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Noether symmetries and conserved quantities of fractional nonconservative singular systems

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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.
Journal of Physics: Conference Series
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Noether symmetries and conserved quantities of fractional
nonconservative singular systems
To cite this article: Mingliang Zheng 2018 J. Phys.: Conf. Ser. 1053 012083
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1234567890 ‘’“”
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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ICPMS2018 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1053 (2018) 012083 doi :10.1088/1742-6596/1053/1/012083
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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IOP Conf. Series: Journal of Physics: Conf. Series 1053 (2018) 012083 doi :10.1088/1742-6596/1053/1/012083
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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Fractional derivative modeling of mechanics and engineering problems
  • W Sun
  • H Sun
  • X C Li
Sun W, Sun H G and Li X C 2010 Fractional derivative modeling of mechanics and engineering problems (in Chinese) (Beijing: Science Press) 3