• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Fractional Action-Like Variational Problem and Its Noether Symmetries for a Nonholonomic System

    2015-11-24 06:57:33ZhangYi張毅LongZixuan龍梓軒
    關(guān)鍵詞:張毅

    Zhang Yi(張毅),Long Zixuan(龍梓軒)

    1.College of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,P.R.China

    2.College of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou 215009,P.R.China

    Fractional Action-Like Variational Problem and Its Noether Symmetries for a Nonholonomic System

    Zhang Yi(張毅)1*,Long Zixuan(龍梓軒)2

    1.College of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,P.R.China

    2.College of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou 215009,P.R.China

    Eor an in-depth study on the symmetric properties for nonholonomic non-conservative mechanical systems,the fractional action-like Noether symmetries and conserved quantities for nonholonomic mechanical systems are studied,based on the fractional action-like approach for dynamics modeling proposed by El-Nabulsi.Eirstly,

    nonholonomic system;fractional action-like variational problem;symmetric transformation;Noether theorem;conserved quantity

    0 Introduction

    The fractional calculus has provided a powerful mathematical tool for a great number of problems in different fields of science and engineering,and has made many break-through results in mathematical physics,classical and quantum mechanics,control theory,nonlinear dynamics,signal and image processing,thermodynamics,bioengineering and other fields[1-2]. Although various fields of application of fractional calculus are already well established,some others have just started.The researches in fractional variational problems and their symmetry and conserved quantity are examples of the latter.

    The study of fractional variational problems began in the work of Riewe[3-4].In 1996,Riewe

    first applied fractional calculus to a non-conservative mechanics modeling,and the fractional Euler-Lagrange equations and the fractional Hamilton equations were formed initially.Since then,the fractional variational problems have become one of the most popular research areas in applied mathematics,physics,dynamics and control,and are increasingly attracting the attention of many scholars:Klimek[5-6], Agrawal[7-9], Atanackovic'[10-11],Jumarie[12],Baleanu[13-15], Torres[16-18],El-Nabulsi[19-24], Cresson[25], Rabei[26], Tarasov[27],and Zhang[28-29],et al.These scholars came up with a variety of fractional models and methods from different views,and established the corresponding fractional Euler-Lagrange equations and fractional Hamilton equations.Erom the point of view of both classical and quantumsystems,the existence of a number of different fractional variational problems and the need for a more precise description of the fractional model,in part,can be interpreted as the nonlocal nature of fractional order differential operators and the corresponding adjoint operators for describing the dynamics.Another reason is that there exist many different fractional integral operators,including Grünwald-Letnikov,Caputo,Riesz,Riemann-Liouville operators,and so on.The Riemann-Liouville operator is one of the most frequently used in the application of fractional calculus operators.

    In order to establish a non-conservative dynamical system model,El-Nabulsi presented a modeling method[19]in 2005,known as the fractional action-like variational approach(also called the El-Nabulsi's fractional model).In his method,the fractional integral about time only needs one parameter,and the resulting fractional Euler-Lagrange equations contain the dissipative forces depending on time.However,there are an arbitrary number of fractional parameters(the order of the derivative)in other fractional models.The novelty of El-Nabulsi dynamics model is that the derived Euler-Lagrange equations are similar to the classical ones,with no fractional derivatives,but the presence of the fractional generalized external force acts on the system.The fractional action-like approach was further extended to the situation of Lagrangian depending on Riemann-Liouville fractional derivatives[20],to the multi-dimensional fractional action-like variational problems[21],the fractional action-like variational problems with holonomic constraints or nonholonomic constraints or dissipative dynamic systems[22],the fractional action-like variational problems with exponential law[23],and the universal fractional action-like Euler-Lagrange equations from a generalized fractional derivative operator[24].Erederico and Torres studied the constant of motion for fractional action-like variational problems,gave Noether's theorem[30]for nonconservative system under El-Nabulsi's fractional model,and extended to the situation of Lagrangian containing higher-order derivatives[31].Recently,authors have obtained the Noether's theorem for Birkhoffian system[32]under El-Nabulsi's fractional model,the Noether's theorems for Lagrange systems[33]and Hamilton systems[34]based on the extended exponentially fractional integral.

    Here the Noether theory for holonomic systems and nonholonomic systems is further studied under the framework of fractional action-like variational approach.The definitions and criteria of fractional action-like Noether symmetric transformations and Noether quasi-symmetric transformations are provided.The fractional action-like Noether theorems of holonomic systems and nonholonomic systems are derived.And the conserved quantities led by the fractional action-like Noether symmetries are given.

    1 Fractional Action-Like Variational Problem

    Assume that the configuration of a mechanical system is determined by generalized coordinates qk(k = 1,…,n),the Lagrangian of the system is L=L(τ,q,.q).With the fractional action-like variational approach for modeling of nonconservative dynamical system presented by El-Nabulsi[19],the fractional variational problem under Riemann-Liouville fractional integrals can be defined as follows.

    Eind the stationary points of the integral function

    with the fixed boundary conditions

    The above variational problem is called the fractional action-like variational problem.Eq.(1)can also be called the fractional action-like Hamilton action.Whenα=1,the problem becomes a classical variational problem of a dynamicalsystem.

    According to the theory of calculus of variations,the necessary condition to achieve extreme for Eq.(1)at qk=qk(τ )is its variation equal to 0,that is,δS=0.Therefore,one can have the following equation

    Using the boundary conditions Eq.(2),one has

    Substituting Eq.(4)into Eq.(3),it becomes

    Since the variationsδqk( k=1,…,n )are independent of each other for a holonomic system,therefore,by the fundamental lemma[35]of the calculus of variations,one obtains

    Erom Eq.(6),one gets

    Eq.(7)are the fractional action-like Lagrange equations of the holonomic system[19].

    Assume that the motion of the system is subjected to g bilateral ideal nonholonomic constraints of Chetaev type

    The restriction of constraints Eq.(8)exerted on the virtual displacements is

    Erom Eq.(5)and the conditions Eq.(9),by using the Lagrange multiplier method,one can obtain

    whereλβare the constraint multipliers.Eq.(10)can be called the fractional action-like differential equations of motion with multipliers for the nonholonomic system.

    Before integrating the equations of motion,by using Eqs.(8,10),one can findλβas the function of t,q and q·.Therefore,Eqs.(10)can be written in the form

    where

    Eqs.(11)are called the equations of motion of the holonomic system corresponding to the nonholonomic system,or the equations of motion of the corresponding holonomic system for short.

    If the initial conditions satisfy the equations of nonholonomic constraints Eq.(8),the motion of the corresponding holonomic system Eq.(12)will give the solution of the nonholonomic system Eqs.(8,10).

    Example 1 Consider a system whose configuration is determined by two generalized coordinates q1,q2.The Lagrangian of the system is

    and its motion is subject to a nonholonomic constraint[36]

    Erom Eqs.(10),one has

    where the first term of the right side of each equation of Eqs.(15)can be viewed as a generalized external force acting on the system,and the second one is the force corresponding to the nonholonomic constraint Eq.(14).Erom Eqs.(14,15),one can find the multiplier

    Then Eqs.(15)can be written as

    Eqs.(17)are the fractional action-like differential equations of motion of the holonomic system corresponding to the nonholonomic system Eqs.(14,15).Ifα =1,Eqs.(17)give the equations of motion in classic situation[36].

    2 Variation of Fractional Action-Like Hamilton Action

    Introduce the infinitesimal transformations of r-parameters finite transformation group

    or their expansion formulae

    whereεσ(σ=1,…,r )are the infinitesimal parameters,andhe generators or generating functions for the infinitesimal transformations.

    The difference of the fractional action-like Hamilton action Eq.(1)before and after transformation is

    whereγis the given curve andγ-a neighbor curve. Denoting the main linear part of Eq.(20)forεσ,i.e.,the part accurate to the first-order infinitesimal,asΔS,one has

    Eor an arbitrary function E,the relation between the non-isochronous variationΔand the isochronous variationδis[36]

    Therefore one has

    Erom Eq.(23),Eq.(21)can be expressed as

    Erom Eqs.(19,23),Eq.(24)can be further expressed as

    Eqs.(21,25)are basic formulae for the variation of fractional action-like Hamilton action.

    3 Fractional Action-Like Symmetric Transformation

    In this section,one establishes the definitions and criteria of fractional action-like Noether symmetric transformations and quasi-symmetric transformations.

    Definition 1 If the fractional action-like Hamilton action Eq.(1)is an invariant of the infinitesimal transformations of the group in Eq.(18),that is,for each of the infinitesimal transformations,the formula

    ΔS=0 (26)holds,the infinitesimal transformations are called the fractional action-like Noether symmetric transformations.

    Erom Definition 1 and Eq.(21),one can obtain the following criterion.

    Criterion 1 Eor the infinitesimal transformations of the group in Eq.(18),if the condition

    is satisfied,the infinitesimal transformations are the fractional action-like Noether symmetric transformations.

    Condition Eq.(27)can also be expressed as

    When r=1,Eq.(28)may be called the fractional action-like Noether identity.

    Erom Definition 1 and Eq.(25),one can obtain the criterion as follows.

    Criterion 2 Eor the infinitesimal transformations of the group in Eq.(19),if the conditions

    are satisfied,the infinitesimal transformations are the fractional action-like Noether symmetric transformations.

    Subsequently,one establishes the definition and criteria of the fractional action-like Noether quasi-symmetry transformations.

    Suppose that L'is another Lagrangian,if the infinitesimal transformations(Eq.(18))accurate to the first-order infinitesimal satisfy the condition

    this invariance is called the quasi-invariance of the fractional action-like Hamilton action Eq.(1)under the infinitesimal transformations of the group in Eq.(18).The functions L'and L determined by Eq.(30)satisfy the same differential equations of motion.Hence the transformations are called the fractional action-like Noether quasi-symmetric transformations,and one has

    Substituting Eq.(31)into Eq.(30),one has

    The left-hand of Eq.(32)is a first-order infinitesimal under the transformations (Eq.(18)). Therefore,the right-hand should be an infinitesimal of the same-order.G can be replaced byΔG,and thus

    Hence,one has

    Definition 2 If the fractional action-like Hamilton action Eq.(1)is a quasi-invariant under the infinitesimal transformations of group (18),i.e.for each of the infinitesimal transformations,the formula

    holds,where G =G(τ,q,q·),the infinitesimal transformations are called the fractional actionlike Noether quasi-symmetric transformations.

    Erom Definition 2 and Eq.(25),one can get the following criterion.

    Criterion 3 Eor the infinitesimal transformations of group(18),if the condition

    is satisfied,the infinitesimal transformations are the fractional action-like Noether quasi-symmetric transformations.

    Condition Eq.(35)can also be expressed as

    Erom Definition 2 and Eq.(25),one can suggest the criterion as follows

    Criterion 4 Eor the infinitesimal transformations of the group in Eq.(19),if the conditions

    are satisfied,the infinitesimal transformations are the fractional action-like Noether quasi-symmetric transformations.

    By using Criterions 1,2,one can determine the fractional action-like Noether symmetry. Likewise,by using Criterions 3,4,one can define the fractional action-like Noether quasi-symmetry.

    4 Fractional Action-Like Noether Theorem of Holonomic System

    The conserved quantity of a holonomic system under the El-Nabulsi's fractional model is firstly defined.

    Definition 3 A function I(τ,q,q·)is said to be a conserved quantity of a holonomic system under El-Nabulsi's fractional model if

    is along all the solution curves of the fractional action-like Lagrange equations.(7).

    Eor a holonomic system,if one can find a fractional action-like Noether symmetric transformation or a Noether quasi-symmetric transformation,one can find a corresponding conserved quantity.Here is the obtained theorem.

    Theorem 1 Eor the holonomic system Eq.(7),if the infinitesimal transformations of group Eq.(19)are the fractional action-like Noether symmetric transformations under Definition 1,the system has r linear and independent conserved quantities,that is

    Proof Since the infinitesimal transformations of group are the fractional action-like Noether symmetric transformations of the system.By Definition 1,one hasΔS=0,namely

    Substituting Eq.(7)into Eq.(40),and considering the independence of parametersεσ,one has

    Integrating it,Eq.(39)is obtained,and then it ends.

    Theorem 2 Eor the holonomic system Eq.(7),if the infinitesimal transformations of the group in Eq.(19)are the fractional action-like Noether quasi-symmetric transformations under Definition 2,the system exists r linear independent conservation quantities,such as

    Theorems 1,2 can be called the fractional action-like Noether theorem for the holonomic system.According to the Noether theorem,for the holonomic system under El-Nabulsi's fractional model,if one can find a fractional actionlike Noether symmetric transformation or a quasisymmetric transformation,one can get a conserved quantity of the system.

    Example 2 The Lagrangian of the planar Kepler problem is

    Here one tries to study the fractional action-like Noether symmetries and conserved quantities of the system.

    Eirst,one finds the fractional action-like Noether quasi-symmetric transformations.Eractional action-like generalized Noether identity Eq.(36)gives

    Eq.(44)has the following solutions

    Eq.(47)is a conserved quantity led by the fractional action-like Noether symmetry Eq.(45)of the system.Whenα=1,Eq.(47)gives

    This is the conserved quantity of a classical Kepler problem[36].

    Erom the generator Eq.(46),according to Theorem 2,one obtains

    Therefore,the infinitesimal transformation corresponding to the generator Eq.(46)is trivial.

    The generator Eq.(45)is corresponding to a fractional action-like Noether symmetric transformation of the system,and the generator Eq.(46)is corresponding to a fractional action-like Noether quasi-symmetric transformation of the system.

    Erom the generator Eq.(45),according to Theorem 1,one has

    5 Fractional Action-Like Noether Theorem of Nonholonomic System

    The definition of fractional action-like Noether quasi-symmetric transformations of the nonholonomic system is firstly given.

    Notice that

    Substituting Eq.(50)into Eq.(9),and considering the independence ofεσ,one has

    This is the restriction of nonholonomic constraints exerted on the generating function of infinitesimal transformations,called the Appell-Chetaev conditions.Thus one has

    Definition 4 Eor the nonholonomic system Eqs.(8,10),if the infinitesimal transformations of the group in Eq.(19)are the fractional actionlike Noether quasi-symmetric transformations,satisfying the Appell-Chetaev conditions Eq.(51),the transformations are called the fractional action-like Noether quasi-symmetric transformations of the nonholonomic system.

    Secondly,one gives the definition of a conserved quantity of a nonholonomic system under El-Nabulsi's fractional model.

    Definition 5 A function I(τ,q,q·)is said to be a conserved quantity of a nonholonomic system under El-Nabulsi's fractional model if

    is along all the solution curves of the fractional action-like differential equations of motion of the nonholonomic system Eqs.(8,10).

    Einally,one establishes the fractional actionlike Noether theorem of the nonholonomic system.

    Theorem 3 Eor the nonholonomic system Eqs.(8,10),if the infinitesimal transformations of the group in Eq.(19)are the Noether quasisymmetric transformations under Definition 4,the system has r linear independent conserved quantities

    Proof Since the infinitesimal transformations of group are the fractional action-like Noether quasi-symmetric transformations of the system,by Definition 2,one has

    Eq.(34)can also be written as

    Since the infinitesimal transformations satisfy the Appell-Chetaev conditions Eq.(51),one has

    Adding Eq.(55)and Eq.(54)together,one gets

    Substituting Eq.(10)into Eq.(56),and considering the independence ofεσ,one obtains

    Integrating it,one obtains Eq.(53),and the theorem is thus proved.

    Theorem 3 can be called the fractional actionlike Noether theorem of the nonholonomic system.By the theorem,one can find a conserved quantity from a known Noether symmetry.

    If the nonholonomic constraints do not exist,then Theorem 3 degenerates to Theorem 2,and if at the same time Gσ=0 is satisfied,Theorem 3 degenerates to Theorem 1.

    Example 3 Let us study the fractional action-like Noether symmetries and the conserved quantities of the nonholonomic system discussed in Example 1.

    Eirst,one tries to find the fractional actionlike Noether quasi-symmetric transformations satisfying the Appell-Chetaev conditions.The fractional action-like generalized Noether identity Eq.(36)gives

    and the Appell-Chetaev conditions Eq.(51)give

    Eqs.(58,59)have the following solutions

    The generators Eqs.(60,61)are both corresponding to the fractional action-like Noether quasi-symmetric transformation of the nonholonomic system.By Theorem 3,the conserved quantity Eq.(53)gives

    Therefore,the infinitesimal transformation corresponding to the generator Eq.(60)is trivial.And whenα=1,the conserved quantity Eq.(63)gives

    This is a classical conserved quantity[36].

    6 Conclusions

    In recent decades,the fractional calculus has been successfully used in various fields of science and engineering.It has also been used in dynamics modeling for a non-conservative or dissipative system and so on,where some complex problems can be solved difficultly with integer order derivatives.Here the fractional action-like variational problem is further studied,based upon the fractional modeling presented by El-Nabulsi.The fractional action-like differential equations of motion for both holonomic and nonholonomic systems are established.The definitions and criteria of both fractional action-like Noether symmetric transformations and Noether quasi-symmetric transformations are given,and the fractional action-like Noether theorems of the systems are established.The presented methods and its results are of universal significance.They can be further applied to various types of constrained mechanical systems.It is noteworthy that classical Noether theory for the circumstance of integer order is a special case of this paper.

    Acknowledgement

    This work was supported by the National Natural Science Eoundation of China(No.11272227).

    [1] Podlubny I.Eractional differential equations [M]. San Diego:Academic Press,1999.

    [2] Herrmann R.Eractional calculus:An introduction for physicists[M].Singapore:World Scientific,2011.

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

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

    [5] Klimek M.Eractional sequential mechanics-modelswith symmetric fractional derivative[J].Czechoslovak Journal of Physics,2001,51(12):1348-1354.

    [6] Klimek M.Lagrangian and Hamiltonian fractional sequential mechanics [J].Czechoslovak Journal of Physics,2002,52(11):1247-1253.

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

    [8] Agrawal O P.Generalized Euler-Lagrange equations and transversality conditions for EVPs in terms of the Caputo derivative[J].Journal of Vibration and Control,2007,13(9/10):1217-1237.

    [9] Agrawal O P,Muslih S I,Baleanu D.Generalized variational calculus in terms of multi-parameters fractional derivatives[J].Communications in Nonlinear Science and Numerical Simulation,2011,16(12):4756-4767.

    [10]Atanackovic'T M,Konjik S,Pilipovic'S.Variational problems with fractional derivatives:Euler-Lagrange equations[J].Journal of Physics A:Mathematical and Theoretical,2008,41(9):095201.

    [11]Atanackovic'T M,Konjik S,Pilipovic'S,et al.Variational problems with fractional derivatives:Invariance conditions and Noether's theorem [J].Nonlinear Analysis:Theory, Methods & Applications,2009,71(5/6):1504-1517.

    [12]Jumarie G.Eractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost functions[J].Journal of Applied Mathematics and Computing,2007,23(1/2):215-228.

    [13]Baleanu D,Avkar T.Lagrangians with linear velocities within Riemann-Liouville fractional derivatives[J].Nuovo Cimento B,2003,119(1):73-79.

    [14]Baleanu D,Trujillo J I.A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives[J].Communications in Nonlinear Science and Numerical Simulation,2010,15(5):1111-1115.

    [15]Baleanu D,Muslih S I,Rabei E M,et al.On fractional Hamiltonian systems possessing first-class constraints within Caputo derivatives[J].Romanian Reports in Physics,2011,63(1):3-8.

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

    [17]Odzijewicz T,Malinowska A B,Torres D E M. Eractional variational calculus with classical and combined Caputo derivatives [J].Nonlinear Analysis:Theory,Methods & Applications,2012,75(3),1507-1515.

    [18]Malinowska A B,Torres D E M.Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative[J].Computers and Mathematics with Applications,2010,59(9):3110-3116.

    [19]El-Nabulsi A R.A fractional approach to nonconservative Lagrangian dynamical systems[J].Eizika A,2005,14(4):289-298.

    [20]El-Nabulsi A R.Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α,β)[J].Mathematical Models & Methods in Applied Sciences,2007,30(15):1931-1939.

    [21]El-Nabulsi A R,Torres D E M.Eractional actionlike variational problems[J].Journal of Mathematical Physics,2008,49(5):053521.

    [22]El-Nabulsi A R.Eractional action-like variational problems in holonomic,non-holonomic and semi-holonomic constrained and dissipative dynamical systems[J].Chaos,Solitons and Eractals,2009,42(1):52-61.

    [23]El-Nabulsi A R.Eractional variational problems from extended exponentially fractional integral[J].Applied Mathematics and Computation,2011,217(22):9492-9496.

    [24]El-Nabulsi R A.Universal fractional Euler-Lagrange equation from a generalized fractional derivate operator[J].Central European Journal of Physics,2011,9(1):250-256.

    [25]Cresson J.Eractional embedding of differential operators and Lagrangian systems[J].Journal of Mathematical Physics,2007,48(3):033504.

    [26]Rabei E M,Rawashdeh I M,Muslih S,et al.Hamilton-Jacobi formulation for systems in terms of Riesz's fractional derivatives[J].International Journal of Theoretical Physics,2011,50(5):1569-1576.

    [27]Tarasov V E.Eractional dynamics[M].Beijing:Higher Education Press,2010.

    [28]Zhou Y,Zhang Y.Eractional 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.

    [29]Zhou Y,Zhang Y.Noether's theorems of a fractional Birkhoffian system within Riemann-Liouville deriva-tives [J].Chinese Physics B,2014,23(12):124502.

    [30]Erederico G SE,Torres D E M.Constants of motion for fractional action-like variational problems[J].International Journal of Applied Mathematics,2006,19(1):97-104.

    [31]Erederico G S E,Torres D E M.Non-conservative Noether's theorem for fractional action-like problems with intrinsic and observer times[J].International Journal of Ecological Economics&Statistics,2007,9(E07):74-82.

    [32]Zhang Y,Zhou Y.Symmetries and conserved quantities for fractional action-like Pfaffian variational problems[J].Nonlinear Dynamics,2013,73(1-2):783-793.

    [33]Long Z X,Zhang Y.Eractional Noether theorem based on extended exponentially fractional integral[J].International Journal of Theoretical Physics,2014,53(3):841-855.

    [34]Long Z X,Zhang Y.Noether's theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space[J].Acta Mechanica,2014,225(1):77-90.

    [35]Goldstein H,Poole C,Safko J.Classical Mechanics[M].Third Edition.Beijing:Higher Education Press,2005.

    [36]Mei E X ,Wu H B.Dynamics of constrained mechanical systems[M].Beijing:Beijing Institute of Technology Press,2009.

    (Executive editor:Zhang Tong)

    O316 Document code:A Article ID:1005-1120(2015)04-0380-10

    *Corresponding author:Zhang Yi,Professor,E-mail:zhy@mail.usts.edu.cn.

    How to cite this article:Zhang Yi,Long Zixuan.Eractional action-like variational problem and its Noether symmetries for a nonholonomic system[J].Trans.Nanjing U.Aero.Astro.,2015,32(4):380-389.

    http://dx.doi.org/10.16356/j.1005-1120.2015.04.380

    (Received 14 August 2014;revised 5 December 2014;accepted 14 December 2014)

    the fractional action-like variational problem is established,and the fractional action-like Lagrange equations of holonomic system and the fractional action-like differential equations of motion with multiplier for nonholonomic system are given;secondly,according to the invariance of fractional action-like Hamilton action under infinitesimal transformations of group,the definitions and criteria of fractional action-like Noether symmetric transformations and quasi-symmetric transformations are put forward;finally,the fractional action-like Noether theorems for both holonomic system and nonholonomic system are established,and the relationship between the fractional action-like Noether symmetry and the conserved quantity is given.

    猜你喜歡
    張毅
    二月二—龍?zhí)ь^
    當代作家(2023年3期)2023-04-23 21:26:58
    張士卿基于敏濕熱瘀辨治過敏性紫癜經(jīng)驗
    《秋水共長天一色》
    青年生活(2019年6期)2019-09-10 17:55:38
    Isolation and callus formation of Gracilariopsis bailiniae(Gracilariales, Rhodophyta) protoplasts*
    Noether Symmetry and Conserved Quantities of Fractional Birkhoffian System in Terms of Herglotz Variational Problem?
    隨便走走(短篇小說)
    當代小說(2017年11期)2018-01-08 09:31:32
    “執(zhí)著”的代價
    檢察風云(2017年10期)2017-06-12 13:50:02
    宮“?!彪u丁
    性格變更
    Perturbation to Noether Symmetries and Adiabatic Invariants for Generalized Birkhoff Systems Based on El-Nabulsi Dynamical Model
    中文字幕人成人乱码亚洲影| 国产乱人伦免费视频| 国产亚洲精品一区二区www| 成人一区二区视频在线观看| 美女 人体艺术 gogo| 国产成人精品久久二区二区91| 午夜福利在线在线| 国产一区在线观看成人免费| 亚洲av电影不卡..在线观看| 狠狠狠狠99中文字幕| 搡老妇女老女人老熟妇| 最近最新中文字幕大全免费视频| 中亚洲国语对白在线视频| av女优亚洲男人天堂 | 亚洲午夜精品一区,二区,三区| 亚洲 国产 在线| 中文字幕人妻丝袜一区二区| 俺也久久电影网| 亚洲男人的天堂狠狠| 免费看光身美女| 久久久久亚洲av毛片大全| 国产成人福利小说| 99热6这里只有精品| 精品国产乱子伦一区二区三区| 老司机在亚洲福利影院| 999久久久精品免费观看国产| 国产男靠女视频免费网站| 国产精品日韩av在线免费观看| 国产亚洲精品久久久com| www.精华液| 真人做人爱边吃奶动态| 久久精品人妻少妇| 99国产综合亚洲精品| av女优亚洲男人天堂 | 亚洲精品国产精品久久久不卡| 久久久久久久精品吃奶| 亚洲男人的天堂狠狠| 两性午夜刺激爽爽歪歪视频在线观看| 少妇人妻一区二区三区视频| 综合色av麻豆| 首页视频小说图片口味搜索| 在线十欧美十亚洲十日本专区| 国产野战对白在线观看| 中文字幕熟女人妻在线| 免费看日本二区| 亚洲专区字幕在线| 国产精品av视频在线免费观看| 欧美日韩乱码在线| 亚洲av日韩精品久久久久久密| 亚洲 国产 在线| 久久精品综合一区二区三区| 男女床上黄色一级片免费看| 亚洲人成网站高清观看| 美女 人体艺术 gogo| 噜噜噜噜噜久久久久久91| www.999成人在线观看| 九九热线精品视视频播放| 最近最新中文字幕大全免费视频| 精品熟女少妇八av免费久了| 欧美黄色淫秽网站| 看黄色毛片网站| 国产极品精品免费视频能看的| 午夜免费观看网址| 一本一本综合久久| 两个人看的免费小视频| 此物有八面人人有两片| 变态另类丝袜制服| 精品一区二区三区视频在线 | 人妻丰满熟妇av一区二区三区| 又紧又爽又黄一区二区| 最近在线观看免费完整版| 精品久久蜜臀av无| 无限看片的www在线观看| 午夜福利18| 在线观看美女被高潮喷水网站 | 最近在线观看免费完整版| e午夜精品久久久久久久| 在线国产一区二区在线| 国产一区在线观看成人免费| 日本一二三区视频观看| 看黄色毛片网站| 久久久久久九九精品二区国产| 日韩 欧美 亚洲 中文字幕| 日本熟妇午夜| 少妇的逼水好多| 夜夜看夜夜爽夜夜摸| 精品电影一区二区在线| 制服丝袜大香蕉在线| 亚洲中文日韩欧美视频| 18禁观看日本| 亚洲国产精品合色在线| 一进一出抽搐gif免费好疼| 床上黄色一级片| 国产真实乱freesex| 国产精品影院久久| 母亲3免费完整高清在线观看| 欧美国产日韩亚洲一区| 12—13女人毛片做爰片一| 女生性感内裤真人,穿戴方法视频| 成人性生交大片免费视频hd| 熟妇人妻久久中文字幕3abv| 亚洲中文av在线| 免费看十八禁软件| 中文字幕高清在线视频| 国产一区在线观看成人免费| 亚洲成av人片免费观看| 搡老熟女国产l中国老女人| 成人三级黄色视频| 国产精华一区二区三区| 午夜福利在线观看吧| or卡值多少钱| 成人无遮挡网站| 首页视频小说图片口味搜索| 窝窝影院91人妻| 啦啦啦观看免费观看视频高清| 最新美女视频免费是黄的| 美女 人体艺术 gogo| 国产精品 欧美亚洲| 搡老熟女国产l中国老女人| 色哟哟哟哟哟哟| 国产免费av片在线观看野外av| 搡老妇女老女人老熟妇| 亚洲第一电影网av| 男人舔奶头视频| 欧美性猛交╳xxx乱大交人| 99久久国产精品久久久| 99久国产av精品| 午夜成年电影在线免费观看| 免费观看精品视频网站| 婷婷精品国产亚洲av| av国产免费在线观看| 亚洲狠狠婷婷综合久久图片| 丰满人妻熟妇乱又伦精品不卡| 不卡一级毛片| 黄色视频,在线免费观看| 成年女人毛片免费观看观看9| 国语自产精品视频在线第100页| 黄片小视频在线播放| 一级作爱视频免费观看| 久久久久久九九精品二区国产| 国产又黄又爽又无遮挡在线| 国产精品永久免费网站| 精品久久久久久久人妻蜜臀av| 亚洲中文字幕日韩| 五月伊人婷婷丁香| 亚洲九九香蕉| 嫁个100分男人电影在线观看| 午夜久久久久精精品| 国产亚洲精品久久久久久毛片| 亚洲国产欧洲综合997久久,| 99久久综合精品五月天人人| 麻豆一二三区av精品| 欧美日韩国产亚洲二区| 久久久久久久久中文| 国产精品99久久99久久久不卡| 村上凉子中文字幕在线| 午夜视频精品福利| 黄色 视频免费看| 日韩成人在线观看一区二区三区| 国产精品日韩av在线免费观看| 国产精华一区二区三区| 欧洲精品卡2卡3卡4卡5卡区| 欧美日韩国产亚洲二区| 男人舔女人的私密视频| 两个人视频免费观看高清| 午夜久久久久精精品| 亚洲专区中文字幕在线| 中国美女看黄片| 国产欧美日韩精品一区二区| 搡老岳熟女国产| 丝袜人妻中文字幕| 丁香六月欧美| 国产主播在线观看一区二区| 99久久精品国产亚洲精品| 老司机在亚洲福利影院| 国产精品九九99| 亚洲精品456在线播放app | 岛国在线免费视频观看| 成人精品一区二区免费| 午夜免费观看网址| 精品一区二区三区四区五区乱码| 在线观看66精品国产| 国产午夜精品久久久久久| 久久精品国产99精品国产亚洲性色| 亚洲专区国产一区二区| 国产精品女同一区二区软件 | 变态另类丝袜制服| 国内久久婷婷六月综合欲色啪| 999久久久国产精品视频| 两个人视频免费观看高清| 欧美日韩黄片免| 在线观看美女被高潮喷水网站 | 黄色片一级片一级黄色片| 免费看光身美女| 久久久久久久久久黄片| 两人在一起打扑克的视频| 国产精品一区二区精品视频观看| 国产av不卡久久| 精品久久蜜臀av无| 此物有八面人人有两片| 男人和女人高潮做爰伦理| 99国产精品一区二区蜜桃av| 97超级碰碰碰精品色视频在线观看| 丁香六月欧美| 国产黄色小视频在线观看| 亚洲乱码一区二区免费版| 久久久久久国产a免费观看| 亚洲男人的天堂狠狠| 国产高清视频在线播放一区| 午夜福利在线观看吧| 中文字幕高清在线视频| 日本精品一区二区三区蜜桃| www.精华液| 又粗又爽又猛毛片免费看| cao死你这个sao货| 亚洲无线观看免费| 亚洲av成人av| 亚洲国产中文字幕在线视频| 99精品欧美一区二区三区四区| 桃色一区二区三区在线观看| 悠悠久久av| 欧美zozozo另类| 一级毛片精品| 97超视频在线观看视频| 亚洲精品乱码久久久v下载方式 | 日韩欧美三级三区| 国产激情久久老熟女| 老熟妇仑乱视频hdxx| 日韩av在线大香蕉| 好看av亚洲va欧美ⅴa在| 在线免费观看的www视频| 婷婷亚洲欧美| 免费大片18禁| 精品久久久久久久人妻蜜臀av| 国产伦精品一区二区三区视频9 | 一本精品99久久精品77| 色噜噜av男人的天堂激情| 欧美日韩福利视频一区二区| 99视频精品全部免费 在线 | 精品久久久久久久毛片微露脸| 精品久久久久久久久久久久久| 大型黄色视频在线免费观看| 久久午夜亚洲精品久久| 在线观看免费视频日本深夜| 91在线精品国自产拍蜜月 | 久久精品综合一区二区三区| 久久久精品大字幕| 色综合亚洲欧美另类图片| 最新在线观看一区二区三区| 国产爱豆传媒在线观看| 久久久久国内视频| 国产精品亚洲美女久久久| bbb黄色大片| 国产黄片美女视频| 亚洲精品中文字幕一二三四区| 动漫黄色视频在线观看| av在线天堂中文字幕| 黑人欧美特级aaaaaa片| 婷婷六月久久综合丁香| 少妇的丰满在线观看| 免费高清视频大片| 国产精品 欧美亚洲| 国产野战对白在线观看| 老司机午夜十八禁免费视频| 日韩欧美 国产精品| 久久亚洲真实| 757午夜福利合集在线观看| 国产精品日韩av在线免费观看| 亚洲天堂国产精品一区在线| 久久婷婷人人爽人人干人人爱| 国产视频内射| 我的老师免费观看完整版| 国产精品久久久久久人妻精品电影| 一区二区三区国产精品乱码| 五月玫瑰六月丁香| 久久热在线av| 免费电影在线观看免费观看| 亚洲第一电影网av| 国产精品av视频在线免费观看| 丁香六月欧美| 五月伊人婷婷丁香| 美女 人体艺术 gogo| 亚洲欧美精品综合一区二区三区| 久久精品影院6| 国产欧美日韩精品一区二区| 免费看a级黄色片| 日本一二三区视频观看| 亚洲专区国产一区二区| 成人午夜高清在线视频| 中文资源天堂在线| 亚洲无线在线观看| 看片在线看免费视频| 丰满人妻一区二区三区视频av | 成人18禁在线播放| 男人舔女人下体高潮全视频| 成熟少妇高潮喷水视频| 村上凉子中文字幕在线| 国产97色在线日韩免费| 婷婷精品国产亚洲av在线| 青草久久国产| 麻豆成人av在线观看| 日本a在线网址| netflix在线观看网站| 国产成人福利小说| 偷拍熟女少妇极品色| 成人永久免费在线观看视频| 免费观看人在逋| 日韩大尺度精品在线看网址| 亚洲激情在线av| www国产在线视频色| 国产极品精品免费视频能看的| 天天添夜夜摸| 91九色精品人成在线观看| 久久久水蜜桃国产精品网| 国产 一区 欧美 日韩| 色噜噜av男人的天堂激情| 亚洲欧美日韩高清专用| 一级毛片高清免费大全| 国产精品99久久99久久久不卡| www日本黄色视频网| 免费看十八禁软件| 9191精品国产免费久久| 国产三级中文精品| 性色av乱码一区二区三区2| xxx96com| 18禁观看日本| 91久久精品国产一区二区成人 | 国产伦人伦偷精品视频| 午夜福利成人在线免费观看| 麻豆成人av在线观看| 国产精品av视频在线免费观看| 99在线视频只有这里精品首页| 成人av一区二区三区在线看| 免费看美女性在线毛片视频| 精品久久久久久成人av| 亚洲中文字幕一区二区三区有码在线看 | 久久久成人免费电影| 桃色一区二区三区在线观看| 亚洲自拍偷在线| x7x7x7水蜜桃| 成人一区二区视频在线观看| 国产高清视频在线观看网站| 人妻丰满熟妇av一区二区三区| 久久久久久久久久黄片| 怎么达到女性高潮| 亚洲精品国产精品久久久不卡| 一级毛片女人18水好多| 免费高清视频大片| 国产精品野战在线观看| 欧美一区二区国产精品久久精品| 亚洲av片天天在线观看| 亚洲性夜色夜夜综合| 国产亚洲精品综合一区在线观看| 国产精品日韩av在线免费观看| 欧美成人免费av一区二区三区| 亚洲午夜精品一区,二区,三区| 一级a爱片免费观看的视频| 99riav亚洲国产免费| 中文在线观看免费www的网站| 日韩欧美 国产精品| 日本a在线网址| 亚洲在线自拍视频| 亚洲自拍偷在线| 亚洲五月婷婷丁香| 搡老熟女国产l中国老女人| 国产免费av片在线观看野外av| 国产美女午夜福利| 国产精品一区二区三区四区免费观看 | 最近最新中文字幕大全电影3| 在线视频色国产色| 日本黄色片子视频| 国产一区在线观看成人免费| svipshipincom国产片| avwww免费| 熟女少妇亚洲综合色aaa.| 巨乳人妻的诱惑在线观看| 婷婷丁香在线五月| 男女下面进入的视频免费午夜| 校园春色视频在线观看| 神马国产精品三级电影在线观看| 高清毛片免费观看视频网站| 性欧美人与动物交配| 亚洲精品久久国产高清桃花| aaaaa片日本免费| 欧美日韩国产亚洲二区| 黄色 视频免费看| 琪琪午夜伦伦电影理论片6080| 亚洲人成伊人成综合网2020| 老司机深夜福利视频在线观看| 亚洲国产欧洲综合997久久,| 欧美中文日本在线观看视频| 国产午夜福利久久久久久| 免费观看人在逋| 国产aⅴ精品一区二区三区波| 亚洲精品在线观看二区| 这个男人来自地球电影免费观看| 国内精品久久久久久久电影| 日韩精品中文字幕看吧| 麻豆国产97在线/欧美| 国产激情久久老熟女| 伊人久久大香线蕉亚洲五| 中国美女看黄片| 一本综合久久免费| 男女床上黄色一级片免费看| 国内精品久久久久精免费| 久久久水蜜桃国产精品网| 小蜜桃在线观看免费完整版高清| 91av网站免费观看| 曰老女人黄片| 嫁个100分男人电影在线观看| 亚洲中文字幕一区二区三区有码在线看 | 久久国产精品人妻蜜桃| 国产一区二区在线av高清观看| 国产精品久久电影中文字幕| 精品乱码久久久久久99久播| 窝窝影院91人妻| 国产精品爽爽va在线观看网站| 中文资源天堂在线| 亚洲国产精品成人综合色| 黄色日韩在线| 午夜精品在线福利| 欧美又色又爽又黄视频| 久久久国产成人免费| 每晚都被弄得嗷嗷叫到高潮| 99久久精品国产亚洲精品| 一级a爱片免费观看的视频| 亚洲国产中文字幕在线视频| 久久久国产精品麻豆| 国产欧美日韩精品一区二区| 免费大片18禁| 黄色成人免费大全| 高清毛片免费观看视频网站| 国产爱豆传媒在线观看| 欧美中文日本在线观看视频| 丰满的人妻完整版| 99精品在免费线老司机午夜| 舔av片在线| 亚洲欧美日韩东京热| 久久久久久久精品吃奶| 精品国产超薄肉色丝袜足j| 在线观看免费午夜福利视频| 亚洲精品一区av在线观看| 老司机深夜福利视频在线观看| 啪啪无遮挡十八禁网站| 动漫黄色视频在线观看| 99热6这里只有精品| 日韩国内少妇激情av| 久久久精品欧美日韩精品| 99国产精品一区二区三区| 天天躁日日操中文字幕| 老汉色∧v一级毛片| x7x7x7水蜜桃| 国内揄拍国产精品人妻在线| 久久久久久久久免费视频了| 中文字幕熟女人妻在线| 香蕉久久夜色| 国产一区二区三区在线臀色熟女| 国产精品永久免费网站| 国产久久久一区二区三区| 国产爱豆传媒在线观看| 色播亚洲综合网| 欧美日本视频| 天堂√8在线中文| www.熟女人妻精品国产| 巨乳人妻的诱惑在线观看| 女警被强在线播放| 亚洲一区二区三区色噜噜| 最近最新免费中文字幕在线| 欧美日韩国产亚洲二区| 国产精品爽爽va在线观看网站| 丁香六月欧美| 久久精品影院6| 国产精品一区二区免费欧美| 草草在线视频免费看| 国产v大片淫在线免费观看| 免费看a级黄色片| 免费在线观看成人毛片| 欧美+亚洲+日韩+国产| 好男人在线观看高清免费视频| 网址你懂的国产日韩在线| 欧美激情在线99| 国产精品乱码一区二三区的特点| 亚洲性夜色夜夜综合| 亚洲午夜理论影院| 亚洲精品中文字幕一二三四区| 国产亚洲精品久久久久久毛片| 可以在线观看的亚洲视频| 搡老岳熟女国产| 最近最新中文字幕大全电影3| 欧美日韩中文字幕国产精品一区二区三区| 久久久久国产精品人妻aⅴ院| 18禁裸乳无遮挡免费网站照片| 精品电影一区二区在线| 亚洲av电影不卡..在线观看| 天堂√8在线中文| 国产精品99久久久久久久久| 男人的好看免费观看在线视频| 日韩欧美国产一区二区入口| 美女扒开内裤让男人捅视频| 天堂√8在线中文| 久久精品亚洲精品国产色婷小说| 国产成人影院久久av| 此物有八面人人有两片| 日本在线视频免费播放| 久久久久久久午夜电影| 曰老女人黄片| 日韩有码中文字幕| 久久精品夜夜夜夜夜久久蜜豆| 国内精品久久久久精免费| 欧美日韩乱码在线| 国产99白浆流出| 成人鲁丝片一二三区免费| 宅男免费午夜| 亚洲一区二区三区不卡视频| 男插女下体视频免费在线播放| 精品人妻1区二区| 黄色丝袜av网址大全| 亚洲中文av在线| 白带黄色成豆腐渣| 国产欧美日韩一区二区精品| 欧美在线黄色| 亚洲 欧美一区二区三区| 亚洲av第一区精品v没综合| 国产一级毛片七仙女欲春2| 欧美3d第一页| 亚洲,欧美精品.| 欧美日韩国产亚洲二区| 禁无遮挡网站| 天天躁狠狠躁夜夜躁狠狠躁| 久久国产精品人妻蜜桃| 日韩欧美免费精品| 亚洲精品在线美女| 女人被狂操c到高潮| 久久久久久久久久黄片| 亚洲自偷自拍图片 自拍| 亚洲成av人片免费观看| 少妇人妻一区二区三区视频| 亚洲欧美日韩高清在线视频| 91在线精品国自产拍蜜月 | 国产精品久久久av美女十八| 国产在线精品亚洲第一网站| 我要搜黄色片| 久久精品91蜜桃| 久久精品国产99精品国产亚洲性色| 美女高潮喷水抽搐中文字幕| 国产伦一二天堂av在线观看| 毛片女人毛片| 成人三级黄色视频| 中文字幕久久专区| 亚洲黑人精品在线| 久久欧美精品欧美久久欧美| 国产高清videossex| 舔av片在线| 国产精品久久久久久久电影 | 岛国在线观看网站| 女警被强在线播放| 欧美午夜高清在线| 无限看片的www在线观看| 国产精品久久视频播放| 欧美一区二区精品小视频在线| 国产激情久久老熟女| 视频区欧美日本亚洲| 在线观看66精品国产| 国内精品久久久久久久电影| 国产精品,欧美在线| 熟妇人妻久久中文字幕3abv| 国产淫片久久久久久久久 | 国产精品1区2区在线观看.| 在线免费观看不下载黄p国产 | 亚洲国产日韩欧美精品在线观看 | 精品一区二区三区av网在线观看| 亚洲av第一区精品v没综合| 每晚都被弄得嗷嗷叫到高潮| 午夜成年电影在线免费观看| 成在线人永久免费视频| 999久久久精品免费观看国产| 精品午夜福利视频在线观看一区| 亚洲国产欧美人成| 在线观看免费视频日本深夜| 国产成人欧美在线观看| 九色成人免费人妻av| 国产精品1区2区在线观看.| av片东京热男人的天堂| 中文字幕人妻丝袜一区二区| 亚洲av五月六月丁香网| av女优亚洲男人天堂 | 欧美丝袜亚洲另类 | 欧美丝袜亚洲另类 | 亚洲在线自拍视频| 90打野战视频偷拍视频| 亚洲自偷自拍图片 自拍| 中亚洲国语对白在线视频| 一区二区三区激情视频| 好男人在线观看高清免费视频| 男人舔奶头视频| 又爽又黄无遮挡网站| 草草在线视频免费看| 亚洲人成网站高清观看| 欧美性猛交黑人性爽| 亚洲自偷自拍图片 自拍| 精品久久久久久久末码| 1000部很黄的大片| 国产视频内射| 男人舔女人下体高潮全视频| 国产aⅴ精品一区二区三区波| 老汉色av国产亚洲站长工具| 国产伦在线观看视频一区| 亚洲黑人精品在线| 999久久久精品免费观看国产| 日韩欧美国产在线观看| 精品99又大又爽又粗少妇毛片 | 日本 欧美在线| 免费在线观看影片大全网站| 两人在一起打扑克的视频| av欧美777|