- Email: [email protected]
Algebraic structure and Poisson method for a weakly nonholonomic system
THEORETICAL & APPLIED MECHANICS LETTERS 2, 023001 (2011)
Algebraic structure and Poisson method for a weakly nonholonomic system Fengxiang Mei and Huibin Wua) Faculty of Science, Beijing Institute of Technology, Beijing 100081, China
(Received 10 November 2010; accepted 26 January 2011; published online 10 March 2011) Abstract The algebraic structure and the Poisson method for a weakly nonholonomic system are studied. The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed. The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system. An example is given to illustrate c 2011 The Chinese Society of Theoretical and Applied Mechanics. the application of the result. ⃝ [doi:10.1063/2.1102301] Keywords weakly nonholonomic system, algebraic structure, Poisson method Nonholonomic systems are mechanical systems which are subject to nonintegrable differential constraints. Researches on the systems not only have important theoretical significance, but also noticeable value for applications.1–8 There is a special nonholonomic system whose constraint equations contain a small parameter. Such a system is called a weakly nonholonomic system. The “weakly property” means that the system becomes a holonomic system when µ = 0, or its dynamical properties are very close to those of a holonomic system when µ is very small. Initial progress has been obtained about the research on weakly nonholonomic systems, such as Refs. 9–12. Reference 13 discussed the Lie algebra structure and the Poisson method for different kinds of constraint mechanical systems. Following Refs. 9–13 for the weakly nonholonomic system, we will study its algebraic structure and Poisson method, then give its differential equations of motion and their first degree approximation, and furthermore, work on the algebraic structure of the equations and the corresponding Poisson theory for integration. Suppose that the position of a mechanical system is determined by the n generalized coordinates qs (s = 1, 2, . . . , n) , and its motion is subject to the g ideal bilateral linear homogeneous nonholonomic constraints
is nonsingular, then before integrating the equations of motion one can solve λβ from Eqs. (1),(2) as functions of q, q, ˙ t and µ, and Eq. (2) can be written as d ∂L ∂L − = Qs + Λs , dt ∂ q˙s ∂qs
(3)
where Λs = Λs (q, q, ˙ t, µ) = λβ
∂fβ ∂ q˙s
(4)
are generalized constraint forces. Equations (3) are called equations of the holonomic system corresponding to the weakly nonholonomic system (1),(2). It has been proved that the solution of the corresponding holonomic system (3) gives the motion of the weakly nonholonomic system if the initial conditions of motion satisfy constraint Eq. (1).2,14 In order to discuss the approximate solution of the weakly nonholonomic system, we expand generalized constraint forces Λs as a power-series in the parameter µ Λs = Λs0 (q, q, ˙ t)+µΛs1 (q, q, ˙ t)+µ2 Λs2 (q, q, ˙ t)+· · · . (5) Then the first-order approximation of Eqs. (3) is
fβ =q˙ε+β − µBε+β,σ (qs , t)q˙σ = 0,
(1)
where µ is a small parameter; β = 1, 2, . . . , g; σ = 1, 2, . . . , ε; ε = n − g; s = 1, 2, . . . , n. When µ = 0, the constraint Eqs. (1) are holonomic. And then the differential equations of motion of the system can be expressed as ∂L ∂fβ d ∂L − = Qs + λβ , dt ∂ q˙s ∂qs ∂ q˙s
author. Email: [email protected].
(6)
For studying the algebraic structures of Eq. (3) and Eq. (6), we now introduce the generalized momentum to be ps =
(2)
where L = L(q, q, ˙ t) is the Lagrangian of the system, Qs = Qs (q, q, ˙ t) are non-potential generalized forces, λβ are undetermined multipliers. Suppose that the system a) Corresponding
d ∂L ∂L − = Qs + Λs0 + µΛs1 . dt ∂ q˙s ∂qs
∂L , ∂ q˙s
and the Hamiltonian to be H = ps q˙s − L. Then Eq. (3) can be expressed as q˙s =
∂H , ∂ps
p˙s = −
∂H e ′s , +Q ∂qs
(7)
023001-2 F.X. Mei and H.B. Wu
Theor. Appl. Mech. Lett. 2, 023001 (2011)
where
Proposition 1 The necessary and sufficient condition under which I(a, t) = const. is a first integral of Eqs. (10) is
e ′s (q, p, t, µ) = Q e s (q, p, t) + Λ e s (q, p, t, µ), Q e s = Qs (q, q(q, Q ˙ p, t), t),
∂I + I ◦ H = 0, ∂t
e s = Λs (q, q(q, Λ ˙ p, t), t, µ)
which is called the generalized Poisson condition. Proposition 2 The necessary and sufficient condition under which the Hamiltonian is a first integral of Eqs. (7) is
and Eq. (6) can be written as q˙s =
∂H , ∂ps
p˙s = −
(14)
∂H e ′′s , +Q ∂qs
(8) ∂H e ′s q˙s = 0, +Q ∂t
where
(15)
e ′′s (q, p, t, µ) =Qs (q, q(q, Q ˙ p, t), t)+ Λs0 (q, q(q, ˙ p, t), t)+ µΛs1 (q, q(q, ˙ p, t), t).
and that of Eqs. (8) is
Let
Proposition 3 If Eqs. (10) possess a first integral I(a, t) involving time t which is not explicitly contained in S µν and H, then ∂I/∂t, ∂ 2 I/∂t2 , . . . are also first integrals of the system. Proposition 4 If Eqs. (10) possess a first integral I(a, t) involving variable aρ which is not explicitly con2 tained in S µν and H, then ∂I/∂aρ , ∂ 2 I/∂(aρ ) , . . . are also first integrals of the system. According to Propositions 1 and 2, we can judge the integrals of the system. And using Propositions 3 and 4, we can find new integrals from the known ones. In the following, we give an example to illustrate the application of the above results. A weakly nonholonomic system is as follows
{ qµ , (µ = 1, . . . , n), a = pµ−n , (µ = n + 1, . . . , 2n). µ
∂H e ′′s q˙s = 0. +Q ∂t
(9)
Then Eqs. (7) can be written in the contravariant algebra form a˙ µ − S µν
∂H =0 ∂aν
(µ, ν = 1, 2, . . . , 2n),
(10)
where S µν =ω µν + T µν , ( ) 0n×n In×n (ω µν )= , −In×n 0n×n ( (T µν )=
0n×n 0n×n 0n×n (−Ωkk )n×n
e ′s =−Ωsk ∂H , Q ∂pk
L=
f = q˙2 −µtq˙1 = 0. (17)
,
(11)
(12)
Define the derivative of a function A(a) with respect to time t according to Eqs. (10) to be a product ∂A µν ∂H def A˙ = S = A ◦ H, ∂aµ ∂aν
Q1 = Q2 = 0,
)
and Eqs. (8) can also be written in form (10) in which e ′′s = −Ωsk ∂H . Q ∂pk
1 2 2 (q˙ +q˙ ), 2 1 2
(16)
(13)
which satisfies the right and left distributive laws and the scalar laws. So, it possesses a consistent algebraic structure. In general, it has the Lie-admissible algebra structure but not the Lie. The Poisson integration method for conservative holonomic systems can be generalized as follows
Find its integrals by using the Poisson integration method. First of all, we establish Eqs. (10). Equations (2) yield q¨1 = −λµt,
q¨2 = λ.
Solving λ and substituting it into the above equations, we acquire q¨1 = −
µ2 t q˙1 , 1 + µ 2 t2
q¨2 =
µ q˙1 . 1 + µ2 t2
(18)
Take the generalized momentums and the Hamiltonian to be p1 = q˙1 ,
p2 = q˙2 ,
H=
1 2 (p + p22 ). 2 1
respectively. Then Eqs. (7) become q˙1 = p1 ,
q˙2 = p2 ,
e ′1 , p˙1 = Q
e ′2 , p˙2 = Q
(19)
023001-3
Algebraic structure and Poisson method
Theor. Appl. Mech. Lett. 2, 023001 (2011)
where
From Eq. (24), it is easy to see that
e ′1 = − Q
µ2 t p1 , 1 + µ2 t2
e ′2 = Q
µ p1 1 + µ2 t2
and Eqs. (8) give q˙1 = p1 ,
e ′′1 , p˙1 = Q
q˙2 = p2 ,
e ′′2 , p˙2 = Q
(20)
e ′′ = µp1 . Q 2
0 0 (S µν ) = −1 0
0 0 (S µν ) = −1 0
0 1 0 0 µ2 t 0 − 1+µ 2 t2 −1 0
0 0 0 −1
1 0 0 0
0 1 0
(21)
µ p1 1+µ2 t2 p2 ,
0 1 0
(22)
µp1 p2
for Eqs. (20). The generalized Poisson condition (14) becomes ∂I ∂I ∂I ∂I + p1 + p2 + ∂t ∂q1 ∂q2 ∂p1 ( ) µ ∂I p1 = 0 ∂p2 1 + µ2 t2
( −
) µ2 t p1 + 1 + µ2 t2
(23) for Eqs. (19), and ∂I ∂I ∂I ∂I + p1 + p2 + µp1 = 0 ∂t ∂q1 ∂q2 ∂p2
(24)
for Eqs. (20). For Eqs. (19), the Hamiltonian H=
1 2 (p + p22 ) 2 1
(25)
is a first integral which can be judged by condition (23), and also by condition (15). By using condition (23), it can be distinguished that I = µq2 − p1 − µtp2 is a first integral.
(28)
Eqs. (20). Since S µν and H do not contain q1 , by using Proposition 4 and Eq. (28), the following integral can be found ∂I2 = µp1 . ∂q1
(29)
Taking the derivatives of expressions (27) and (29) with respect to t according to Eqs. (19), we obtain
for Eqs. (19), and
I2 = p1 (µq1 − p2 )
I3 =
Furthermore, we have
(27)
are first integrals of the first-order approximation
where e ′′ = 0, Q 1
I1 = p2 − µq1 ,
(26)
I˙1 = O(µ3 ),
I˙3 = O(µ3 ).
It is clear that the first-order approximate solution is very close to the corresponding exact solution. The classical Poisson integration theory is only suitable to mechanical systems which possess the Lie algebra structure. And a weakly nonholonomic system, in general, does not have the Lie algebra structure. The 4 propositions given in this paper are generalizations and applications of the Poisson integration theory to the weakly nonholonomic system. This work was supported by the National Natural Science Foundation of China(10772025, 10932002, 10972031) and the Beijing Municipal Key Disciplines Fund for General Mechanics and Foundation of Mechanics.
1. J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, (AMS, Providence, RI, 1972). 2. F. X. Mei, Foundations of Mechanics of Nonholonomic Systems, (Beijing Institute of Technology Press, Beijing, 1985, (in Chinese)). 3. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and R. M. Murray, Arch. Rat. Mech. Anal. 136, 21 (1996). 4. J. G. Papastavridis, Appl. Mech. Rev. 51, 239 (1998). 5. S. Ostrovskaya, J. Angels, Appl. Mech. Rev. 51, 415 (1998). 6. F. X. Mei, Appl. Mech. Rev. 53, 283 (2000). 7. Y. X. Guo, S. K. Luo, F. X. Mei, Advances in Mechanics 34, 477 (2004)(in Chinese). 8. S. A. Zegzhda, Sh. Kh. Soltakhanov and M. P. Yushkov, Equations of Motion of Nonholonomic Systems and Variational Principles of Mechanics. New kind of Control Problems,(FIMATLIT, Moscow, 2002, (in Russian)). 9. F. X. Mei, J. Beijing Institute of Technology 9, 10 (1989)(in Chinese). 10. F. X. Mei, Chin. Sci. Bull. 38, 281 (1993)(in Chinese). 11. F. X. Mei, J. Beijing Institute of Technology 15, 237 (1995)(in Chinese). 12. W. K. Ge, Acta Phys. Sin. 58, 6729 (2009)(in Chinese). 13. F. X. Mei, Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems, (Science Press, Beijing, 1999, (in Chinese)). 14. V. S. Novoselov, Variational Methods in Mechanics, (LGU Press, Leningrad, 1966, (in Russian)).
Algebraic structure and Poisson method for a weakly nonholonomic system Fengxiang Mei and Huibin Wua) Faculty of Science, Beijing Institute of Technology, Beijing 100081, China
(Received 10 November 2010; accepted 26 January 2011; published online 10 March 2011) Abstract The algebraic structure and the Poisson method for a weakly nonholonomic system are studied. The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed. The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system. An example is given to illustrate c 2011 The Chinese Society of Theoretical and Applied Mechanics. the application of the result. ⃝ [doi:10.1063/2.1102301] Keywords weakly nonholonomic system, algebraic structure, Poisson method Nonholonomic systems are mechanical systems which are subject to nonintegrable differential constraints. Researches on the systems not only have important theoretical significance, but also noticeable value for applications.1–8 There is a special nonholonomic system whose constraint equations contain a small parameter. Such a system is called a weakly nonholonomic system. The “weakly property” means that the system becomes a holonomic system when µ = 0, or its dynamical properties are very close to those of a holonomic system when µ is very small. Initial progress has been obtained about the research on weakly nonholonomic systems, such as Refs. 9–12. Reference 13 discussed the Lie algebra structure and the Poisson method for different kinds of constraint mechanical systems. Following Refs. 9–13 for the weakly nonholonomic system, we will study its algebraic structure and Poisson method, then give its differential equations of motion and their first degree approximation, and furthermore, work on the algebraic structure of the equations and the corresponding Poisson theory for integration. Suppose that the position of a mechanical system is determined by the n generalized coordinates qs (s = 1, 2, . . . , n) , and its motion is subject to the g ideal bilateral linear homogeneous nonholonomic constraints
is nonsingular, then before integrating the equations of motion one can solve λβ from Eqs. (1),(2) as functions of q, q, ˙ t and µ, and Eq. (2) can be written as d ∂L ∂L − = Qs + Λs , dt ∂ q˙s ∂qs
(3)
where Λs = Λs (q, q, ˙ t, µ) = λβ
∂fβ ∂ q˙s
(4)
are generalized constraint forces. Equations (3) are called equations of the holonomic system corresponding to the weakly nonholonomic system (1),(2). It has been proved that the solution of the corresponding holonomic system (3) gives the motion of the weakly nonholonomic system if the initial conditions of motion satisfy constraint Eq. (1).2,14 In order to discuss the approximate solution of the weakly nonholonomic system, we expand generalized constraint forces Λs as a power-series in the parameter µ Λs = Λs0 (q, q, ˙ t)+µΛs1 (q, q, ˙ t)+µ2 Λs2 (q, q, ˙ t)+· · · . (5) Then the first-order approximation of Eqs. (3) is
fβ =q˙ε+β − µBε+β,σ (qs , t)q˙σ = 0,
(1)
where µ is a small parameter; β = 1, 2, . . . , g; σ = 1, 2, . . . , ε; ε = n − g; s = 1, 2, . . . , n. When µ = 0, the constraint Eqs. (1) are holonomic. And then the differential equations of motion of the system can be expressed as ∂L ∂fβ d ∂L − = Qs + λβ , dt ∂ q˙s ∂qs ∂ q˙s
author. Email: [email protected].
(6)
For studying the algebraic structures of Eq. (3) and Eq. (6), we now introduce the generalized momentum to be ps =
(2)
where L = L(q, q, ˙ t) is the Lagrangian of the system, Qs = Qs (q, q, ˙ t) are non-potential generalized forces, λβ are undetermined multipliers. Suppose that the system a) Corresponding
d ∂L ∂L − = Qs + Λs0 + µΛs1 . dt ∂ q˙s ∂qs
∂L , ∂ q˙s
and the Hamiltonian to be H = ps q˙s − L. Then Eq. (3) can be expressed as q˙s =
∂H , ∂ps
p˙s = −
∂H e ′s , +Q ∂qs
(7)
023001-2 F.X. Mei and H.B. Wu
Theor. Appl. Mech. Lett. 2, 023001 (2011)
where
Proposition 1 The necessary and sufficient condition under which I(a, t) = const. is a first integral of Eqs. (10) is
e ′s (q, p, t, µ) = Q e s (q, p, t) + Λ e s (q, p, t, µ), Q e s = Qs (q, q(q, Q ˙ p, t), t),
∂I + I ◦ H = 0, ∂t
e s = Λs (q, q(q, Λ ˙ p, t), t, µ)
which is called the generalized Poisson condition. Proposition 2 The necessary and sufficient condition under which the Hamiltonian is a first integral of Eqs. (7) is
and Eq. (6) can be written as q˙s =
∂H , ∂ps
p˙s = −
(14)
∂H e ′′s , +Q ∂qs
(8) ∂H e ′s q˙s = 0, +Q ∂t
where
(15)
e ′′s (q, p, t, µ) =Qs (q, q(q, Q ˙ p, t), t)+ Λs0 (q, q(q, ˙ p, t), t)+ µΛs1 (q, q(q, ˙ p, t), t).
and that of Eqs. (8) is
Let
Proposition 3 If Eqs. (10) possess a first integral I(a, t) involving time t which is not explicitly contained in S µν and H, then ∂I/∂t, ∂ 2 I/∂t2 , . . . are also first integrals of the system. Proposition 4 If Eqs. (10) possess a first integral I(a, t) involving variable aρ which is not explicitly con2 tained in S µν and H, then ∂I/∂aρ , ∂ 2 I/∂(aρ ) , . . . are also first integrals of the system. According to Propositions 1 and 2, we can judge the integrals of the system. And using Propositions 3 and 4, we can find new integrals from the known ones. In the following, we give an example to illustrate the application of the above results. A weakly nonholonomic system is as follows
{ qµ , (µ = 1, . . . , n), a = pµ−n , (µ = n + 1, . . . , 2n). µ
∂H e ′′s q˙s = 0. +Q ∂t
(9)
Then Eqs. (7) can be written in the contravariant algebra form a˙ µ − S µν
∂H =0 ∂aν
(µ, ν = 1, 2, . . . , 2n),
(10)
where S µν =ω µν + T µν , ( ) 0n×n In×n (ω µν )= , −In×n 0n×n ( (T µν )=
0n×n 0n×n 0n×n (−Ωkk )n×n
e ′s =−Ωsk ∂H , Q ∂pk
L=
f = q˙2 −µtq˙1 = 0. (17)
,
(11)
(12)
Define the derivative of a function A(a) with respect to time t according to Eqs. (10) to be a product ∂A µν ∂H def A˙ = S = A ◦ H, ∂aµ ∂aν
Q1 = Q2 = 0,
)
and Eqs. (8) can also be written in form (10) in which e ′′s = −Ωsk ∂H . Q ∂pk
1 2 2 (q˙ +q˙ ), 2 1 2
(16)
(13)
which satisfies the right and left distributive laws and the scalar laws. So, it possesses a consistent algebraic structure. In general, it has the Lie-admissible algebra structure but not the Lie. The Poisson integration method for conservative holonomic systems can be generalized as follows
Find its integrals by using the Poisson integration method. First of all, we establish Eqs. (10). Equations (2) yield q¨1 = −λµt,
q¨2 = λ.
Solving λ and substituting it into the above equations, we acquire q¨1 = −
µ2 t q˙1 , 1 + µ 2 t2
q¨2 =
µ q˙1 . 1 + µ2 t2
(18)
Take the generalized momentums and the Hamiltonian to be p1 = q˙1 ,
p2 = q˙2 ,
H=
1 2 (p + p22 ). 2 1
respectively. Then Eqs. (7) become q˙1 = p1 ,
q˙2 = p2 ,
e ′1 , p˙1 = Q
e ′2 , p˙2 = Q
(19)
023001-3
Algebraic structure and Poisson method
Theor. Appl. Mech. Lett. 2, 023001 (2011)
where
From Eq. (24), it is easy to see that
e ′1 = − Q
µ2 t p1 , 1 + µ2 t2
e ′2 = Q
µ p1 1 + µ2 t2
and Eqs. (8) give q˙1 = p1 ,
e ′′1 , p˙1 = Q
q˙2 = p2 ,
e ′′2 , p˙2 = Q
(20)
e ′′ = µp1 . Q 2
0 0 (S µν ) = −1 0
0 0 (S µν ) = −1 0
0 1 0 0 µ2 t 0 − 1+µ 2 t2 −1 0
0 0 0 −1
1 0 0 0
0 1 0
(21)
µ p1 1+µ2 t2 p2 ,
0 1 0
(22)
µp1 p2
for Eqs. (20). The generalized Poisson condition (14) becomes ∂I ∂I ∂I ∂I + p1 + p2 + ∂t ∂q1 ∂q2 ∂p1 ( ) µ ∂I p1 = 0 ∂p2 1 + µ2 t2
( −
) µ2 t p1 + 1 + µ2 t2
(23) for Eqs. (19), and ∂I ∂I ∂I ∂I + p1 + p2 + µp1 = 0 ∂t ∂q1 ∂q2 ∂p2
(24)
for Eqs. (20). For Eqs. (19), the Hamiltonian H=
1 2 (p + p22 ) 2 1
(25)
is a first integral which can be judged by condition (23), and also by condition (15). By using condition (23), it can be distinguished that I = µq2 − p1 − µtp2 is a first integral.
(28)
Eqs. (20). Since S µν and H do not contain q1 , by using Proposition 4 and Eq. (28), the following integral can be found ∂I2 = µp1 . ∂q1
(29)
Taking the derivatives of expressions (27) and (29) with respect to t according to Eqs. (19), we obtain
for Eqs. (19), and
I2 = p1 (µq1 − p2 )
I3 =
Furthermore, we have
(27)
are first integrals of the first-order approximation
where e ′′ = 0, Q 1
I1 = p2 − µq1 ,
(26)
I˙1 = O(µ3 ),
I˙3 = O(µ3 ).
It is clear that the first-order approximate solution is very close to the corresponding exact solution. The classical Poisson integration theory is only suitable to mechanical systems which possess the Lie algebra structure. And a weakly nonholonomic system, in general, does not have the Lie algebra structure. The 4 propositions given in this paper are generalizations and applications of the Poisson integration theory to the weakly nonholonomic system. This work was supported by the National Natural Science Foundation of China(10772025, 10932002, 10972031) and the Beijing Municipal Key Disciplines Fund for General Mechanics and Foundation of Mechanics.
1. J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, (AMS, Providence, RI, 1972). 2. F. X. Mei, Foundations of Mechanics of Nonholonomic Systems, (Beijing Institute of Technology Press, Beijing, 1985, (in Chinese)). 3. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and R. M. Murray, Arch. Rat. Mech. Anal. 136, 21 (1996). 4. J. G. Papastavridis, Appl. Mech. Rev. 51, 239 (1998). 5. S. Ostrovskaya, J. Angels, Appl. Mech. Rev. 51, 415 (1998). 6. F. X. Mei, Appl. Mech. Rev. 53, 283 (2000). 7. Y. X. Guo, S. K. Luo, F. X. Mei, Advances in Mechanics 34, 477 (2004)(in Chinese). 8. S. A. Zegzhda, Sh. Kh. Soltakhanov and M. P. Yushkov, Equations of Motion of Nonholonomic Systems and Variational Principles of Mechanics. New kind of Control Problems,(FIMATLIT, Moscow, 2002, (in Russian)). 9. F. X. Mei, J. Beijing Institute of Technology 9, 10 (1989)(in Chinese). 10. F. X. Mei, Chin. Sci. Bull. 38, 281 (1993)(in Chinese). 11. F. X. Mei, J. Beijing Institute of Technology 15, 237 (1995)(in Chinese). 12. W. K. Ge, Acta Phys. Sin. 58, 6729 (2009)(in Chinese). 13. F. X. Mei, Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems, (Science Press, Beijing, 1999, (in Chinese)). 14. V. S. Novoselov, Variational Methods in Mechanics, (LGU Press, Leningrad, 1966, (in Russian)).