Perturbation to Symmetries and Adiabatic Invariants of Nonholonomic Dynamical System of Relative Motion

Transcription

1 Commun. Theo. Phys. Beijing, China) 43 25) pp c Intenational Academic Publishes Vol. 43, No. 4, Apil 15, 25 Petubation to Symmeties and Adiabatic Invaiants of Nonholonomic Dynamical System of Relative Motion CHEN Xiang-Wei, WANG Ming-Quan, and WANG Xin-Min Depatment of Physics, Shangqiu Teaches College, Shangqiu 476, China Received Mach 9, 24; Revised Septembe 1, 24) Abstact Bed on the theoy of symmeties and conseved quantities, the exact invaiants and adiabatic invaiants of nonholonomic dynamical system of elative motion ae studied. The petubation to symmeties fo the nonholonomic dynamical system of elative motion unde small excitation is discussed. The concept of high-ode adiabatic invaiant is pesented, and the fom of exact invaiants and adiabatic invaiants well the conditions fo thei existence ae given. Then the coesponding invese poblem is studied. PACS numbes: 2.2.Sv, 2.2.Qs, 45.2.Jj Key wods: nonholonomic dynamical system of elative motion, petubation, exact invaiant, adiabatic invaiant 1 Intoduction People have paid moe and moe attention to the eseach of symmeties and conseved quantities of mechanical systems and many esults have spung up duing ecent yeas. [1 12] These esults develop the knowledge of people to the essence of mechanical system. In 1917, Buges fistly poposed adiabatic invaiants which w efeed to a special type of Hamilton system. [13] The adiabatic invaiants mean that they ae almost not changed when the paamete vaies vey slowly and they play a vey impotant ole in the eseach into qui-integability of mechanical system. In fact, if the paamete vaies vey slowly then it is equivalent to the action of a small petubation, we conduct ou eseach unde the action of a small petubation in this pape. Owing to the development of moden science and technology, the study on dynamics of complicated mechanical systems becomes moe and moe impotant. [14] The motion of these complicated systems includes the motion of the caie and the motion of the caied caie elative to the caie. The motion of complicated systems can be studied in the inetial coodinate system, also in the mobile coodinate system closely linked to the caie. The study on dynamics of elative motion of mechanical system by the method of analytical mechanics not only eaches unity in expession but also shows supeioity fo the complicated systems. The study of adiabatic invaiants h become a popula subject in mechanics, [15 18] atomic and molecula physics, [19] etc. In this pape, we discuss the adiabatic invaiants of nonholonomic dynamical system of elative motion fom the view of symmeties and conseved quantities. Fist, we constuct the equations of the nonholonomic dynamical system of elative motion, and popose the concept of highe ode adiabatic invaiants. Second, we pove the conditions fo the existence of the exact invaiants and adiabatic invaiants, and give thei foms. Thid, we study the invese poblems of the petubation to symmeties of the system. Finally, we pesent an example to illustate these esults. 2 Dynamical Equations of Relative Motion Suppose that the system be composed of a caie and a caied caie. The velocity v of the be point O and the angula velocity ω of the caie ae the known functions of time. The caied caie is composed of N paticles, and the configuation of the system is detemined by the n genealized coodinates q s s = 1,..., n). Suppose the motion of the caied caie does not affect the oiginal motion law of the caie. The dynamical equations of elative motion of the holonomic mechanical system can be witten in the fom [14] d T T = Q s V + V ω ) + Qs ω + Γ s dt s = 1,..., n), 1) whee T is the elative motion kinetic enegy of the system, Q s genealized foces, and V potential enegy of unifom foce field, V = Mv + ω v ) c, 2) whee M is the total ms of the system, c the adius vecto of the ms cente in the efeence system of motion, and V ω the potential enegy of centifugal foce, V ω = 1 2 ω θ ω, 3) The poject suppoted by National Natual Science Foundation of China unde Gant No ; the Natual Science Foundation of Henan Povince of China unde Gant No

2 578 CHEN Xiang-Wei, WANG Ming-Quan, and WANG Xin-Min Vol. 43 whee θ is the inetia tenso in the point O of the system, Qs ω the genealized inetial foce of gyation, Q ω s = ω m i i) i. 4) Γ s is the genealized gyoscopic foce ) i Γ s = γ sk q k, γ sk = 2ω m i i. 5) q k The genealized foce Q s is divided into the potential pat Q s and the nonpotential pat Q s, i.e. Let Q s = Q s + Q s, Q s = V. 6) L = T V V V ω, 7) then equation 1) can be witten in the fom s + Qs ω + Γ s s = 1,..., n), 8) dt Suppose the system is subjected to g ideal nonholonomic constaints of Chetaev type, f β t, q, q) = β = 1,..., g). 9) Limitations of the nonholonomic constaints 9) to the vitual displacements ae shown δq s = β = 1,..., g). 1) So the equations of motion of the system can be witten s + Q ω s + Γ s + λ β dt s = 1,..., n), 11) whee λ β is constaint multiplie. Suppose that the system is not singula, i.e. 2 L ) deth sk ) = det. 12) Befoe we integate the diffeential equations of motion, we can obtain λ β fom Eqs. 9) and 11), and expess it a function of t, q, and q. Thus, equation 11) can be expessed whee s + Qs ω + Γ s + Λ s dt s = 1,..., n), 13) Λ s = Λ s t, q, q) = λ β. 14) Equations 13) ae called the equations of motion of holonomic system coesponding to the nonholonomic system 9) and 11). All the genealized acceleations can be solved fom Eqs. 13), q s = α s t, q, q) s = 1,..., n). 15) 3 Symmeties and Exact Invaiants Intoducing the infinitesimal tansfomations of goups of time and genealized coodinates, t = t + t, q st ) = q s t) + q s s = 1,..., n), 16) and thei expansions t = t + ετ t, q, q), q st ) = q s t) + εξ st, q, q), s = 1,..., n), 17) whee ε is an infinitesimal paamete, τ and ξs ae infinitesimal geneatos. Intoduce an infinitesimal geneato vecto X ) = τ t + ξ s, 18) and its fist polongation X 1) = X ) + ξ s q s τ ). 19) The invaiance of Eq. 15) unde the infinitesimal tansfomations 17) leads to the following detemining equations: ξ s q s τ 2 τ α s = X 1) α s ), s = 1,..., n). 2) Definition 1 [14] If the geneatos τ and ξs satisfy the detemining equations 2), then the coesponding tansfomations ae called the Lie symmetical tansfomations, and the coesponding symmeties ae called the Lie symmeties. The invaiance of constaints equation 9) unde the infinitesimal tansfomations 17) leads to the following estiction equations: X 1) f β t, q, q)) = β = 1,..., g). 21) Definition 2 If the geneatos τ and ξ s satisfy the detemining equations 2) and the estiction equations 21), then the coesponding symmeties ae called the weak Lie symmeties. The limitation equations 1) to the vitual displacement added on the infinitesimal tansfomation geneatos leads to the following additive estiction equations fo the geneato: ξ s q s τ ) = β = 1,..., g). 22) Definition 3 If the geneatos τ and ξs satisfy the detemining equations 2), the estiction equations 21) and the additive estiction equations 22), then the coesponding symmeties ae called the stong Lie symmeties. Poposition 1 Fo the infinitesimal geneatos τ and ξs satisfying Eq. 2), if thee exists a gauge function G = G t, q, q) satisfying the following stuctue equation: L τ + X 1) L ) + Q s + Q ω s + Γ s + Λ s )ξ s q s τ )

3 No. 4 Petubation to Symmeties and Adiabatic Invaiants of Ġ =, 23) then the nonholonomic dynamical system of elative motion h the following exact invaiant of Lie symmeties: Poof I = L τ + L ξ s q s τ ) + G = const. 24) di dt = L τ + L τ + L ξ s q s τ q s τ ) + ξs q s τ ) dt L τ X 1) L ) s + Q ω s + Γ s + Λ s )ξ s q s τ ) = ξ s q s τ ) L ) s Qs ω Γ s Λ s =. dt Similaly, we can pove the following popositions. Poposition 2 Fo the infinitesimal geneatos τ and ξs satisfying the detemining equations 2) and the estiction equations 21), if thee exists a gauge function G = G t, q, q) satisfying the stuctue equation 23), then the nonholonomic dynamical system of elative motion h the exact invaiant of weak Lie symmeties Eq. 24). Poposition 3 Fo the infinitesimal geneatos τ and ξs satisfying the detemining equations 2), the estiction equations 21), and the additive estiction equations 22), if thee exists a gauge function G = G t, q, q) satisfying the following stuctue equation: L τ + X 1) L ) + Q s + Q ω s + Γ s )ξ s q s τ ) + Ġ =, 25) then the nonholonomic dynamical system of elative motion h the exact invaiant of stong Lie symmeties Eq. 24). 4 Petubation to Symmeties and Adiabatic Invaiants Fistly we give the concept of high-ode adiabatic invaiants. Definition 4 If I z t, q, q, ε) is a physical quantity including ε with the highest powe z in a mechanical system, and its deivative with espect to time t is in diect popotion to ε z+1, then I z is called a z-th ode adiabatic invaiant of the mechanical system. Suppose the nonholonomic dynamical system of elative motion coesponding to Eq. 13) is petubed by small quantities εw s, then the equations of motion of the system become s + Qs ω + Γ s + Λ s + εw s, dt s = 1,..., n). 26) Due to the action of εw s, the pimay symmeties and invaiants of the system may vay. Suppose the vaiation is a small petubation to the symmetical tansfomation of the system without petubation, and τt, q, q) and ξ s t, q, q) expess the geneatos of time and space espectively afte being petubed, then and they satisfy whee τ = τ + ετ 1 + ε 2 τ 2 +, ξ s = ξ s + εξ 1 s + ε 2 ξ 2 s +, 27) L τ + X 1) L ) + Q s + Q ω s + Γ s + Λ s )ξ s q s τ) + εw s ξ s + q s τ) + Ġ =, 28) X 1) = X ) + ξ s q s τ) = τ t + ξ s + q ξ s q s τ). s And G in Eq. 28) is a gauge function. Let G = G + εg 1 + ε 2 G 2 +, 29) X 1)k = τ k t + ξk s + q ξ s k q s τ k ). 3) s Substituting Eqs. 3) and 29) into Eq. 28), we obtain L τ k + X 1)k L ) + Q s + Q ω s + Γ s + Λ s )ξ k s q s τ k ) + W s ξ k 1 s + q s τ k 1 ) + Ġk =, k =, 1, 2,..., z), 31) if k =, the condition W s = holds. Poposition 4 Fo the dynamical systems of elative motion petubed by small quantities εw s, denote the gauge function by G k t, q, q), if the geneatos τ k t, q, q) and ξs k t, q, q) unde infinitesimal tansfomations satisfy Eq. 31), then I z = ε k[ L τ k + L ξ k s q s τ k ) + G k], k =, 1,..., z) 32) is a z-th ode adiabatic invaiant of the mechanical system. Poof dt = εk[ L τ k + L τ k + L ξ k s q s τ k q s τ k ) + ξs k q s τ k ) L τ k X 1)k L ) dt

4 58 CHEN Xiang-Wei, WANG Ming-Quan, and WANG Xin-Min Vol. 43 s + Q ω s + Γ s + Λ s )ξ k s q s τ k ) ] W s ξs k 1 q s τ k 1 ) = ε k[ L ) s Qs ω Γ s Λ s dt ] ξs k q s τ k ) W s ξs k 1 q s τ k 1 ). Using Eqs. 26), we have dt = εk[ εw s ξ k s q s τ k ) W s ξ k 1 s q s τ k 1 ) ]. Expanding the above fomula and making summation, we have dt = εz+1 W s ξ z s q s τ z ). 33) This shows that /dt is in diect popotion to ε z+1, so I z is a z-th ode adiabatic invaiant of the mechanical system. 5 Invese Poblem Suppose that the system h a fist ode adiabatic invaiant follows: I 1 = λ t, q, q) + ελ 1 t, q, q), 34) diffeentiating this with espect to t, we have di 1 dt = λ t + λ q s + λ ) q s [ λ1 + ε t + λ1 q s + λ )] 1 q s. 35) Multiplying Eq. 35) by ξ s q s τ, and taking the summation ove s, we have ξ s q s τ) L ) Q s Qs ω Γ s Λ s εw s =. dt 36) Adding Eqs. 35) and 36), and consideing Eqs. 33), we have λ t + λ q s + λ ) [ λ1 q s + ε t + λ1 q s + λ )] 1 q s + ξ s q s τ) s dt ) Qs ω Γ s Λ s εw s = ε 2 W s ξ 1 s q s τ 1 ), 37) whee the expessions of τ and ξ s ae shown in Eqs. 27). We seek the geneatos τ and ξs of the infinitesimal tansfomations without petubation. Sepaating the tems not containing ε, the tems containing q k in Eq. 37) and taking the coefficients of q k zeos, we have λ 2 L ξk q k τ ) = k = 1,..., n). 38) Denoting the Hess matix by 2 L / = H sk, h sk = H sk ) 1, the above equations become Let ξ k q k τ = h sk λ. 39) λ = L τ + L ξ k q k τ ) + G. 4) Substituting it into Eqs. 39), we can obtain the items without petubation: τ = L 1 λ λ h sk G ), ξk λ = h sk + q k L 1 λ h sk λ G ). 41) Futhe we can obtain the esults of petubation items of geneato functions unde small distubance: τ 1 = L 1 λ 1 λ 1 h sk G 1), ξk 1 λ 1 = h sk + q k L 1 λ 1 h sk λ 1 G 1). 42) 6 An Illustated Example Conside a dynamical system of elative motion T = 1 2 q2 1 + q q 2 3), V =, V ω = 1 2 ω2 q q 2 2), Γ 1 = 2ω q 2, Γ 2 = 2ω q 1, Γ 3 =, Q s =, Q ω s =, s = 1, 2, 3). 43) Suppose the system is subjected to a nonholonomic constaint f = q 2 q 3 q 1 =. 44) Ty to study the exact invaiants and adiabatic invaiants of the system. Fistly, we seek the exact invaiant. Equations 11) give the dynamical equations of elative motion of the system q 1 ω 2 q 1 = 2ω q 2 λq 3, q 2 ω 2 q 2 = 2ω q 1 + λ, q 3 =. 45) Fom Eqs. 44) and 45), we can obtain the constaint multiplie: λ = 1 { ω 2 } 2 q 3 q 1 q 2 ) + 2ω q 1 + q 3 q 2 ) + q 1 q 3. 46) Substituting Eq. 46) into Eq. 45), we obtain q 1 = 1 { ω 2 } 2 q 1 + q 2 q 3 ) q 3 q 1 q 3 = α1, q 2 = 1 { ω 2 2 q 2 q3 2 } + q 3 q 1 ) + q 1 q 3 = α2, q 3 = = α 3. 47)

5 No. 4 Petubation to Symmeties and Adiabatic Invaiants of 581 The detemining equations 2) gives ξ 1 q 1 τ 2 τ α 1 = X 1) α 1 ), with solutions ξ 2 q 2 τ 2 τ α 2 = X 1) α 2 ), ξ 3 q 3 τ =, 48) τ = 1, ξ 1 = ξ 2 = ξ 3 =. 49) They coespond to the Lie symmeties of the system. The estiction equations 21) give ξ 2 q 2 τ q 3 ξ 1 q 1 τ ) q 1 ξ 3 =. 5) The additive estiction equations 22) give ξ 2 q 2 τ q 3 ξ 1 q 1 τ ) =. 51) Obviously, the geneatos 49) satisfy the Eqs. 5) and 51), so they coespond to the stong Lie symmeties of the system. The stuctue equation 23) gives 2ω q 2 λq 3 ) q 1 2ω q 1 + λ) q 2 + Ġ =, 52) which h a solution G =. 53) So we can obtain the following exact invaiant I = 1 2 q2 1 + q q 2 3) ω2 q q 2 2) = const., 54) which is a genealized enegy integal of the system. Next we study the adiabatic invaiants of the system. Suppose the system is petubed by small quantities εw 1 = εq 1, εw 2 = εq 2, εw 3 = εq 3. 55) The stuctue equation 31) gives 2ω q 2 λq 3 ) q 1 2ω q 1 + λ) q 2 + W 1 ξ 1 q 1 τ ) It h solutions +W 2 ξ 2 q 2 τ ) + W 3 ξ 3 q 3 τ ) + Ġ1 =.56) τ 1 = 1, ξ 1 1 = ξ 1 2 = ξ 1 3 =, G 1 = 1 2 q q q ) So we can obtain the following fist ode adiabatic invaiant by poposition 4: I 1 = 1 2 q2 1 + q q 2 3) ω2 q q 2 2) [ + ε 1 2 q2 1 + q q 3) ω2 q1 2 + q2) ] 2 q2 1 + q2 2 + q3) 2. 58) Moeove we can futhe obtain highe-ode adiabatic invaiants. Refeences [1] B.D. Vujanovic, Int. J. Non-linea Mech ) 783. [2] M. Lutzky, J. Phys A: Math. Gen ) 973. [3] M. Lakshmanan and V.M. Senthil, J. Phys A: Math. Gen ) [4] F.X. Mei, Y.F. Zhang, and M. Shang, Mech. Res. Commun ) 7. [5] Y.X. Guo, Chin. Phys. 1 21) 181. [6] R.C. Zhang, X.W. Chen and F.X. Mei, Chin. Phys. 1 21) 12. [7] F.X. Mei, Chin. Phys. 1 21) 177. [8] Y.F. Qiao, Acta Phys. Sin. 5 21) 811 in Chinese). [9] S.K. Luo, Chin. Phys ) 841. [1] S.K. Luo and J.L. Cai, Chin. Phys ) 357. [11] S.K. Luo, Commun. Theo. Phys. Beijing, China) 39 22) 257. [12] S.K. Luo, Commun. Theo. Phys. Beijing, China) 4 23) 265. [13] J.M. Buges, Annalen de Physik ) 195. [14] F.X. Mei, The application of Lie goups and Lie algeba to constained mechanical systems, Science Pess, Beijing 1999) in Chinese). [15] J.H. Hu, SIAM J. Appl. Math ) 322. [16] Y.Y. Zhao and F.X. Mei, Acta Mech. Sin ) 27 in Chinese). [17] X.W. Chen and F.X. Mei, Chin. Phys. 9 2) 721. [18] X.W. Chen and Y.M. Li, Chin. Phys ) 936. [19] V.N. Ostovsky and N.V. Pudov, J. Phys. B: Atomic, Molecula and Optical Phys ) 4435.