Discrete Fractional Lagrange Equations of NonconservativeSystems

SONG Chuanjing，ZHANG Yi

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

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.

Key words: discrete Lagrange equation;time scale;fractional difference operator;nonconservative system

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［6-22］，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)=inf{s ∈T：s ＞t}is called the forward jump operator.

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

（4）Tκ=T((supT )，supT ] when sup T ＜∞；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) | ≤ε|σ(t) -ω| for all ω ∈N，then fΔ(t) is called the delta derivative of f at t.

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.

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

where

Hence

Fractional summation by parts is given as

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

2 Discrete Equation

Assume that the configuration of a mechanical system is determined by the generalized coordinates qσi，i=1，2，…，n，the kinetic energy function isThe Hamilton's principle for the nonconservative system with fractional difference operators of Riemann-Liouville type has the following form

From Eqs（.6）and（7），we have

Considering

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

Since the value of δqσjis arbitrary，we obtain

in Eq.（15），Qjcontains the conservative for ce Q′jand the nonconservative force Q″j. if Q′jis potential，that is，there exists a function V=V(t，qσi)such that

Since

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

if Q′jhas the generalized potential，that is，there exists a function U=U(such that

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

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

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

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

3 Discrete Equation with Dynamic Constraint

We assume that the motion of the nonconserva-tive system is subjected to the following ideal dynamic constraint

which satisfies

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

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

it follows from Eqs.（26）and（27）that

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

in Eq.（29），Qjcontains the conservative force Q′jand the nonconservative force Q″j. if Q′jis potential，that is，there exists a function V=V(t，qσi)such that

Since

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

if Q′jhas generalized potential，that is，there exists a function U=U(t)such that

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

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

Remark 2 if α=1，L and hkdo not depend ontΔβbqi，and the discrete Lagrange equation of the nonconservative system with dynamic constraint can be obtained

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

4 Examples

Example 1 Consider the following nonconservative system

with dynamic constraint

From Eq.（36），we have

From Eq.（39），we have

it follows from Eqs.（40）and（41）that

Specially，when α=β=1，we obtain

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

the dynamic constraint is

From Eq.（36），we have

From Eq.（45），we have

it follows from Eqs.（46）and（47）that

Specially，when α=β=1，we obtain

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.