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Noether symmetries and conserved quantities of fractional

nonconservative singular systems

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ICPMS2018 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1053 (2018) 012083 doi :10.1088/1742-6596/1053/1/012083

Noether symmetries and conserved quantities of fractional

nonconservative singular systems

Mingliang Zheng1

Shcool of mechanical and electrical, Taihu University of Wuxi, Wuxi, 214064, China

1Email: zhmlwxcstu@163.com

Abstract. The Noether theory of fractional nonconservative singular systems is studied based

on fractional factor derivative method in form space. The Lagrange equations with fractional

factor are established through the variational principle. The criterion equation and the

conserved quantities are further studied according to the fractional order Hamilton action

quantity maintain invariance under the infinitesimal transformation. Finally, an example is

given to illustrate the application. The results show that comparing with the conservative

systems, the nonconservative forces have impact on the Noether identity, but because of

enhancing the invariance condition, it does not change the form of Noether type conserved

quantities, at the same time, we use fractional factor method to study the nonconservative

singular systems, some conclusions are highly natural consistent with the classical integer

order singular systems, so the fractional factor can establish the connection between the

fractional order systems and the integer order systems.

1. Introduction

Fractional order dynamics plays an important role in the study of complex classical dynamics theory

and quantum mechanics theory, especially in the field of chaos and micro environment. At present,

there are two main forms of research on fractional dynamics in the world. One is the fractional order

dynamics of sequence form, represented by the physicist Fred Riewe [1-2]. The other is fractional

order dynamics of the order of alpha form, represented by mathematicians Om.P.Agrawal [3] and

Vasily E.Tarasov [4]. In the dynamic analysis, the study of symmetries and conserved quantities of

fractional variational problems is an important aspect of fractional order dynamic system. In recent

years, the fractional Lagrange system [5-6] and fractional Hamilton system [7-8], fractional order

Birkhoff system [9], fractional nonconservative dynamical systems [10], fractional order generalized

Hamilton system [11-12] and fractional order nonholonomic mechanical systems such as [13] have

achieved certain results.Khalil [14] and Abdeljawad [15] proposed a new method of fractional calculus

recently. The definition of this derivative is the limit form, The fractional order can be transformed

into integer order by using polynomial function. Fu [16-17] respectively studied the Noether, Lie

symmetries and conserved quantities of fractional order Lagrange and Hamilton systems based on the

joint Caputo derivatives and the uniform fractional derivatives. Fu [18] obtains some new results on

the equations of motion and integral factors of holonomic fractional Lagrange systems based on

fractional factor.

Study on the symmetries of fractional nonsingular systems have been obtained some results,

However, under the Legendre transformation, when the singular Lagrange system transits to the phase

space and is described by the Hamilton system, there exists an inherent constraint between its

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IOP Conf. Series: Journal of Physics: Conf. Series 1053 (2018) 012083 doi :10.1088/1742-6596/1053/1/012083

canonical variables, which is called the constrained Hamilton system [19]. Many important dynamical

systems in reality are constrained Hamilton system model [20], such as supersymmetry, supergravity,

electromagnetic field, relativistic motion of the particle, superstring and Yang-Mills field etc..

However, research on the variational problem and the symmetries of fractional constraint Hamilton

system is rarely reported, almost at the beginning stage.In this paper, a new definition of fractional

derivative of fractional factor is given, and then the theory of Noether symmetry for fractional singular

systems is established.

2. Fractional factor and fractional derivative

As everyone knows, Riemann-Liouville fractional derivative, Grunwald-Letnikov fractional derivative

and Caputo fractional derivative are the integral form of the definition, it has only linear optimality,

but its basic properties of calculus with integer order calculus is not a natural consistency. Recently, a

novel fractional derivative whose definition and important properties follows [18].

The

order derivative (

10

) of function

)(tfy

, which is defined with fractional factor:

tdedt

tet

td

tdf

t

tftetf

tffD

t

t

t

t

)1(

)1(

)1(

0

)(

)()(

lim)()(

(1)

Fractional integral based on fractional factor can be used as：

b

a

t

ab

n

i

i

t

ab

dttfetdtftffI

i

)()()(lim)(

)1(

1

0}max{

(2)

The exchange relations between isochronous variational and fractional order operators, and the

fractional differential rule of composite functions are：

fgDgfDfgD

dt

qd

eqD

dt

dq

eqD tt

)()()(

)( )1()1(

(3)

3. The motion e

q

uations of fractional nonconservative sin

g

ular s

y

stems

The form of fractional order mechanical systems is determine by generalized coordinates

),...,2,1( nsqs, the Lagrange function is

),,(

ss

qDqtL

)10(

, the non-potential and non-

conservative force is ),,( sss qDqtQ

. The Hamilton variational principle of fractional order systems with

nonconservative forces is:

0,0

0]),,([]),,([

)1(

btsats

b

assss

t

b

assss

qqt

dtqQqDqtLetdqQqDqtLS

(4)

According to the formula (3), we have:

b

as

s

ss

s

tdqD

qD

L

qQ

q

L

S

])[(

(5)

And because:

s

s

s

s

s

s

q

qD

L

Dq

qD

L

DqD

qD

L

)()(

(6)

The formula (6) is substituted into the formula (5), we have:

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ICPMS2018 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1053 (2018) 012083 doi :10.1088/1742-6596/1053/1/012083

b

ass

ss

t

ts

s

b

ass

ss

b

as

s

b

ass

ss

b

as

s

b

ass

ss

tdqQ

qD

L

D

q

L

q

qD

L

tdqQ

qD

L

D

q

L

tdq

qD

L

td

d

tdqQ

qD

L

D

q

L

tdq

qD

L

DtdqQ

qD

L

D

q

L

S

])([(

)][(])([(

)][(])([(

)]([])([(

2

1

(7)

Due to the s

q

is arbitrary, so:

0)(

s

ss

Q

q

L

qD

L

D

(8)

It is completely equivalent to the motion equations of fractional order nonconservative systems are

obtained through the D'Alembert-Lagrange principle.

Spreading the formula (8), and the fractional order generalized acceleration are obtained:

tqD

L

q

qqD

L

Qe

q

L

eqD

dt

d

qDqD

L

s

k

ks

s

t

s

t

k

ks

22

)1()1(

2

][

(9)

Because of the singularity, the generalized accelerations are not all expressed, but a part of them can be

solved, so it can be recorded:

),...,2,1)(,,,( rkDtAqD

dt

d

kk qqq

(10)

And the (n- r) relationships are:

),...,2,1(0),,,( rnjDt

j

qqq

(11)

4. Noether s

y

mmetr

y

of fractional nonconservative sin

g

ular s

y

stems

Set

as an infinitesimal parameter,

s

,

0

are the generating elements of infinitesimal transformation,

the infinitesimal transformations contain time and generalized coordinates:

),,,(),,,,(

*

0

*

sssssssss

qDqqtqqqDqqttt

(12)

Under the (12), the Hamilton action quantity of the system is changed to:

][

}{

dt

td

LqD

qD

L

q

q

L

t

t

L

I

dt

td

LLIS

sa

s

s

s

ab

ab

(13)

Be aware：

)(

)(

0

11

0

0

DqD

tqDqDtqDqDqD

qtqqq

t

ss

sssss

ssssss

(14)

Below, we introduce the definition and criterion equations (Noether identities) of the Noether

symmetric transformation for fractional order singular systems.

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Definition: If the Hamilton action quantity of the fractional order singular systems is the invariant

under the infinitesimal transformation (12), that is, for each generating element

s

,

0

,itisalways

established:

])([

ssN

ab

qQG

td

d

IS

(15)

The

),,,,(

sssN

qDqqtG

is a gauge function, then, the infinitesimal transformation is the Noether

generalized quasi symmetric transformation.

Simultaneous the formula (13), (14) and (15), we can obtain the criterion equation (Noether identity)

of the Noether quasi symmetric transformation:

0)()(

0000

Nsssss

s

s

s

GDqQLDqD

qD

L

q

L

t

L

(16)

The invariance of the intrinsic constraint equations (11) under the infinitesimal transformation (12)

are the following limit equations:

0])()([()(

0000

s

j

sss

s

j

ss

s

j

s

j

qD

qD

dt

d

qDD

q

q

qt

(17)

5. The conservation of Noether s

y

mmetr

y

There is a close and profound relationship between the symmetry and the conserved quantity of the

mechanical systems, and the Noether symmetry can directly lead to the conservation.

According to the relation between total variation and isochronous variation, the formula (13) can

also be expressed as：

s

s

as

s

s

s

tab

t

s

s

s

s

ab

q

qD

L

Dq

q

L

q

qD

L

Le

td

d

I

tLeqD

qD

L

q

q

L

I

tSSS

)[

0

)1(

)1(

(18)

Simultaneous the formula (8), (14) and (15), the formula (18) can be reduced to:

0]})([{ 00

)1(

Nss

s

tab Gq

qD

L

Le

td

d

I

(19)

Sum deduced from above, we can easily obtain the Noether theorem for fractional order

singular systems:

Theorem: If the generating elements

),(),,(

0sssssss

qDqqtqDqqt

,,,,

and the gauge function,

),,,,(

sssN

qDqqtG

satisfy the criterion equation (16) and the limit equations (17) of Noether

generalized quasi ymmetric transformation for fractional singular Lagrange systems, then the system

have the Noether type conserved quantities:

constGq

qD

L

LeI

Nss

s

t

)(

00

)1(

(20)

6. Illustrated exam

p

le

The Lagrange function of fractional order systems with three degrees of freedom is:

321

2

2

2

1)(])()[(

2

1qDqqqDqDL

(21)

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10

. The nonconservative generalized forces are:

0,, 31221

QqDQqDQ

(22)

Please study the Noether symmetry and conserved quantity.

According to the equations (10), the system's differential equations are:

3

)1(

1

)1(

2

3

)1(

2

)1(

1

qDeqDeqD

dt

d

qDeqDeqD

dt

d

tt

tt

(23)

The rank of hessian matrix of

L

is

r

= 2, thus, there is one inherent constraint:

0

21 qq

(24)

According to the Noether generalized quasi symmetric criterion equation (16) , we have：

0)()())((

)()(

02210112003321

022201112313

N

GDqqDqqDLDqDqq

DqDqDDqDqDqDqD

(25)

According to the limit equations(17) , we have:

0)()(

022011

qq

(26)

The infinitesimal generating element and the gauge function in (25) and (26) may be considered as

the following solution:

0,0,1,1,0

3210

N

G

(27)

According to (20), the Noether generalized quasi symmetry leads to the conserved quantity is：

constqDqDI

21

(28)

7. Conclusions

In order to solve the large number of singular Lagrange function problems in modern physics and

mechanics, in this paper, the Noether symmetry theory has been extended to fractional order

nonconservative singular systems based on a fractional factor, a series of new results have been given .

The fractional order motion differential equations are add to a mixture of the first derivative and the

fractional partial derivative in comparing with the integer order singular systems. A part of the

generalized accelerations can be solved, the remaining part can expressed as constraints, so the

singularity have impact on the Noether symmetry, the generating elements need to maitanin the

constraints for invariant. The generating elements function and gauge function are related to the

nonconservative forces, but the nonconservative forces don't appear in the Noether type conserved

quantities, the form between the classical Noether and the fractional Noether conserved quantities is

coincident, it only adds to a fractional factor, it shows that the biggest feature of the fractional factor

derivative method is that the fractional order systems can be transformed into an integer order systems.

The method in this paper can be applied to other fractional order dynamics problems, such as stability,

symmetric perturbation and adiabatic invariants, etc.

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