordinate q 2 of the ytem i conervational when the controllable parameter u(t)! 0, i.e., it i a conervative holonomic ytem, which i very important in phyic. The ytem expreed by Eq.(25) and (26) i a nonholonomic ytem in the ordinary cae. To tudy the Lie ymmetry of the ytem, we firt tudy the direct problem of the Lie ymmetrie. The equation of motion of the ytem are a follow q 1 = u; q 2 = k: (27) From Eq.(26) and (27), we obtain = (k + u _u _q 1 )=(1 + u 2 ): (28) Subtituting it into Eq.(27) yield q 1 = ku + u _u _q 1 1+u 2 ; q 2 = u _u _q 1 ku 2 1+u 2 : (29) Then ubtituting Eq.(29) into the determining Eq. (16), we obtain 8 ο 1 _q 1 ο0 2 ο _ 0 ku + u _u _q 1 1+u 2 = ο 0 k(1 u2 )_u (1 + u 2 ) uü 2 1+u _q 2 1 _u2 (1 u 2 ) (1 + u 2 ) _q 2 1 >< >: +( ο _ 1 _q 1 ο0 _ u _u ) 1+u ; 2 ο 2 _q 2 ο0 2 _ ο 0 u _u _q1 1+u 2 ku2 1+u 2 = ο 0 uü 1+u 2 _q 1 + _u2 u 2 (1 + u 2 ) 2 _q 1 _u 2 (1 + u 2 ) 2 _q 1 2 2ku _u (1 + u 2 ) +( ο _ 1 _q _ u _u 1 ο0 ) 1+u : 2 (30) Equation (30) ha the following olution ο 0 = 0; ο 1 = 0; ο 2 = 1; (31) ο 0 = 0; ο 1 = 1; ο 2 = 0; (32) ο 0 = 0; ο 1 = 1; ο 2 = 1; (33) ο 0 = 0; ο 1 = 1; ο 2 = t; (34) ο 0 = 0; ο 1 = 0; ο 2 = t: (35) They have Lie ymmetrie correponding to the controllable holonomic ytem. By ubtituting generator ο 0 and ο into Eq.(13), we have _u _q 1 ο 0 ( _ ο 1 _q 1 _ ο0 )u +( _ ο 2 _q 2 _ ο0 ) = 0: (36) The generator (34) and (35) do not atify retrictive Eq. (36), o there are no weak Lie ymmetrie in the controllable nonholonomic ytem. Wherea the generator (31), (32) and (34) atify retrictive Eq. (36), o there are weak Lie ymmetrie for the nonholonomic ytem. From Eq.(16) and (17) we obtain gauge function and conerved quantitie for the generator (31), (32) and (34) a follow G g = kt; I g = _q 2 + kt = contant:; (37) G h = 0; I h = _q 1 = contant:; (38) G j = kt; I j = _q 1 + kt = contant: (39) They are the gauge function and conerved quantitie for weak Lie ymmetrie of Eq.(25) and (26), i.e., they are uual gauge function and conerved quantitie of

5 No. 7 Lie ymmetrie and conerved quantitie of the Lie ymmetrie. Under the infiniteimal tranformation (31), (32) and (33), ome concluion can be obtained: form (37) expree that the momentum i not conervative along the generalized coordinate q 2, form (38) indicate that the momentum i conervative along the generalized coordinate q 1, and form (39) expree that the momenta of the ytem i not conervative. Subtituting contraint (26) into Eq.(15), we obtain u(t)(ο 1 _q 1 )+(ο 2 _q 2 ) = 0: (40) Eq. (40) cannot be atified when ubtituting the generator (31), (32) and (34) into Eq.(40), o the ymmetrie are jut weak Lie ymmetrie. Now we tudy the invere problem. Suppoe the ytem ha a conerved quantity I = _q 1 + kt: (41) Subtituting Eq.(41) into Eq.(22) and (24) lead to and then ο 1 = 1; ο 2 = 1; Lο 0 + _q 1 ο 1 ο 2 + G = _q 1 + kt; (42) ο 0 = 1 L (kt G); ο 1 = 1+ _q 1 ο 0 ; ο 2 = 1 ο 0 : (43) Subtituting Eq.(39) into Eq.(43), we obtain ο 0 = 0; ο = 1; ο 2 = 1: (44) The generator (44) atify the retrictive Eq. (36), but they do not atify the upplementary retrictive Eq. (40), o the ymmetry (44) i weak Lie ymmetrie, not trong Lie ymmetrie. Reference [1] Noether A E 1918 Nachr, Akad. Wi. Göttingen Math. Phy. KI, II 235 [2] Li Z P 1993 Claical and Quantum Contrained Sytem and their Symmetrical Propertie (Beijing: Beijing Polytechnic Univerity Pre) [3] Mei F X 1993 Chin. Sci. Serie A (in Chinee) [4] Liu D 1990 Chin. Sci. Serie A [5] Liu D 1989 Acta Mech. Sin [6] Zhang J F 1991 Chin. Sin. Bull (in Chinee) [7] Li Z P 1981 Acta Phy. Sin (in Chinee) [8] Fu J L, Chen X W and Luo S K 2001 Acta Solid Mech. Sin (in Chinee) [9] Ge W K 2002 Acta Phy. Sin (in Chinee) [10] Lutzky M 1979 J. Phy. A: Math. Gen [11] Zhao Y Y 1994 Acta Mech. Sin [12] Zhao Y Y and Mei F X 1993 Adv. Mech (in Chinee) [13] Mei F X, Wu R H and Zhang Y F 1998 Acta Mech. Sin [14] Fu J L and Wang X M 2000 Acta Phy. Sin (in Chinee) [15] Fu J L, Chen X W and Luo S K 2000 Appl. Math. Mech [16] Mei F X 2000 Acta Phy. Sin (in Chinee) [17] Mei F X and Shang M 2000 Acta. Phy. Sin (in Chinee) [18] Zhang Y, Shang M and Mei F X 2000 Chin. Phy [19] Fu J L and Liu R W 2000 Acta Math. Phy. Sin (in Chinee) [20] Zhang R C and Mei F X 2000 Chin. Phy [21] Qiao Y F, Li R J and Zhao S H 2001 Acta Phy. Sin (in Chinee) [22] Li Y C et al 2001 Chin. Phy [23] Mei F X 1999 Application of Lie Group and Lie Algebra to Contrained Mechanic Sytem (Beijing: Science Pre) (in Chinee) [24] Katuhiko Ogate 1998 Modern Control Engineering 3rd edn (Englewood Cliff, NJ: Prentice Hall) [25] You C D 1996 The Modern Control Theory Baic (Beijing: Electronic Indutry Pre) (in Chinee) [26] Mei F X 1985 Mechanical Foundation of Nonholonomic Sytem (Beijing: Beijing Intitute of Technology Pre) (in Chinee) [27] Mei F X, Liu D and Luo Y 1991 Advanced Analytical Mechanic (Beijing: Beijing Intitute of Technology Pre) (in Chinee)