Exact Invariants and Adiabatic Invariants of Raitzin s Canonical Equations of Motion for Nonholonomic System of Non-Chetaev s Type

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1 Commun. Theor. Phys. Beiing, China pp c International Academic Publishers Vol. 43, No. 6, June 15, 2005 Exact Invariants and Adiabatic Invariants of Raitzin s Canonical Equations of Motion for Nonholonomic System of Non-Chetaev s Type QIAO Yong-Fen 1,2 and ZHAO Shu-Hong 2 1 Department of Mechanical Engineering and Automation, Zheiang Sci-Tch University, Hangzhou , China 2 Engineeing College of Northeast Agricultural University, Harbin , China Received November 9, 2004; Revised January 18, 2005 Abstract The exact invariants and the adiabatic invariants of Raitzin s canonical equations of motion for the nonholonomic system of non-chetaev s type are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher order adiabatic invariant of mechanical system with the action of a small perturbation, the form of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results. PACS numbers: Kk, Sv Key words: nonholonomic system, Raitzin s canonical equation, symmetry, perturbation, exact invariant, adiabatic invariant 1 Introduction The conservation laws for a dynamical system have great mathematical significance and very deep physical background. Furthermore, the study of them has become one of the most important directions in modern analytical mechanics. At present, some results have been obtained. [111 In 1917, Burgers first proposed adiabatic invariants, which referred to a special kind of Hamilton system. [12 The adiabatic invariants mean that they are almost not changed when the parameter varies very slowly and they play a very important role in the research on quasiintegrability of a mechanical system. In fact, that the parameter varies very slowly is equivalent to the action of a small perturbation, so we perform research using the latter model in this paper. The study of adiabatic invariants has become a popular subect in mechanics, atomic and molecular physics, [13 and some important results were obtained. [1417 An example frictional brake staff of linear nonholonomic constraint of non-chetaev s type was given by V.S. Novoselov in Afterward, differential equation of motion of a system was studied by him. [18 V.V. Rumyatsev pointed out that if using non-inertial servo-system to replace for the example, the nonlinear nonholonomic constraint of non-chetaev s type can be realized. [19 The main character of nonholonomic system of non-chetaev s type is that it cannot directly get equation of virtual displacement by equation of nonholonomic constraint. [3 In this paper, the perturbation to symmetries and adiabatic invariants of Raitzin s canonical equations [2022 for the nonholonomic system of non-chetaev s type are studied. 2 Raitzin s Canonical Equations of Nonholonomic System of Non-Chetave s Type We consider a mechanical system composed of N particles, and its configuration is determined by n generalized coordinates q 1, q 2,..., q n. The motion of the system is subected to the following g ideal bilateral nonholonomic constraints of non-chetaev s type, f β t, q, q = 0, β = 1, 2,..., g. 1 The equations of virtual displacements are F β t, q, qδq = 0, β = 1, 2,..., g; = 1, 2,..., n. 2 As a general rule, F β is nothing to f β / q. When F β = f β q β = 1, 2,..., g; = 1, 2,..., n, 3 the constraints of non-chetaev s type become constraints of Chetaev s type. Thus, the nonholonomic constraints of non-chetaev s type have more generality. The equations of motion of the system can be written in Routh s form, [3 d dt L L = Q λ β F β, q q β = 1, 2,..., g; = 1, 2,..., n, 4 where L is the Lagrangian of the system, Q are the nonpotential generalized forces and λ β are the constrained multipliers. Now, we introduce the Raitzin s canonical variables and function [20 as follows: r = L, s = q, 5 q Rt, r, s = L r q. 6 The proect supported by Natural Science Foundation of Heilongiang Province of China under Grant No. 9507

2 988 QIAO Yong-Fen and ZHAO Shu-Hong Vol. 43 According to the Raitzin s variables 5, the nonholonomic constraints 1 and constrained multipliers λ β can be written as f β t, r, s = f β t, qt, r, s, s = 0, λ β t, r, s = λ β t, qt, r, s, s. 7 Then from the nonholonomic system 4 6, we obtain the Raitzin s canonical equations of the nonholonomic system of non-chetaev s type as follows: s = d, q =, dt r r r = d dt s Q λ β Fβ, = 1, 2,..., n; β = 1, 2,..., g. 8 Here, the symbol is the expression obtained by substituting t, r, s for q, q. Noticing I = δi I t = = { r q δr 3 Infinitesimal Transformations and Exact Invariants We study the variation of the Raitzin s canonical action. Introduce the Raitzin s canonical action I = Ldt = and the infinitesimal transformations R r q dt, 9 t = t ετ 0 t, r, s, q = q εξ 0 t, r, s, r = r εη 0 t, r, s, 10 where ε is an infinitesimal parameter, τ 0, ξ 0, and η0 are infinitesimal generators. Under the infinitesimal transformations 10, the variation of Raitzin s canonical action is [ δs δr q δr r δq dt R r q t t 1 s r r d δq d [ } δq R r q t dt. 11 dt s dt s t = ετ 0, δq = q s t = εξ 0 s τ 0, δr = r ṙ t = εη 0 ṙ τ 0, 12 the formula 11 can be expressed as { I = ε q η 0 ṙ τ 0 r d ξ 0 s τ 0 d [ ξ 0 s τ 0 R r q τ 0} dt. 13 r dt s dt s This is the fundamental formula of the variation of the Raitzin s canonical action 9. Definition 1 For the infinitesimal transformations 10, if the variation of the Raitzin s canonical action 9 satisfies I = [ d dt G0 Q δq dt, 14 where Q = Q t, r, s are generalized non-potential forces, G 0 = G 0 t, r, s is gauge function, then the transformations 10 are called the generalized quasi-symmetrical transformations of the given system. Theorem 1 For the nonholonomic system of non-chetaev s type given by Eqs. 7 and 8, if infinitesimal transformations 10 are generalized quasi-symmetrical transformations and they satisfy the non-chetaev conditions F β ξ 0 s τ 0 = 0, β = 1, 2,..., g; = 1, 2,..., n, 15 then the system possesses the following exact invariant: I 0 = s ξ 0 s τ 0 R r q τ 0 G 0 = const. 16 Proof If the infinitesimal transformations 10 are generalized quasi-symmetrical transformations, then I = [ d dt G0 Q δq dt, where G 0 = εg 0. According to the formula 13, we have { ε q η 0 ṙ τ 0 r d r dt s Q ξ 0 s τ 0 d dt [ r ξ 0 r τ 0 R r q τ 0 G 0} dt = Since infinitesimal transformations 10 satisfy the non-chetaev conditions 10, introducing the Lagrangian multipliers λ β, we have ε λ β Fβ ξ 0 s τ 0 = 0. 18

3 No. 6 Exact Invariants and Adiabatic Invariants of Raitzin s Canonical Equations of Motion for 989 Adding Eqs. 17 and 18, we obtain { ε q η 0 ṙ τ 0 r d r dt s Q λ β Fβ ξ 0 s τ 0 d [ ξ 0 s τ 0 R r q τ 0 G 0} dt = dt s According to Eqs. 8, considering that the integral interval [, t 1 is arbitrary and ε is independent, we have d [ ξ 0 s τ 0 R r q τ 0 G 0 = dt s Integrating this equation, we obtain the exact invariant 16. Theorem 2 For the nonholonomic system of non-chetaev s type given by Eqs. 7 and 8, if the generators τ 0, ξ 0, and η 0 of the infinitesimal transformations 10 and the gauge function G0 satisfy the following conditions λ β Fβ ξ 0 s τ 0 = 0, β = 1, 2,..., g, 21 q η 0 ṙ τ 0 r d r dt s Q ξ 0 s τ 0 d [ ξ 0 s τ 0 R r q τ 0 = dt s Ġ0. 22 then the system possesses the same exact invariants as Eq Perturbation to Symmetries and Adiabtic Invariants First we give the concept of higher-order adiabatic invariants. Definition 2 If I Z t, r, s, ε is a physical quantity including ε in which the highest power is z in a mechanical system, and its derivative with respect to time t is in direct proportion to ε z1, then I Z is called a z-th-order adiabatic invariant of the mechanical system. Suppose the nonholonomic system 7 and 8 of non-chetaev s type is perturbed by small quantities εw, then the equations of motion of the system become s = d, q =, r = d dt r r dt s Q λ β Fβ εw. 23 Due to the action of εw, the primary symmetries and invariants of the system may vary. Suppose the variation is a small perturbation based on the symmetrical transformations of the system without perturbation, and τt, r, s, ξ t, r, s, and η t, r, s express the new generators after being perturbed, then τ = τ 0 ετ 1 ε 2 τ 2, ξ = ξ 0 εξ 1 ε 2 ξ 2, η = η 0 εη 1 ε 2 η 2, 24 and they satisfy q η ṙ τ r d r dt s Q ξ s τ εw ξ s τ d [ ξ s τ R r q τ = Ġ dt s, 25 F β ξ s τ = 0, β = 1, 2,..., g 26 with G in Eq. 25 being a gauge function. Let G = G 0 εg 1 ε 2 G 2 27 Substituting Eqs. 24 and 27 into Eqs. 25 and 26, we obtain q η k ṙ τ k r d r dt s Q ξ k s τ k W ξ k1 s τ k1 d [ ξ k s τ k R r q τ k = dt s Ġk, k = 0, 1, 2,..., z. 28 When k = 0, the condition W = 0 holds, Then we have λ β Fβ ξ k s τ k = Theorem 3 For the nonholonomic system 7 and 23 of non-chetaev s type perturbed by small quantities εw, if the generators τ k t, r, s, ξ kt, r, s, and ηk t, r, s under infinitesimal transformations and the gauge function Gk t, r, s satisfy Eqs. 28 and 29, then I z = ε k[ ξ k s τ k R r q τ k G k 30 s

4 990 QIAO Yong-Fen and ZHAO Shu-Hong Vol. 43 is a z-th-order adiabatic invariant of the mechanical system. Proof Differentiating I z with respect to time t, we have dt = εk[ d ξ k s τ k dt s s ξ k ṡ τ k s τ k Ṙτ k R τ k r q τ k q ṙ τ k q r τ k Ġk. 31 Substituting Eqs. 28 and 29 into the above formula, we obtain dt = εk[ d ξ k s τ k dt s s ξ k ṡ τ k s τ k Ṙτ k R τ k r q τ k q ṙ τ k q r τ k q r η k ṙ τ k r d dt s Q λ β Fβ ξ k s τ k W ξ k1 s τ k1 d ξ k s τ k dt s s ξ k ṡ τ k s τ k Ṙτ k R τ k r q τ k q ṙ τ k r q τ k. 32 Using Eqs. 23, we have dt = εk [εw ξ k s τ k W ξ k1 s τ k1, k = 0, 1, 2,..., z. 33 Expanding the above formula and making summation, we have dt = εz1 W ξ z s τ z. 34 This shows that /dt is in direct proportion to ε z1, so I z is the z-th-order adiabatic invariant of the mechanical system. 5 Inverse Theorem of Perturbation to Symmetries of System Suppose that the nonholonomic system 7 and 23 of non-chetaev s type has a first-order adiabatic invariant as follows: I 1 = ϕ 0 t, r, s ε ϕ 1 t, r, s. 35 In order to make the calculation simple, we use canonical variables 5 and the formula p = / s, then the above formula can be written as I 1 = ϕ 0 t, q, p εϕ 1 t, q, p. 36 So we have di 1 dt = t q ϕ 0 ϕ1 ṗ ε q p t ϕ 1 ϕ 1 q ṗ. 37 q p ṗ = r Q λ β Fβ εw, 38 then we have di 1 dt = t q r q p Q λ [ ϕ1 β Fβ εw ε t ϕ 1 q ϕ 1 r q p Q λ β Fβ εw. 39 From Eqs. 23, we obtain r d dt s Q λ β Fβ εw ξ s τ q η ṙ τ = r By using Eqs. 23, and adding Eqs. 39 and 40, we have t q r q p Q λ [ ϕ1 β Fβ εw ε t ϕ 1 q ϕ 1 r q p Q λ β Fβ εw r d dt s Q λ β Fβ εw ξ s τ q η ṙ τ = ε 2 W ξ 1 s τ r Considering Eqs. 24 and rearranging the above equation, we have t q r q p Q λ d β Fβ εw r ξ 0 s τ 0 dt s Q λ β Fβ εw ξ 0 s τ 0 ε r d dt s Q λ [ β Fβ ξ 1 s τ 1 ϕ1 ε t ϕ 1 q ϕ 1 r q p Q λ β Fβ εw q η ṙ τ Oε = r

5 No. 6 Exact Invariants and Adiabatic Invariants of Raitzin s Canonical Equations of Motion for 991 Now, we seek the generators τ 0, ξ 0, and η0 of the infinitesimal transformations without perturbation. Firstly, separating the terms not containing ε in Eq. 42, then separating the terms containing r and taking their coefficients as zeros, we obtain ξ 0 s τ 0 = p Further let ϕ 0 = s ξ0 s τ 0 R r q τ 0 G From Eqs. 43 and 44, we can obtain τ 0 = ϕ 0 / s / p G 0 R r q, 45 ξ 0 = s ϕ 0 / s / p G R r q p r = εη 0, r = r t t r q i r q q i q i [ r = ε t τ 0 r ξi 0 r q i q ξ i 0 q i τ 0, i therefore η 0 = r t τ 0 r ξi 0 r q i q ξ i 0 q i τ i Further, separating the terms containing ε in Eq. 42, then separating the terms containing r and taking their coefficients as zeros, we have τ 1 = ϕ 1 / s ϕ 1 / p G 1, 48 R r q ξ 1 = s [ ϕ 1 / s ϕ 1 / p G 1 R r q ϕ 1 p, 49 Next, as calculated in the above formula 47, we have η 1 = r t τ 1 r ξi 1 r ξ1 q i q i q i τ i Then we have the following theorem. Inverse Theorem 1 For the nonholonomic system 7 and 23 of non-chetaev s type perturbed by small quantities εw, if it has a first-order adiabatic invariant, then the system has an infinitesimal symmetrical transformations, the items without perturbation and the first-order perturbation items of the infinitesimal generators of the transformations are determined by Eqs and respectively, when the gauge functions G 0 and G 1 are given. 6 Example Suppose that the Lagrangian acting on the system is L = 1 2 q2 1 q 2 2 q 2, 51 the nonholonomic constraint of non-chetaev s type acting on the system is f = q 2 t q 1 = 0, 52 the equation of virtual displacement is δq 1 δq 2 = We study the perturbation to symmetries and adiabatic invariants of the system. The first step is to seek the exact invariants of the system. Equations 4 give q 1 = λ, q 2 1 = λ, 54 p 1 = L q 1 = q 1, p 2 = L q 2 = q From Eqs. 52 and 54, we obtain λ = q 1 1 t Introduce the Raitzin s canonical variables and function as r 1 = L q 1 = 0, r 2 = L q 2 s 1 = q 1, s 2 = q 2, 57 R = L r q = 1 2 s2 1 s Then, the nonholonomic constraint 52 and the constrained multiplier 56 can be written as f = s 2 ts 1 = 0, 59 λ = s 1 1 t λ F 11 ξ 0 1 s 1 τ 0 = s 1 1 t 1 ξ0 1 s 1 τ 0, λ F 12 ξ2 0 s 2 τ 0 = s 1 1 t 1 ξ0 2 s 2 τ 0, 61 then, from Eqs. 21 and 22, we obtain s 1 1 t 1 ξ0 1 s 1 τ 0 s 1 1 t 1 ξ0 2 s 2 τ 0 = 0, 62 q 1 η1 0 q 2 η2 0 ṡ 1 ξ1 0 s 1 τ 0 1 ṡ 2 ξ2 0 s 2 τ 0 d { s 1 ξ1 0 s 1 τ 0 s 2 ξ2 0 s 2 τ 0 dt [ 1 2 s2 1 s 2 2 q 3 τ 0} = Ġ0. 63 Equations 62 and 63 have solutions τ 0 = 0, ξ 0 1 = ξ 0 2 η 0 1 = η 0 2 = 0, G 0 = t. 64 So we can obtain the following exact invariant by Theorem 2, I 0 = s 1 s 2 t = const. 65 The second step is to study the adiabatic invariant of the system. We suppose the system is perturbed by small quantities as follows: εw 1 = ε cos t, εw 2 = ε sin t. 66

6 992 QIAO Yong-Fen and ZHAO Shu-Hong Vol. 43 Then, from Eqs. 28 and 29, we have q 1 η1 1 q 2 η2 1 ṡ 1 ξ1 1 s 1 τ 1 1 ṡ 2 ξ2 1 s 2 τ 1 cos tξ1 0 s 1 τ 0 sin tξ2 0 s 2 τ 0 d { s 1 ξ1 1 s 1 τ 1 dt [ 1 s 2 ξ2 1 s 2 τ 1 2 s2 1 s 2 2 q 3 τ 1} = Ġ1, 67 s 1 1 t 1 ξ1 s 1 τ 1 s 1 1 t 1 ξ1 2 s 2 τ 1 = The solutions of Eqs. 67 and 68 are τ 1 = 0, ξ 1 1 = ξ 1 2 η 1 1 = η 1 2 = So we can obtain the following first-order adiabatic invariant by Theorem 3, I 1 = s 1 s 2 t ε[s 1 s 2 t sin t cos t = const.70 Furthermore we can obtain more higher-order adiabatic invariants. The third step is to study the inverse problem. Suppose the system is perturbed by the small quantities 66, and there exists a first-order adiabatic invariant 70. ϕ 0 = s 1 s 2 t, ϕ 0 = p 1 p 2 t, s 1 = s 1, s 2 = s 2, p 1 p 2 R r q = 1 2 s2 1 s 2 2 q 2. Substituting the above formulas into Eqs. 45 and 46, when G 0 = t, we have τ 0 = 0, ξ 0 1 = ξ 0 2 = Further, substituting formulas 71 into Eq. 47, and considering Eqs. 57, we obtain With the same method, we have η 0 1 = η 0 2 = ϕ 1 = s 1 s 2 t sin t cos t, ϕ 1 = p 1 p 2 t sin t cos t, s 1 = s 1, ϕ 1 p 1 s 2 = s 2, ϕ 1 p 2 R r q = 1 2 ms2 1 s 2 2 q 2. When G 1 = t sin t cos t, from the formulas and considering Eqs. 57, we have τ 1 = 0, ξ 1 1 = ξ 1 2 η 1 1 = η 1 2 = References [1 Y.F. Qiao, J. Meng, and S.H. Zhao, Chin. Phys [2 Z.P. Li, Acta Phys. Sin in Chinese. [3 F.X. Mei, Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems, Science Press, Beiing 1999 p. 90 in Chinese. [4 Y.F. Qiao and S.H. Zhao, Acta Phys. Sin in Chinese. [5 S.H. Zhao, Y.F. Qiao, and Y.S. Ma, J. Jiangxi Normal University in Chinese. [6 F.X. Mei, Chin. Phys [7 Y.F. Qaio, Y.L. Zhang, and G.C. Han, Acta Phys. Sin in Chinese. [8 S.K. Luo, Commun. Theor. Phys. Beiing, China [9 S.K. Luo and L.Q. Jia, Commun. Theor. Phys. Beiing, China [10 S.K. Luo, Commun. Theor. Phys. Beiing, China [11 Y.F. Qiao, S.H. Zhao, and R.J. Li, Chin. Phys [12 J.M. Burgers, Ann Phys. Lpz [13 V.N. Ostrovsky and N.V. Prudov, J. Phys. B: At. Mol. Opt. Phys [14 Y.Y. Zhao and F.X. Mei, Acta Mech. Sin in Chinese. [15 X.W. Chen, R.C. Zhang, and F.X. Mei, Acta Mech. Sin [16 X.W. Chen and F.X. Mei, Chin. Phys [17 X.W. Chen and Y.M. Li, Chin. Phys [18 F.X. Mei, Research on Nonholonomic Dynamics, Beiing Institute of Technology Press, Beiing 1987 p. 277 in Chinese. [19 F.X. Mei, R.H. Wu, and Y.F. Zhang, Acta Mech. Sin [20 C.M. Raitzin, An. Soc. Cient Argent [21 Y.F. Qiao, Acta Mech. Sin in Chinese. [22 Y.F. Qiao, Y.L. Zhang, and S.H. Zhao, Acta, Phys. Sin in Chinese.