Transactions of Nanjing University of Aeronautics & Astronautics
+Advanced Search
引
×

分享给微信好友或者朋友圈

使用微信“扫一扫”功能。

Discrete Fractional Lagrange Equations of Nonconservative Systems

  • SONG Chuanjing1

    Affiliation： School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, P. R. China

    ×
  • ZHANG Yi2

    Affiliation： College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215009, P. R. China

    Role：Corresponding author

    Email：zhy@mail.usts.edu.cn.

    Profile：E-mail address:zhy@mail.usts.edu.cn.

    ×

1. School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, P. R. China； 2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215009, P. R. China

CLC: O316

DOI:http://dx.doi.org/10.16356/j.1005⁃1120.2019.01.017

无

中图分类号: O316

DOI:http://dx.doi.org/10.16356/j.1005⁃1120.2019.01.017

参考文献 1
FORTT . The calculus of variations applied to Nörlund’s sum[J]. Bulletin of the American Mathematical Society, 1937, 43(12):885-887.
查找原文
参考文献 2
ANASTASSIOUG A . Discrete fractional calculus and inequalities[J]. Mathematical and Computer Modelling, 2009, 51(5/6):562-571.
查找原文
参考文献 3
CHENF L, LUOX N, ZHOUY . Existence results for nonlinear fractional difference equation[J]. Advances in Difference Equations, 2011, 2011(1):1-12.
查找原文
参考文献 4
KELLEYW G, PETERSONA C . Difference equations[M]. Boston: Academic Press, 1991.
查找原文
参考文献 5
HILSCHERR, ZEIDANV . Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: Survey [J]. Journal of Difference Equations & Applications, 2005, 11(9):857-875.
查找原文
参考文献 6
BALEANUD . New applications of fractional variational principles[J]. Reports on Mathematical Physics, 2008, 61(2):199-206.
查找原文
参考文献 7
BALEANUD, DEFTERLIO, AGRAWALO P . A central difference numerical scheme for fractional optimal control problems[J]. Journal of Vibration & Control, 2009, 15(4):583-597.
查找原文
参考文献 8
BASTOSN R O, FERREIRAR A C, TORRESD F M . Necessary optimality conditions for fractional difference problems of the calculus of variations[J]. Discrete & Continuous Dynamical Systems, 2011, 29(2):417-437.
查找原文
参考文献 9
RABEIE M, NAWAFLEHAK I, HIJJAWIAR S, et al . The Hamilton formalism with fractional derivatives [J]. Journal of Mathematical Analysis & Applications, 2007, 327(2):891-897.
查找原文
参考文献 10
RABEIE M, TARAWNEHD M, MUSLIHS I, et al . Heisenberg’s equations of motion with fractional derivatives[J]. Journal of Vibration & Control, 2007, 13(9/10):1239-1247.
查找原文
参考文献 11
RIEWEF . Nonconservative Lagrangian and Hamiltonian mechanics[J]. Physical Review E, 1996, 53(2):1890-1899.
查找原文
参考文献 12
RIEWEF . Mechanics with fractional derivatives[J]. Physical Review E, 1997, 55(3):3581-3592.
查找原文
参考文献 13
KLIMEKM . Fractional sequential mechanics-models with symmetric fractional derivative [J]. Czechoslovak Journal of Physics, 2001, 51(12):1348-1354.
查找原文
参考文献 14
AGRAWALO P . Formulation of Euler-Lagrange equations for fractional variational problems[J]. Journal of Mathematical Analysis & Applications, 2002, 272(1):368-379.
查找原文
参考文献 15
AGRAWALO P . Fractional variational calculus and the transversality conditions[J]. Journal of Physics A: Mathematical General, 2006, 39(33):10375-10384.
查找原文
参考文献 16
AGRAWALO P . Fractional variational calculus in terms of Riesz fractional derivatives[J]. Journal of Physics A: Mathematical and Theoretical, 2007, 40(24):6287-6303.
查找原文
参考文献 17
AGRAWALO P, MUSLIHS I, BALEANUD . Generalized variational calculus in terms of multi-parameters fractional derivatives[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(12):4756-4767.
查找原文
参考文献 18
ATANACKOVIĆT M, KONJIKS, PILIPOVIĆS . Variational problems with fractional derivatives: Euler-Lagrange equations[J]. Journal of Physics A: Mathematical and Theoretical, 2011, 41(9):1937-1940.
查找原文
参考文献 19
ALMEIDAR, TORRESD F M . Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(3):1490-1500.
查找原文
参考文献 20
ALMEIDAR . Fractional variational problems with the Riesz-Caputo derivative[J]. Applied Mathematics Letters, 2012, 25(2):142-148.
查找原文
参考文献 21
ZHOUY, ZHANGY . Fractional Pfaff-Birkhoff principle and Birkhoff’s equations in terms of Riesz fractional derivatives[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2014, 31(1): 63-69.
查找原文
参考文献 22
ZHANGY, LONGZ X . Fractional action-like variational problem and its Noether symmetries for a nonholonomic system[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2015 32(4): 380-389.
查找原文
参考文献 23
BASTOSN R O . Fractional calculus on time scales [D]. Aveiro: University of Aveiro, 2012.
查找原文
参考文献 24
ATICIF M, ELOEP W . Initial value problems in discrete fractional calculus[J]. Proceedings of the American Mathematical Society, 2009, 137(3):981-989.
查找原文
参考文献 25
SRIVASTAVAH M, OWA S . Univalent functions, fractional calculus, and their applications[M]. Ellis Horwood: Halsted Press, 1989:139-152.
查找原文
参考文献 26
HILGERS . Ein Maßkettenkalkiil mit Anwendung auf Zentrumsmannigfaltigkeiten[D]. Würzburg: Universität Würzburg, 1988. (in German)
查找原文
参考文献 27
BOHNERM, PETERSONA C . Dynamic equations on time scales―An introduction with applications[M]. Boston: Birkhäuser, 2001.
查找原文
参考文献 28
CAIP P, FUJ L, GUOY X . Noether symmetries of the nonconservative and nonholonomic systems on time scales[J]. Science China: Physics, Mechanics & Astronomy, 2013, 56(5):1017-1028.
查找原文
参考文献 29
MEIF X . The foundation of mechanics of nonholonomic system[M]. Beijing: Beijing Institute of Technology Press, 1985. (in Chinese)
查找原文
目录 contents

    Abstract

    In order to study discrete nonconservative system, Hamilton’s principle within fractional difference operators of Riemann-Liouville type is given. Discrete Lagrange equations of the nonconservative system as well as the nonconservative system with dynamic constraint are established within fractional difference operators of Riemann-Liouville type from the view of time scales. Firstly, time scale calculus and fractional calculus are reviewed. Secondly, with the help of the properties of time scale calculus, discrete Lagrange equation of the nonconservative system within fractional difference operators of Riemann-Liouville type is presented. Thirdly, using the Lagrange multipliers, discrete Lagrange equation of the nonconservative system with dynamic constraint is also established. Then two special cases are discussed. Finally, two examples are devoted to illustrate the results.

    摘要

    暂无

  • 0 Introduction

    In 1937, Fort[1] first introduced the theory for the discrete calculus of variations. Based on this theory, fractional difference operators within Caputo sense were established and used to solve some difference equations[2,3]. Besides, some important results of discrete calculus of variations were summarized in Ref.[4]. Considering the useful applications of the discrete analogues of differential equations[4,5], and intense investigations on the continuous fractional calculus of variations[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22], Bastos[23] started a fractional discrete-time theory of the calculus of variations in 2012. He introduced the fractional difference operators of Riemann-Liouville type on the basis of Refs.[24,25], and achieved the fractional discrete Euler-Lagrange equations. In particular, when α=1 , the classical discrete results of the calculus of variations can be obtained.

    In this paper, we establish discrete Lagrange equations of the nonconservative system and the nonconservative system with dynamic constraint within fractional difference operators of Riemann-Liouville type. We use some properties of time scale calculus for convenience. Time scale T, which is an arbitrary nonempty closed subset of the real numbers, was introduced by Hilger in 1988[26]. It follows from the definition that time scale calculus has the features of unification and extension. From some properties of time scale T, we can obtain the corresponding properties for the continuous analysis when letting T=R . Similarly, we can obtain the corresponding properties for the discrete analysis when letting T=Z . Apart from R and Z, T has many other values, for instance, T=qN0 ( q>1 ). We mainly use the properties of time scale calculus by letting T=Z in this paper.

  • 1 Time Scale Calculus and Fractional Calculus

    We briefly review time scale calculus and fractional calculus. Refs.[23,27-28] provide more details.

    A time scale T is an arbitrary nonempty closed subset of the real number set. Hence, the integer set Z and the real number set R are the special cases of T.

    Let T be a time scale, then

    (1) The mapping σ:T→T , σ(t)=infs∈T:s>t is called the forward jump operator.

    (2) The mapping ρ:T→T , ρ(t)=sups∈T:s<t is called the backward jump operator.

    (3) the mapping θ:T→[0,∞) , θ(t)=σ(t)-t is called the forward graininess function.

    (4) Tκ=T\(ρ(supT),supT] when supT<∞ ; Tκ=T when supT=∞ .

    (5) Let f:T→R , t∈Tκ ,if for any ε>0 , there exists N=(t-δ,t+δ)⋂T for some δ>0 such that (f(σ(t))-f(ω)_-fΔ(t)(σ(t)-ω)≤εσ(t)-ω for all ω∈N , then fΔ(t) is called the delta derivative of f at t .

    When T=R , σ(t)=ρ(t)=t , θ(t)=0 , fΔ(t)=f˙(t) . When T=Z , σ(t)=t+1 , ρ(t)=t-1 , θ(t)=1 , fΔ(t)=f(t+1)-f(t)=fσ-f=Δdf .

    In this paper, a is set to be an arbitrary real number, and the time scale is a,a+1,⋯,b . Then it is easy to obtain Tκ=a,a+1,⋯,b-1 . Let α , β be two arbitrary real numbers such that α,β∈(0,1] , and put μ=1-α , ν=1-β .

    For arbitrary x,y∈R , x(y)=Γ(x+1)Γ(x+1-y) , where Γ is the gamma function.

    The left fractional sum and the right fractional sum are defined as

    aΔtαf(t)=Δd(ΔaΔt-μf(t))=ΔaΔt-μΔdf(t)+
    (t+μ-a)(μ-1)Γ(μ)f(a)
    (1)
    tΔbβf(t)=-Δd(ΔtΔb-νf(t))=-ΔtΔρb-νΔdf(t)+
    νΓ(ν+1)(b+ν-σ(t))(ν-1)f(b)
    (2)

    where

    aΔt-μf(t)=f(t)+
    μΓ(μ+1)∑s=at-1(t+μ-σ(s))(μ-1)f(s)t∈T
    (3)
    tΔb-νf(t)=f(t)+
    νΓ(ν+1)∑s=σtbs+ν-σtν-1ft∈T
    (4)

    Hence

    aΔt0f(t)=ΔtΔb0f(t)=f(t),
    aΔt1f(t)=Δdf(t),tΔb1f(t)=-Δdf(t)
    (5)

    Fractional summation by parts is given as

    ∑t=ab-1f(t)ΔaΔtαg(t)=f(b-1)g(b)-
    f(a)g(a)+∑t=ab-2ΔtΔρbαf(t)gσ(t)+
    μΓ(μ+1)g(a)∑t=ab-1(t+μ-a)(μ-1)f(t)-
    ∑t=σab-1(t+μ-σ(a))(μ-1)f(t)
    (6)
    ∑t=ab-1f(t)ΔtΔbβg(t)=-g(b)ΔaΔt-νf(t)t=ρ(b)+
    g(a)ΔaΔt-νf(t)t=a+∑t=ab-2gσ(t)ΔaΔtβf(t)+
    νg(b)Γ(ν+1)∑t=ab-1(b+ν-σ(t))(ν-1)f(t)
    (7)

    The commutative relations between the isochronous variation and the fractional difference operators are

    δΔaΔtαf(t)=ΔaΔtαδf(t)δΔtΔbβf(t)=ΔtΔbβδf(t)
    (8)

  • 2 Discrete Equation

    Assume that the configuration of a mechanical system is determined by the generalized coordinates qiσ , i=1,2,⋯,n , the kinetic energy function is T˜=T˜(t,qiσ,ΔaΔtαqi,ΔtΔbβqi) . The Hamilton’s principle for the nonconservative system with fractional difference operators of Riemann-Liouville type has the following form

    ∑t=ab-1(δT˜+Qjδqjσ)=0j=1,2,⋯,n
    (9)

    where Qjδqjσ is the virtual work of the generalized force Qj , qj(a)=Aj , qj(b)=Bj .

    From Eqs.(6) and (7), we have

    ∑t=ab-1∂T˜∂ΔaΔtαqj⋅δΔaΔtαq=∑t=ab-1∂T˜∂ΔaΔtαqj⋅ΔaΔtαδq∂T˜∂ΔaΔtαqj(b-1)⋅δqj(b)-∂T˜∂ΔaΔtαqj(a)⋅δqj(a)+∑t=ab-2ΔtΔρbα∂T˜∂ΔaΔtαqj(t)⋅δqjσ+μδqj(a)Γ(μ+1)∑t=ab-1(t+μ-a)(μ-1)∂T˜∂ΔaΔtαqj(t)-∑t=σab-1(t+μ-σ(a))(μ-1)∂T˜∂ΔaΔtαqj(t)
    (10)
    ∑t=ab-1∂T˜∂ΔtΔbβqj⋅δΔtΔbβq=∑t=ab-1∂T˜∂ΔtΔbβqj⋅ΔtΔbβδq-δqjb⋅ΔaΔt-ν∂T˜∂ΔtΔbβqj(t)t=ρ(b)+δqja⋅ΔaΔt-ν∂T˜∂ΔtΔbβqj(t)t=a+∑t=ab-2δqjσ(t)⋅ΔaΔtβ∂T˜∂ΔtΔbβqj(t)+νδqj(b)Γ(ν+1)⋅∑t=ab-1(b+ν-σ(t))(ν-1)∂T˜∂ΔtΔbβqj(t)
    (11)

    Considering

    ∑t=ab-1∂T˜∂qjσ⋅δq=∂T˜∂qjσ⋅δqjσt=b-1∑t=ab-2∂T˜∂qjσ⋅δqjσ
    (12)
    ∑t=ab-1Qj⋅δq=Qj⋅δqjσt=b-1∑t=ab-2Qj⋅δqjσ
    (13)

    and the boundary conditions qj(a)=Aj , qj(b)=Bj , we have

    ∑t=ab-1(δT˜+Qjδqjσ)=∑t=ab-1∂T˜∂qjσ⋅δqjσ+∂T˜∂ΔaΔtαqjδΔaΔtαqj+∂T˜∂ΔtΔbβqj⋅δΔtΔbβqj+Qjδqjσ=∑t=ab-2∂T˜∂qjσ+ΔtΔρbα∂T˜∂ΔaΔtαqj+aΔtβ∂T˜∂ΔtΔbβqj+Qjδqjσ=0
    (14)

    Since the value of δqjσ is arbitrary, we obtain

    ∂T˜∂qjσ+ΔtΔρbα∂T˜∂ΔaΔtαqj+ΔaΔtβ∂T˜∂ΔtΔbβqj+Qj=0
    t∈a,a+1,⋯,b-2
    (15)

    In Eq.(15), Qj contains the conservative force Q'j and the nonconservative force Q″j . If Q'j is potential, that is, there exists a function V=Vt,qiσ such that

    Q'j=-∂V∂qjσ
    (16)

    Since

    ∂V∂ΔaΔtαqj=0,∂V∂ΔtΔbβqj=0
    (17)

    Substituting Eqs.(16) and (17) into Eq.(15), we have

    ∂(T˜-V)∂qjσ+ΔtΔρbα∂(T˜-V)∂ΔaΔtαqj+ΔaΔtβ∂(T˜-V)∂ΔtΔbβqj+Q″j=0t∈a,a+1,⋯,b-2
    (18)

    Let L=T˜-V , Eq.(18) can be written as

    ∂L∂qjσ+ΔtΔρbα∂L∂ΔaΔtαqj+ΔaΔtβ∂L∂ΔtΔbβqj+Q″j=0t∈a,a+1,⋯,b-2
    (19)

    If Q'j has the generalized potential, that is, there exists a function U=U(t,qiσ,ΔaΔtαqi,ΔtΔbβqi) such that

    Q'j=∂U∂qjσ+ΔtΔρbα∂U∂ΔaΔtαqj+ΔaΔtβ∂U∂ΔtΔbβqj
    (20)

    Substituting Eq.(20) into Eq.(15), we have

    ∂(T˜+U)∂qjσ+ΔtΔρbα∂(T˜+U)∂ΔaΔtαqj+ΔaΔtβ∂(T˜+U)∂ΔtΔbβqj+Q″j=0t∈a,a+1,⋯,b-2
    (21)

    Let L=T˜+U=T˜-V , Eq.(21) can be written as

    ∂L∂qjσ+ΔtΔρbα∂L∂ΔaΔtαqj+ΔaΔtβ∂L∂ΔtΔbβqj+Q″j=0t∈a,a+1,⋯,b-2
    (22)

    Eq.(22) is called discrete fractional Lagrange equation of the nonconservative system.

    Remark 1 If α=1 , L does not depend on tΔbβqi , and the discrete Lagrange equation of the nonconservative system can be obtained

    ∂L(t,qi(t+1),Δdqi)∂qj(t+1)-Δd∂L∂Δdqj+Q″j=0t∈a,a+1,⋯,b-2
    (23)

    Eq.(23) is consistent with the result in Ref.[28].

  • 3 Discrete Equation with Dynamic Constraint

    We assume that the motion of the nonconservative system is subjected to the following ideal dynamic constraint

    hk(t,qiσ,ΔaΔtαqi,ΔtΔbβqi)=0
    k=1,2,⋯,g;i=1,2,⋯,n
    (24)

    which satisfies

    ∂hk∂ΔaΔtαqjδqjσ=0,∂hk∂ΔtΔbβqjδqjσ=0
    (25)

    In the sequel, we study the d’Alembert-Lagrange principle with fractional difference operators. By virtue of Eq.(15), the universal d’Alembert-Lagrange principle with fractional difference operators can be expressed as

    ∂T˜∂qjσ+ΔtΔρbα∂T˜∂ΔaΔtαqj+ΔaΔtβ∂T˜∂ΔtΔbβqj+Qj⋅[δqσj=0][t∈a,a+1,⋯,b-2]
    (26)

    Introducing the Lagrange multipliers λk , k=1,2,⋯,g , from Eq.(25), we obtain

    λk∂hk∂ΔaΔtαqjδqjσ=0,λk∂hk∂ΔtΔbβqjδqjσ=0
    (27)

    It follows from Eqs.(26) and (27) that

    ∂T˜∂qjσ+ΔtΔρbα∂T˜∂ΔaΔtαqj+ΔaΔtβ∂T˜∂ΔtΔbβqj+Qj+λk∂hk∂ΔaΔtαqj+λk∂hk∂ΔtΔbβqj⋅δqjσ=0
    (28)

    Similarly, considering the arbitrariness of the value of δqjσ , we have

    ∂T˜∂qjσ+ΔtΔρbα∂T˜∂ΔaΔtαqj+ΔaΔtβ∂T˜∂ΔtΔbβqj+Qj+λk∂hk∂ΔaΔtαqj+λk∂hk∂ΔtΔbβqj=0
    (29)

    In Eq.(29), Qj contains the conservative force Q'j and the nonconservative force Q″j . If Q'j is potential, that is, there exists a function V=V(t,qiσ) such that

    Q'j=-∂V∂qjσ
    (30)

    Since

    ∂V∂ΔaΔtαqj=0,∂V∂ΔtΔbβqj=0
    (31)

    Substituting Eqs.(30) and (31) into Eq.(29), we have

    ∂(T˜-V)∂qjσ+ΔtΔρbα∂(T˜-V)∂ΔaΔtαqj+ΔaΔtβ∂(T˜-V)∂ΔtΔbβqj+
    λk∂hk∂ΔaΔtαqj+λk∂hk∂ΔtΔbβqj+Q″j=0
    (32)

    Let L=T˜-V , Eq.(32) can be written as

    ∂L∂qjσ+ΔtΔρbα∂L∂ΔaΔtαqj+ΔaΔtβ∂L∂ΔtΔbβqj+λk∂hk∂ΔaΔtαqj+λk∂hk∂ΔtΔbβqj+Q″j=0
    (33)

    If Q'j has generalized potential, that is, there exists a function U=U(t,qiσ,ΔaΔtαqi,ΔtΔbβqi) such that

    Q'j=∂U∂qjσ+ΔtΔρbα∂U∂ΔaΔtαqj+ΔaΔtβ∂U∂ΔtΔbβqj
    (34)

    Substituting Eq.(34) into Eq.(29), we have

    ∂(T˜+U)∂qjσ+ΔtΔρbα∂(T˜+U)∂ΔaΔtαqj+ΔaΔtβ∂(T˜+U)∂ΔtΔbβqj+λk∂hk∂ΔaΔtαqj+λk∂hk∂ΔtΔbβqj+Q″j=0
    (35)

    Let L=T˜+U=T˜-V , Eq.(35) can be written as

    ∂L∂qjσ+ΔtΔρbα∂L∂ΔaΔtαqj+ΔaΔtβ∂L∂ΔtΔbβqj+λk∂hk∂ΔaΔtαqj+λk∂hk∂ΔtΔbβqj+Q″j=0
    (36)

    Eq.(36) is called discrete Lagrange equation with multipliers of the nonconservative system with dynamic constraint. From Eqs.(36) and (24), λk and qi can be solved.

    Remark 2 If α=1 , L and hk do not depend on tΔbβqi , and the discrete Lagrange equation of the nonconservative system with dynamic constraint can be obtained

    ∂L(t,qi(t+1),Δdqi)∂qj(t+1)-Δd∂L∂Δdqj+λk∂hk∂Δdqj+
    Q″j=0t∈a,a+1,⋯,b-2
    (37)

    Eq.(37) is consistent with the result in Ref.[28].

  • 4 Examples

    Example 1 Consider the following nonconservative system

    L=12(ΔaΔtαq)2+qσΔaΔtαqQ″=-ΔaΔtαq+ΔtΔbβq+qσ
    (38)

    with dynamic constraint

    h=(ΔaΔtαq)2+qσ=0
    (39)

    From Eq.(36), we have

    tΔρbα(qσ+ΔaΔtαq)+2λΔaΔtαq+
    tΔbβq+qσ=0
    (40)

    From Eq.(39), we have

    tΔρbαh=2ΔaΔtαq⋅ΔtΔρbα(ΔaΔtαq)+tΔρbαqσ=0
    (41)

    It follows from Eqs.(40) and (41) that

    λ=tΔρbαqσ4(ΔaΔtαq)2-tΔρbαqσ+ΔtΔbβq+qσ2ΔaΔtαq
    (42)

    Specially, when α=β=1 , we obtain

    λ=-Δdqσ4(Δdq)2+Δdqσ+Δdq-qσ2Δdq
    (43)

    Example 2 There is a well-known example called Appell-Hamel[29]. We discuss the example of Appell-Hamel within fractional difference operators of Riemann-Liouville type.

    The Lagrangian is

    L=12m[(ΔaΔtαq1)2+(ΔaΔtαq2)2+(ΔaΔtαq3)2]-
    mgq3σ
    (44)

    the dynamic constraint is

    h=b2a2[(ΔaΔtαq1)2+(ΔaΔtαq2)2]-(ΔaΔtαq3)2=0
    (45)

    From Eq.(36), we have

    mΔtΔρbαΔaΔtαq1+λ2b2a2ΔaΔtαq1=0mΔtΔρbαΔaΔtαq2+λ2b2a2ΔaΔtαq2=0-mg+mΔtΔρbαΔaΔtαq3-2λΔaΔtαq3=0
    (46)

    From Eq.(45), we have

    tΔρbαh=2b2a2ΔaΔtαq1⋅ΔtΔρbα(ΔaΔtαq1)+2b2a2ΔaΔtαq2⋅ΔtΔρbα(ΔaΔtαq2)-2ΔaΔtαq3⋅ΔtΔρbα(ΔaΔtαq3)=0
    (47)

    It follows from Eqs.(46) and (47) that

    λ=-a4gΔaΔtαq32b4[(ΔaΔtαq1)2+(ΔaΔtαq2)2]+2a4(ΔaΔtαq3)2
    (48)

    Specially, when α=β=1 , we obtain

    λ=-a4gΔdq32b4[(Δdq1)2+(Δdq2)2]+2a4(Δdq3)2
    (49)

  • 5 Conclusions

    Using the properties of the time scale calculus, discrete Lagrange equations of the nonconservative system and the nonconservative system with dynamic constraint in terms of fractional difference operators of Riemann-Liouville type are obtained. Two special cases are given. In addition, the proposed method can also be applied to study other mechanical systems, such as the Hamiltonian system and the Birkhoffian system.

    In addition, we will conduct further research in symmetry and conserved quantity, perturbation to symmetry and adiabatic invariants within fractional difference operators of Riemann-Liouville type of constrained mechanical systems.

  • References

    • 1

      FORT T . The calculus of variations applied to Nörlund’s sum[J]. Bulletin of the American Mathematical Society, 1937, 43(12):885-887.

    • 2

      ANASTASSIOU G A . Discrete fractional calculus and inequalities[J]. Mathematical and Computer Modelling, 2009, 51(5/6):562-571.

    • 3

      CHEN F L, LUO X N, ZHOU Y . Existence results for nonlinear fractional difference equation[J]. Advances in Difference Equations, 2011, 2011(1):1-12.

    • 4

      KELLEY W G, PETERSON A C . Difference equations[M]. Boston: Academic Press, 1991.

    • 5

      HILSCHER R, ZEIDAN V . Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: Survey [J]. Journal of Difference Equations & Applications, 2005, 11(9):857-875.

    • 6

      BALEANU D . New applications of fractional variational principles[J]. Reports on Mathematical Physics, 2008, 61(2):199-206.

    • 7

      BALEANU D, DEFTERLI O, AGRAWAL O P . A central difference numerical scheme for fractional optimal control problems[J]. Journal of Vibration & Control, 2009, 15(4):583-597.

    • 8

      BASTOS N R O, FERREIRA R A C, TORRES D F M . Necessary optimality conditions for fractional difference problems of the calculus of variations[J]. Discrete & Continuous Dynamical Systems, 2011, 29(2):417-437.

    • 9

      RABEI E M, NAWAFLEHA K I, HIJJAWIA R S, et al . The Hamilton formalism with fractional derivatives [J]. Journal of Mathematical Analysis & Applications, 2007, 327(2):891-897.

    • 10

      RABEI E M, TARAWNEH D M, MUSLIH S I, et al . Heisenberg’s equations of motion with fractional derivatives[J]. Journal of Vibration & Control, 2007, 13(9/10):1239-1247.

    • 11

      RIEWE F . Nonconservative Lagrangian and Hamiltonian mechanics[J]. Physical Review E, 1996, 53(2):1890-1899.

    • 12

      RIEWE F . Mechanics with fractional derivatives[J]. Physical Review E, 1997, 55(3):3581-3592.

    • 13

      KLIMEK M . Fractional sequential mechanics-models with symmetric fractional derivative [J]. Czechoslovak Journal of Physics, 2001, 51(12):1348-1354.

    • 14

      AGRAWAL O P . Formulation of Euler-Lagrange equations for fractional variational problems[J]. Journal of Mathematical Analysis & Applications, 2002, 272(1):368-379.

    • 15

      AGRAWAL O P . Fractional variational calculus and the transversality conditions[J]. Journal of Physics A: Mathematical General, 2006, 39(33):10375-10384.

    • 16

      AGRAWAL O P . Fractional variational calculus in terms of Riesz fractional derivatives[J]. Journal of Physics A: Mathematical and Theoretical, 2007, 40(24):6287-6303.

    • 17

      AGRAWAL O P, MUSLIH S I, BALEANU D . Generalized variational calculus in terms of multi-parameters fractional derivatives[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(12):4756-4767.

    • 18

      ATANACKOVIĆ T M, KONJIK S, PILIPOVIĆ S . Variational problems with fractional derivatives: Euler-Lagrange equations[J]. Journal of Physics A: Mathematical and Theoretical, 2011, 41(9):1937-1940.

    • 19

      ALMEIDA R, TORRES D F M . Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(3):1490-1500.

    • 20

      ALMEIDA R . Fractional variational problems with the Riesz-Caputo derivative[J]. Applied Mathematics Letters, 2012, 25(2):142-148.

    • 21

      ZHOU Y, ZHANG Y . Fractional Pfaff-Birkhoff principle and Birkhoff’s equations in terms of Riesz fractional derivatives[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2014, 31(1): 63-69.

    • 22

      ZHANG Y, LONG Z X . Fractional action-like variational problem and its Noether symmetries for a nonholonomic system[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2015 32(4): 380-389.

    • 23

      BASTOS N R O . Fractional calculus on time scales [D]. Aveiro: University of Aveiro, 2012.

    • 24

      ATICI F M, ELOE P W . Initial value problems in discrete fractional calculus[J]. Proceedings of the American Mathematical Society, 2009, 137(3):981-989.

    • 25

      SRIVASTAVA H M, OWA S . Univalent functions, fractional calculus, and their applications[M]. Ellis Horwood: Halsted Press, 1989:139-152.

    • 26

      HILGER S . Ein Maßkettenkalkiil mit Anwendung auf Zentrumsmannigfaltigkeiten[D]. Würzburg: Universität Würzburg, 1988. (in German)

    • 27

      BOHNER M, PETERSON A C . Dynamic equations on time scales―An introduction with applications[M]. Boston: Birkhäuser, 2001.

    • 28

      CAI P P, FU J L, GUO Y X . Noether symmetries of the nonconservative and nonholonomic systems on time scales[J]. Science China: Physics, Mechanics & Astronomy, 2013, 56(5):1017-1028.

    • 29

      MEI F X . The foundation of mechanics of nonholonomic system[M]. Beijing: Beijing Institute of Technology Press, 1985. (in Chinese)

    • 贡献声明和致谢

    Dr. SONG Chuanjing wrote the manuscript. Prof. ZHANG Yi contributed to the discussion of the study. All authors commented on the manuscript draft and approved the submission.

    This work was supported by the National Natural Science Foundation of China (Nos.11802193, 11572212, 11272227), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (18KJB130005), the Science Research Foundation of Suzhou University of Science and Technology (331812137), and Natural Science Foundation of Suzhou University of Science and Technology.

    利益冲突

    The authors declare no competing interests.

  • 基本信息

    DOI:http://dx.doi.org/10.16356/j.1005⁃1120.2019.01.017

    中图分类号: O316

    文献标识码: A

    引用信息

    SONG Chuanjing, ZHANG Yi. Discrete Fractional Lagrange Equations of Nonconservative Systems[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2019,36(1):175-180.

    稿件历史

    纸刊出版 : 2019-03-30
    收稿日期 : 2018-02-13
    修回日期 : 2018-04-26
    录用日期 : 2018-09-05

    • SONGChuanjing

    Affiliation：School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, P. R. China

    Profile：Ms. SONG Chuanjing received the B.S. degree in mathematics from Henan University, Kaifeng, China, in 2010. Since 2013, she has been working for her doctorate degree in Nanjing University of Science and Technology. Her research has focused on analytical mechanics.

    ZHANGYi

    Affiliation： College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215009, P. R. China

    角 色：通讯作者

    Role：Corresponding author

    邮 箱：zhy@mail.usts.edu.cn.

    Profile：E-mail address:zhy@mail.usts.edu.cn.

    image /

  • References

    • 1

      FORT T . The calculus of variations applied to Nörlund’s sum[J]. Bulletin of the American Mathematical Society, 1937, 43(12):885-887.

    • 2

      ANASTASSIOU G A . Discrete fractional calculus and inequalities[J]. Mathematical and Computer Modelling, 2009, 51(5/6):562-571.

    • 3

      CHEN F L, LUO X N, ZHOU Y . Existence results for nonlinear fractional difference equation[J]. Advances in Difference Equations, 2011, 2011(1):1-12.

    • 4

      KELLEY W G, PETERSON A C . Difference equations[M]. Boston: Academic Press, 1991.

    • 5

      HILSCHER R, ZEIDAN V . Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: Survey [J]. Journal of Difference Equations & Applications, 2005, 11(9):857-875.

    • 6

      BALEANU D . New applications of fractional variational principles[J]. Reports on Mathematical Physics, 2008, 61(2):199-206.

    • 7

      BALEANU D, DEFTERLI O, AGRAWAL O P . A central difference numerical scheme for fractional optimal control problems[J]. Journal of Vibration & Control, 2009, 15(4):583-597.

    • 8

      BASTOS N R O, FERREIRA R A C, TORRES D F M . Necessary optimality conditions for fractional difference problems of the calculus of variations[J]. Discrete & Continuous Dynamical Systems, 2011, 29(2):417-437.

    • 9

      RABEI E M, NAWAFLEHA K I, HIJJAWIA R S, et al . The Hamilton formalism with fractional derivatives [J]. Journal of Mathematical Analysis & Applications, 2007, 327(2):891-897.

    • 10

      RABEI E M, TARAWNEH D M, MUSLIH S I, et al . Heisenberg’s equations of motion with fractional derivatives[J]. Journal of Vibration & Control, 2007, 13(9/10):1239-1247.

    • 11

      RIEWE F . Nonconservative Lagrangian and Hamiltonian mechanics[J]. Physical Review E, 1996, 53(2):1890-1899.

    • 12

      RIEWE F . Mechanics with fractional derivatives[J]. Physical Review E, 1997, 55(3):3581-3592.

    • 13

      KLIMEK M . Fractional sequential mechanics-models with symmetric fractional derivative [J]. Czechoslovak Journal of Physics, 2001, 51(12):1348-1354.

    • 14

      AGRAWAL O P . Formulation of Euler-Lagrange equations for fractional variational problems[J]. Journal of Mathematical Analysis & Applications, 2002, 272(1):368-379.

    • 15

      AGRAWAL O P . Fractional variational calculus and the transversality conditions[J]. Journal of Physics A: Mathematical General, 2006, 39(33):10375-10384.

    • 16

      AGRAWAL O P . Fractional variational calculus in terms of Riesz fractional derivatives[J]. Journal of Physics A: Mathematical and Theoretical, 2007, 40(24):6287-6303.

    • 17

      AGRAWAL O P, MUSLIH S I, BALEANU D . Generalized variational calculus in terms of multi-parameters fractional derivatives[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(12):4756-4767.

    • 18

      ATANACKOVIĆ T M, KONJIK S, PILIPOVIĆ S . Variational problems with fractional derivatives: Euler-Lagrange equations[J]. Journal of Physics A: Mathematical and Theoretical, 2011, 41(9):1937-1940.

    • 19

      ALMEIDA R, TORRES D F M . Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(3):1490-1500.

    • 20

      ALMEIDA R . Fractional variational problems with the Riesz-Caputo derivative[J]. Applied Mathematics Letters, 2012, 25(2):142-148.

    • 21

      ZHOU Y, ZHANG Y . Fractional Pfaff-Birkhoff principle and Birkhoff’s equations in terms of Riesz fractional derivatives[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2014, 31(1): 63-69.

    • 22

      ZHANG Y, LONG Z X . Fractional action-like variational problem and its Noether symmetries for a nonholonomic system[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2015 32(4): 380-389.

    • 23

      BASTOS N R O . Fractional calculus on time scales [D]. Aveiro: University of Aveiro, 2012.

    • 24

      ATICI F M, ELOE P W . Initial value problems in discrete fractional calculus[J]. Proceedings of the American Mathematical Society, 2009, 137(3):981-989.

    • 25

      SRIVASTAVA H M, OWA S . Univalent functions, fractional calculus, and their applications[M]. Ellis Horwood: Halsted Press, 1989:139-152.

    • 26

      HILGER S . Ein Maßkettenkalkiil mit Anwendung auf Zentrumsmannigfaltigkeiten[D]. Würzburg: Universität Würzburg, 1988. (in German)

    • 27

      BOHNER M, PETERSON A C . Dynamic equations on time scales―An introduction with applications[M]. Boston: Birkhäuser, 2001.

    • 28

      CAI P P, FU J L, GUO Y X . Noether symmetries of the nonconservative and nonholonomic systems on time scales[J]. Science China: Physics, Mechanics & Astronomy, 2013, 56(5):1017-1028.

    • 29

      MEI F X . The foundation of mechanics of nonholonomic system[M]. Beijing: Beijing Institute of Technology Press, 1985. (in Chinese)

    • Articles

    WeChat

    Mobile website

    Website Copyright © Transactions of Nanjing University of Aeronautics & Astronautics
    Supported by:Beijing E-Tiller Technology Development Co., Ltd.