Lie symmetry and Mei conservation law of continuum system

Transcription

1 Chin. Phys. B Vol. 20, No Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive 7 July 200; revise manuscript receive 2 September 200 Lie symmetry an Mei conservation law of continuum Lagrange system are stuie in this paper. The equation of motion of continuum system is establishe by using variational principle of continuous coorinates. The invariance of the equation of motion uner an infinitesimal transformation group is etermine to be Lie-symmetric. The conition of obtaining Mei conservation theorem from Lie symmetry is also presente. An example is iscusse for applications of the results. Keywors: continuum mechanics, Lie symmetry, Mei symmetry, conservation law PACS:.30. j, 83.0.Ff DOI: 0.088/ /20/2/020. Introuction The symmetry plays an important role not only in unerstaning the behaviours of the mechanical systems, whether iscrete or continuous meium, but also in ientifying conserve quantities. Many important conclusions of theoretical mechanics can be state in the forms of conservation laws which inicate uner what conitions the mechanical quantities are constant with respect to the time. 4 There exist three symmetries we always use for searching invariant quantities. The Lie symmetry has been etermine to be an invariance of the ifferential equation of motion uner a continuous group now known as Lie group use to unify various methos of solving orinary ifferential equations as its original purpose. 5 0 The Noether symmetry is an invariance of the action integral uner Lie transformation group. The Noether theory has become a useful tool in many subfiels of mathematics, physics an mechanics, since Noether showe that there was a close relationship between the mathematical symmetries of a physical system an its conserve quantities. 5 These two types of symmetries an their applications have obtaine many results in iscrete meium constraine mechanical systems. 2 4 Recently, Mei an his collaborators have introuce a new type of symmetry theory referre to as Mei symmetry, relate to the form invariances of the ifferential equations of motion uner ynamical transformations in which the ynamical quantities such as Lagrangian, non-potential generalize forces, generalize constraine forces, etc. are replace by the switche ones which are transforme by the Lie transformation groups. Investigations on the Mei symmetry an its conserve quantities in iscrete meium constraine systems have evolve to being a vast fiel an obtaine many important evelopments. 2 4,6 20 The purpose of this paper is to exten the Lie symmetry an the Mei symmetry, an their relationships to the continuous meium systems. We construct the Euler Lagrange equations of motion for continuum system an etermine Lie symmetry an Mei symmetry. We also erive the criteria for obtaining Mei conservation laws from Lie symmetry. 2. Equation of motion of continuum system Let u α x i, α =, 2,..., p; i =, 2, 3, 4 be fiel quantities in a continuous meium system where x = t enotes time parameter an x 2 = x, x 3 = y, x 4 = z refer to position parameters. We consier the continuum system enote by its Lagrangian ensity L x i, u α, u α,i where u α,i = /, an the Lagrangian of the system to be L = L x j j = 2, 3, 4. Project supporte by the National Natural Science Founation of China Grant No an the Natural Science Founation of Zhejiang Province of China Grant No. Y Corresponing author. hzshishenyang@sina.com c 200 Chinese Physical Society an IOP Publishing Lt

2 Chin. Phys. B Vol. 20, No The action integral is efine as I = L x k k =, 2, 3, 4. 2 The calculation of the variation of action functional 2 yiels δi = δ = L x k L δu α + L δu α,i Using two relation equations: an x k. 3 δu α,i = δ δu α 4 L δu α,i L δu α equation 3 becomes δ L x k = + L δu α, 5 L L δu α L δu α x k. 6 The quantities x i, i =, 2, 3, 4 are inepenent of each other, therefore the partial ifferential are equivalent to the total ifferential. Aopting the fixe bounary conition in u space δu α = δu α 2 = 0, the last term on the right-han sie of Eq. 6 vanishes. In consequence of the variational principle of Hamilton of continuous coorinates, we obtain the Euler Lagrange equation for continuum system to be L L = 0, 7 where there are as many equations as the ifferent values of α. 3. Lie symmetry of the continuum Lagrange system Let the infinitesimal transformations of continuous coorinates an fiel quantities be x i = x i + εξ i x i, u α, u α = u α + εη α x i, u α, 8 where ε is a group parameter, an ξ i an η α are generators of the infinitesimal transformations. The vector fiel of generators is X 0 = ξ i + η α, 9 which can be extene to the l-th-orer form: X l = ξ i + η α + η α,i u + η 2 α,i α u i 2 α,i i η l α,i i 2...i l, 0 i 2...i l where u α,ii 2...i m m u α / 2... m, i m =, 2, 3, 4, for m =, 2,..., l, enote all the l-th orer partial erivatives of u α with respect to x i. In Eq. 0, there are the following recursions: η α,i = D i η α D i ξ n u α,n, 2 η 2 α,i i 2 = D i2 η α,i D i2 ξ n u α,in, 3 η l α,i i 2...i l where... = D il η l α,i i 2...i l D il ξ n u α,ii 2...i l n, n =, 2, 3, 4, D im = m + u α,im + u α,ni m u m α,n u α,nn 2...n l i m,nn 2...n l 5 enotes the total erivative operator. We now rewrite Eq. 7 as F α x i, u α, u α,i, u α,ii 2,..., u α,ii 2...i l = 0. 6 Applying prolongation 0 to Eq. 7, we obtain X l α F α x i, u α, u α,i, u α,ii 2,..., u α,ii 2...i l = 0. 7 Criterion If the infinitesimal generators ξ i an η α satisfy Eq. 7, then invariance of each equation of motion in expression 7 is the Lie symmetry of the continuum Lagrange system. Equations in expression 7 are thus the etermining equations of Lie symmetry

3 4. Mei conservation theorem Chin. Phys. B Vol. 20, No We perform the ynamical transformation of Lagrangian ensity L an give L = L x i, u α, u α,i = L x i, u α, u α,i + εx L + oε 2 +, 8 where oε 2 + are the secon an the higher orer infinitesimals of L. Substituting the transforme Lagrangian ensity L into Eq. 7 yiels L L = 0. 9 Omit the secon an the higher orer infinitesimals oε 2 +, use Eq. 7 an keep the form of Eq. 7 invariant, then we will have X L X L = X L ξ i + η α u α,n ξ n X L = X L ξ i + ξ i Criterion 2 If the infinitesimal generators ξ i an η α satisfy Eq. 20, then the form invariance of each equation of motion in expression 7 is the Mei symmetry of a continuum system. Equations in expression 20 are thus the etermining equations of Mei symmetry. If the infinitesimal group generators ξ i an η α satisfy Eq. 20, in aition, there exists gauge function G M x i, u α such that the ientity X L ξ i + X X L + G M = 0 2 hols, then the system has the Mei conserve law X L ξ i + η α u α,n ξ n X L + G M = const. 22 Proof Using the total erivative operator 5, equation 22 becomes + G M X L + u α,i X L + u α,ni X L + η α u α,n ξ n X L + X L,n η α u α,n ξ n ξ n u α,ni η α u α,n ξ n X L + η α u α,n ξ n X L + G M = There are two relevant equations since n = i =, 2, 3, 4, i.e., ξ i u α,i X L = ξ n u α,n X L, 24 ξ i u α,ni X L,n = ξ n u α,ni X L, 25 therefore equation 23 reuces to X L ξ i + η α u α,n ξ n X L + G M = X L ξ i X L X L X L + ξ i + η α + η α u α,n ξ n X L + η α u α,n ξ n X L + G M = X L ξ i + X X L + G M. 26 Combining Eq. 26 with Eq. 2, we have the result Eq. 22. Given Criterion, there is the conition for obtaining Mei conservative theorem from Lie symmtry of the continuum system: Criterion 3 If the Lie symmetry generators ξ i an η α satisfy Eq. 2, then the Lie symmetry of the system leas to Mei conserve law Eq

4 5. Example Chin. Phys. B Vol. 20, No We take the transmission line equation as an example to apply this proceure. The Lagrangian ensity of the system is L = 2 Lu2 t 2 C u2 x, 27 where L represents the inuctance an C enotes the capacitance. The equation of motion of the system is Lu tt C u xx = F = 0, 28 Eqs. 0 5 lea to etermining equation of Lie symmetry Lη tt C η xx = Now we will solve Eq. 29 to obtain the Lie symmetry generators. Suppose that the generators are in the forms of ξ = ξ x, t, ξ 2 = ξ 2 x, t, 30 ηx, t, u = fx, tu + gx, t. 3 where we have use Euler Lagrange equation 7. Substituting Eq. 28 into Eq. 7 an using Substituting Eqs. 30 an 3 into Eq. 29 leas to the etermining equation = C 2 g L t 2 u x + t f t 2 u 2 ξ 2 g x f x 2 u 2 ξ 2 x 2 u t + 2 f t 2 ξ 2 t 2 2 f x 2 ξ x 2 u t 2 ξ t u xt + u x 2 ξ 2 x u xt + f 2 ξ 2 u tt t f 2 ξ x u xx, 32 an setting the coefficient of each term of Eq. 32 to be zero yiels ηx, t, u = CC x 2 + L C t 2 + C 2 xt + C 3 u + CC 4 x 2 + L C 4t 2 + C 5 xt + C 6 x + C 7 t + C 8, 33 ξ x, t = C 9 x + L C 0t + C, 34 ξ 2 x, t = CC 0 x + C 9 t + C 2, 35 where C C 2 are constants. As for the Mei conservative law, after substituting Eqs into Mei symmetry etermining equation 2 X L ξ i + X X L + G M = 0, 36 we obtain that ηx, t, u = C 8, 37 ξ x, t = 2 x + C, 38 ξ 2 x, t = 2 t + C 2, 39 then G M x, t, u = Use Eqs an Criteria 2 an 3, we can obtain the Mei conservative law I M = 2 Lu2 t 2 C u2 x 2 x 2 t + C + C 2 + C 6 x + C 7 t + C 8 2 x + C u x 2 t + C 2 u t Lu t C u x = const Conclusions We obtaine four main results in this paper: i the etermining equation 20 of Mei symmetry of the continuum system; ii the conitions of Mei conservative theorems 2 an 22; iii Criterion 3 for obtaining Mei conservative law from Lie symmetry; iv the Lie symmetry generators an the Mei conserve quantity 4 of the transmission line equation. References Marsen J E an Ratiu T S 994 Introuction to Mechanics an Symmetry New York: Springer-Verlag 2 Mei F X 999 Applications of Lie Groups an Lie Algebras to Constraine Mechanical Systems Beijing: Science Press in Chinese 020-4

5 Chin. Phys. B Vol. 20, No Mei F X 2004 Symmetries an Conserve Quantities of Constraine Mechanical Systems Beijing: Beijing Institute of Technology Press in Chinese 4 Lou S K an Zhang Y F 2008 Avances in the Stuy of Dynamics of Constraine Systems Beijing: Science Press in Chinese 5 Armstrong M A 997 Group an Symmetry New York: Springer-Verlag 6 Olver P J 999 Applications of Lie Groups to Differential Equations New York: Springer-Verlag 7 Bluman G W an Anco S C 2004 Symmetry an Integration Methos for Differential Equations New York: Springer-Verlag 8 Haas F an Goeert J 2000 J. Phys. A: Math. Gen Chavarriga J, García I A an Giné J 200 Nonlinearity Aleynikov D V an Tolkachev E A 2003 J. Phys. A: Math. Gen Baker T W an Tavel M A 974 Am. T. Phys Rosen J 972 Ann. Phys Djukić Dj 974 Archives of Mech Sarlet W an Cantrijn F 98 SIAM Rev Bahar L Y an Kanty H G 987 Int. J. Nonlinear Mech Ge W K 2007 Acta Phys. Sin. 56 in Chinese 7 Liu H J, Fu J L an Tang Y F 2007 Chin. Phys Fang J H, Ding N an Wang P 2007 Acta Phys. Sin in Chinese 9 Zheng S W an Jia L Q 2007 Acta Phys. Sin in Chinese 20 Fang J H, Ding N an Wang P 2007 Chin. Phys