Symmetry of Lagrangians of holonomic systems in terms of quasi-coordinates

Transcription

1 Vol 18 No 8, Augut 009 c 009 Chin. Phy. Soc /009/1808/ Chinee Phyic B an IOP Publihing Lt Symmety of Lagangian of holonomic ytem in tem of quai-cooinate Wu Hui-Bin an Mei Feng-Xiang School of Science, Beijing Intitute of Technology, Beijing , China Receive 18 Octobe 008; evie manucipt eceive 1 Decembe 008 Thi pape icue the ymmety of Lagangian of holonomic ytem in tem of quai-cooinate. Fitly, the efinition an the citeion of the ymmety ae given. Seconly, the conition une which thee exit a coneve quantity an the fom of the coneve quantity ae obtaine. Finally, an example i hown to illutate the application of the eult. Keywo: quai-cooinate, holonomic ytem, ymmety of Lagangian, coneve quantity PACC: Intouction The iffeential equation of motion of a mechanical ytem can be expee in tem of eithe genealize cooinate o quai-cooinate. The latte have bette univeality, o thei application get moe attention. [1] Refeence [] tuie the ymmetie an the coneve quantitie of holonomic ytem in tem of quai-cooinate, incluing the Noethe ymmety, the Lie ymmety, the fom invaiance an the coneve quantitie that aie fom them: the Noethe coneve quantity, the Hojman coneve quantity an the new coneve quantity. Refeence [3] [11] icue iffeent kin of ymmetie an coneve quantitie. But all the above belong to the invaiance obtaine une infiniteimal tanfomation of time an cooinate. In 1966, the autho in Ref.[1] tuie the poblem of equivalent Lagangian fo mechanical ytem with a ingle egee of feeom an pointe out that thee exit coneve quantitie in thi cae. In 1981, the autho in Ref.[13] extene the eult to geneal multi-egeeof-feeom ytem. The eult of Ref.[1] an [13] can only be applie to conevative holonomic ytem. Refeence [14] icue the poblem of equivalent ytem of function: L, Q an L, Q fo nonconevative holonomic ytem, an pointe out the conition une which thee exit coneve quantitie an the fomation of a coneve quantity. Refeence [15] aee the ymmety of Lagangian of nonholonomic ytem. Thi pape exten the eult of Ref.[14] to holonomic ytem in tem of quaicooinate.. Diffeential equation of motion of ytem Suppoe that the poition of a mechanical ytem i etemine by the n genealize cooinate q = 1,..., n. Intouce fom elewhee n quaivelocitie ω which ae inepenent an compatible with each othe: ω = a k q q k,, k = 1,..., n. 1 An aume that fom which all genealize velocitie q can be olve a: q = b k qω k. Then the iffeential equation of motion of the ytem have the fom [16] L L t ω k L = P, 3 k π whee k = akm q l a kl b l b m 4 q m ae Boltzmann thee-inex ymbol, atifying k = k, 5 Poject uppote by the National Natual Science Founation of China Gant No an an the Doctoal Pogam Founation of Intitution of Highe Eucation of China Gant No , an the Fun fo Funamental Reeach of Beijing Intitute of Technology Gant No Coeponing autho. huibinwu@bit.eu.cn

2 3146 Wu Hui-Bin et al Vol. 18 an L t, q, ω = Lt, q, b k ω k 6 i the Lagangian in tem of quai-velocitie. The patial eivative with epect to quai-cooinate π i = b k, 7 π q k an P ae non-potential genealize foce in tem of quai-velocitie P t, q k, ω k = Q t, q, b k ω k b Definition an citeion of ymmety of Lagangian Dicu two goup of ynamical function: L, P an L, P. Let L = L L t ω k L P k π = W k ω k π ω t L k ω L π P, 9 whee L = L L t ω k L k π P = W k ω k π ω t L k ω L π P, 10 W k =, Wk =. 11 Definition Fo the ytem 3, if one can obtain L k = 0 k = 1,..., n fom L = 0 = 1,..., n an vice vea, then the ymmety i calle Lagangian ymmety. whee Fom L = 0, olving all ω, we have ω = Ū k P k L π k t ω L π lkω l, 1 W k Ū k = δ. 13 Subtituting Eq.1 into L = 0, we obtain W k Ū kl P l L π l t ω L π mlω m = P L π t ω L π ω k, k,, l, m = 1,..., n. 14 k Then we have the following citeion. Citeion If two goup of ynamical function L, P an L, P atify Eq.14, then the coeponing ymmety i a ymmety of Lagangian of holonomic ytem 3 in tem of quai-cooinate. Equation 14 ae calle the citeion equation fo ymmety of Lagangian of holonomic ytem in tem of quai-cooinate. 4. Coneve quantity euce by ymmety of Lagangian We now euce the coneve quantity by uing the ymmety of Lagangian. Subtituting Eq.14 into Eq.9, an uing Eq.10, 13, we obtain L = W ω W k Ū kl P l L π l t ω L π mlω m = W Ū k W kl ω l ω l π l t L mkω l m L l π P k k = W Ū k L k. 15

3 No. 8 Symmety of Lagangian of holonomic ytem in tem of quai-cooinate 3147 Let W U k = Λ k, 16 then equation 15 can be witten in the fom By vitue of Eq.16, we have L = Λ L. 17 W k = Λ W k. 18 Taking the patial eivative of Eq.18 with epect to ω l, we obtain Λ W k = 3 L Λ 3 L = W l Λ W l = Λ W l. 19 Taking the patial eivative of Eq.18 with epect to π l, multiplying the eult by ω l an umming it ove l, we have Λ W k π l ω l W k ω l = Λ W k ω l. 0 π l π l Alo, citeion equation 14 can be witten in the fom L ω π t L kω k L π P = Λ L ω l π l t L l kω k L P. 1 l π Taking the patial eivative of Eq.1 with epect to ω m, we obtain i.e. 3 L ω π L = Λ Λ π m 3 L t π l ω l t L l kω k L π P 3 L π l ω l π m W m ω W m π t π m = Λ W k ω k Λ Conieing Eq.19, we have Noticing Eq.0, we get 3 L t π Wm π l ω l π m kω k L m m Λ W k ω k = Λ W m ω k = l kω k L l m L kω k P π P P, π W m W lm l π t kω k L m l P Λ t Λ t. Λ ω l W m. 3 π l In aition, we have Λ W m π l ω l = W m ω l Λ W m ω l. 4 π l π l Subtituting Eq.3, 4 an 5 into Eq., we obtain L π m = Λ t W m Λ Λ t W m = W Λ W m. 5 t t π [ L π m L kω k P π L l kω k P ]. 6 Suppoe that two goup of ynamical function atify the following conition: L kω k P = Λ L l kω k P, 7 l

4 3148 Wu Hui-Bin et al Vol. 18 then fom Eq.6 we get L π m π = Λ t W m Λ L π m π. 8 Intoucing two matice T an T whoe element ae epectively T m = T m =, π m π, 9 π m π we can expe Eq.8 in the matix fom Λ = T U ΛT U, 30 whee Λ = Λ an U = U k ae n n matice. Then fo any poitive intege m, conieing the fact that T an T ae antiymmetic, U an W ae ymmetic, an Λ = W U, an uing the popetie of the tace of matice, we get an Thu, i.e. ΛΛ m1 = T U W UT UΛ m1 = T U W U m1 W UT U W U m1 t [ T U W U m1 ] = 0 t [ W UT U W U m1 ] = 0, t ΛΛ m1 = 0. [ ] t [tλm ] = t t Λm = mt ΛΛ m1 = 0, t [tλm ] = 0, 31 fo any poitive intege m, an we have the coneve quantitie tλ m = cont., 3 namely, the tace of all poitive intege powe of Λ ae contant of the motion. Theefoe, we have Popoition Fo the ytem 3, if the conition 7 i atifie, then the ymmety of Lagangian of the ytem will lea to the coneve quantity 3. Taking the genealize velocitie a the quaivelocitie, i.e., ω = q, we have l k = 0, P = Q, L = L, W k = L, Wk = L. q q k q q k Then the citeion equation 14 can be expee a W k Ū kl Q l L L q l q l t L q q l q = Q L L q q t L q, 33 q q an the conition 7 change into Q q k = Λ Q. 34 q k Thu, we have the following coollaie. Coollay 1 Fo a holonomic ytem in tem of genealize cooinate, if two goup of ynamical function: L, Q an L, Q atify Eq.33, 34, then the ymmety of Lagangian of the ytem will lea to the coneve quantity 3. Coollay Fo a Lagange ytem, the ymmety of Lagangian will lea to the coneve quantity 3. Coollay 1 wa obtaine by Ref.[14]. 5. Illutative example In oe to illutate the above eult, we give an example. Now, we tuy the ymmety of Lagangian an a coneve quantity fo the motion in a pace of a fee igi boy. Take the Eule angle ψ, θ, ϕ an cente-of-ma cooinate x c, y c, z c a genealize cooinate q 1 = ψ, q = θ, q 3 = ϕ, q 4 = x c, q 5 = y c, q 6 = z c. Let quai-velocitie ω 1 = q 1 in q in q 3 q co q 3, ω = q 1 in q co q 3 q in q 3, ω 3 = q 1 co q q 3, ω 4 = q 4, ω 5 = q 5, ω 6 = q 6, whee ω 1, ω, ω 3 ae the pojection of angula velocitie on the cental pincipal axe of inetia. Then the Boltzmann thee-inex ymbol ae a follow: 3 1 = 3 1 = 1, 31 = 13 = 1, 1 3 = 1 3 = 1,

5 No. 8 Symmety of Lagangian of holonomic ytem in tem of quai-cooinate 3149 the othe equal zeo. An the Lagangian in tem of quai-velocitie ha a imple fom: L = Bω Cω 3 ω 4 ω5 ω6 mgq6, whee A, B, C ae pincipal moment of inetia of the igi boy, m i it ma. The non-potential genealize foce P = 0 = 1,..., 6. then Chooe othe goup of ynamical function a L = 1 6 mω3 4 1 P = 0, 1 Bω Cω 3 1 m ω 5 ω 6 mgq6, L 4 = ω 4 L 4, L k = L k, k = 1,, 3, 5, 6, Λ 4 4 = ω 4, Λ k k = 1, k = 1,, 3, 5, 6, Λ k = 0 k, P = P, = 1,..., 6. The coneve quantity 3 euce by the ymmety of Lagangian give I = ω 4 5 = cont. an Similaly, chooe L = 1 6 mω Bω Cω 3 1 m ω 4 ω 6 mgq6 L = 1 6 m ω 6 gt m ω 4 ω 5 mgq6 ω 6, 1 Bω Cω 3 the ymmety of Lagangian can lea to the following coneve quantitie epectively I = ω 5 5 = cont., I = ω 6 gt 5 = cont. All the thee integal epeent the conevation of genealize momenta. 6. Concluion Thi pape icue the ymmety of Lagangian an the coneve quantity euce by uing it fo holonomic ytem in tem of quaicooinate. Quai-cooinate ae moe geneal than genealize cooinate; the eult of Ref.[14] i a pecial cae of that of thi pape. Refeence [1] Zegzha S A, Soltakhanov Sh Kh an Yuhkov P M 005 Equation of Motion of Nonholonomic Sytem an Vaiational Pinciple of Mechanic, New Cla Poblem of Contol Mocow: FIZMATLIT in Ruian [] Mei F X 004 Symmetie an Coneve Quantitie of Containe Mechanical Sytem Beijing: Beijing Intitute of Technology Pe in Chinee [3] Lutzky M 1979 J. Phy. A [4] Zhang H B 00 Chin. Phy [5] Fu J L an Chen L Q 003 Phy. Lett. A [6] Fang J H, Liao Y P an Peng Y 004 Chin. Phy [7] Xu X J, Mei F X an Zhang Y F 006 Chin. Phy [8] Wang S Y an Mei F X 00 Chin. Phy [9] Qiao Y F, Zhao S H an Li R J 004 Chin. Phy [10] Wu H B 005 Chin. Phy [11] Luo S K an Mei F X 004 Acta Phy. Sin in Chinee [1] Cuie D G an Saletan E J 1966 J. Math. Phy [13] Hojman S an Haleton H 1981 J. Math. Phy [14] Zhao Y Y an Mei F X 1999 Symmetie an Invaiant of Mechanical Sytem Beijing: Science Pe in Chinee [15] Mei F X an Wu H B 008 Phy. Lett. A [16] Hamel G 1949 Theoetiche Mechanik Belin: Spinge- Velag