Vol.
122 (2012)
ACTA PHYSICA POLONICA A
No. 1
Perturbation to Noether Symmetry and Noether
Adiabatic Invariants of Discrete Dierence
Variational Hamilton Systems
M.-J. Zhanga,b,∗ and J.-H. Fangc
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China
Key Laboratory of Space Object and Debris Observation, PMO, CAS, Nanjing 210008, China
a
b
c
College of Science, China University of Petroleum, Qingdao 266555, China
(Received April 15, 2011; in nal form March 2, 2012)
The perturbation to the Noether symmetry and the Noether adiabatic invariants of discrete dierence
variational Hamilton systems are investigated. The discrete the Noether exact invariant induced directly by the
the Noether symmetry of the system without perturbation is given. The concept of discrete high-order adiabatic
invariant is presented, the criterion of the perturbation to the Noether symmetry is established, and the discrete
the Noether adiabatic invariant induced directly by the perturbation to the Noether symmetry is obtained. Lastly,
an example is discussed to illustrate the application of the results.
PACS: 02.20.Sv, 11.30.−j, 45.05.+x
1. Introduction
adiabatic invariants play a very important role in the research on the quasi-integrability of mechanical systems.
A classical adiabatic invariant is a certain physical quantity that changes more slowly than some slowly-varying
parameter of the system [17]. In fact, the parameter
varying very slowly is equivalent to the action of a small
disturbance. At present, studies in this eld have become
very active, and many important results have been obtained [1823]. But all these studies have focused on perturbation to symmetries and adiabatic invariants of the
continuous mechanical systems. Recently, Zhang et al.
[24] presented the concept of discrete high-order adiabatic invariant, and studied the perturbation to the the
Noether symmetry and the Noether adiabatic invariant
of the general discrete holonomic system. Wang and Zhu
[25] further discussed perturbation to symmetry and adiabatic invariants of general discrete holonomic dynamical systems on a uniform lattice. However, perturbation
to symmetries and adiabatic invariants of the discrete
Hamilton systems has never been studied so far.
In this paper, based on the concept of discrete
high-order adiabatic invariant, the perturbation to the
Noether symmetry and the Noether adiabatic invariants
of discrete dierence variational Hamilton systems are
studied. The discrete the Noether exact invariant induced directly by the the Noether symmetry of the system without perturbation is given. The criterion of the
perturbation to the Noether symmetry is established, and
the discrete the Noether adiabatic invariant induced directly by the perturbation to the Noether symmetry is
obtained. Meanwhile, an example is discussed to illustrate the application of the results.
The research on symmetries and conserved quantities of a mechanical system possesses important mathematical and physical signicance, and is also an important development direction in analytical mechanics. In
Refs. [1, 2] symmetries and conserved quantities of constrained mechanical systems have been thoroughly investigated. Moreover, symmetries and conserved quantities of more general systems (optimal control problems)
are also studied in depth [35]. Recently, researches
on symmetries and conserved quantities have been extended to a general time scale (including, as particular
cases, both discrete and continuous settings). Levi et al.
[68] rst extended the Lie symmetry to discrete systems; Dorodnitsyn [9] adapted the Noether's theory to
discrete Lagrangian system; Torres et al. [1012] studied
the Noether symmetry theorems for an arbitrary time
scale; Shi et al. [13, 14] investigated Lie symmetry and the
Noether conserved quantities (or exact invariants) of discrete non-conservative mechanical systems and discrete
dierence variational Hamilton systems without perturbations; based upon the property of the discrete models entirely inheriting the symmetry of the continuous
systems, Fu et al. [15] presented the Hojman conserved
quantities and Lie symmetries of discrete mechanico-electrical coupling systems by the Lie groups of transformations of continuous systems.
As we know, even a tiny disturbance acting on the mechanical systems, that we can call a perturbation, may
inuence the original symmetries and conserved quantities of mechanical systems. Pioneered in this area, Burgers [16] proposed adiabatic invariants for a special kind of
the Hamilton systems. Perturbation to symmetries and
∗ corresponding author; e-mail:
2. the Noether symmetry and discrete the
Noether exact invariant
For brevity of notation, we consider a one-dimensional discrete dierence variational Hamil-
mjzhang@pmo.ac.cn
(25)
M.-J. Zhang, J.-H. Fang
26
ton system.
The Hamiltonian of the system is
Hd (tk , tk+1 , qk , qk+1 , pk , pk+1 ) (k = 0, 1, . . . , N − 1), tk ,
qk , and pk are discrete time, discrete generalized coordinates, and discrete generalized momenta, respectively.
The discrete equations of motion of the system can be
written as [14]:
1
(pk−1 − pk+1 ) − D3 Hd (φk )(tk+1 − tk )
2
− D4 Hd (φk−1 )(tk − tk−1 ) = 0,
1
2
(1)
(qk+1 − qk−1 ) − D5 Hd (φk )(tk+1 − tk )
− D6 Hd (φk−1 )(tk − tk−1 ) = 0,
(2)
Hd (φk ) − Hd (φk−1 ) − D1 Hd (φk )(tk+1 − tk )
− D2 Hd (φk−1 )(tk − tk−1 ) = 0,
(3)
where Dj is the partial derivative of the discrete function with respect to the argument j and
φk = (tk , tk+1 , qk , qk+1 , pk , pk+1 ) represents the discrete
sequence.
We introduce the innitesimal transformations with
respect to discrete time tk , discrete generalized coordinates qk , and discrete generalized momenta pk as
t∗k = tk + ∆tk = tk + ετk0 (tk , qk , pk ),
qk∗ = qk + ∆qk = qk + εξk0 (tk , qk , pk ),
p∗k = pk + ∆pk = qk + εηk0 (tk , qk , pk ),
(4)
τk0 ,
where ε is an innitesimal group parameter,
and
ηk0 are the discrete innitesimal generators. The vector
eld of generators is
∂
∂
∂
(0)
X0,d = τk0
+ ξk0
+ ηk0
,
(5)
∂tk
∂qk
∂pk
which can be prolonged to the two-point scheme
∂
∂
(1)
(0)
0
0
X0,d = X0,d + τk+1
+ ξk+1
∂tk+1
∂qk+1
∂
0
+ ηk+1
.
(6)
∂pk+1
ξk0
The recursive and derivative operators of discrete systems for any discrete variable or function are represented
as
R± f (zk ) = f (zk±1 ),
(7)
R+ f (zk ) − f (zk )
.
(8)
tk+1 − tk
The the Noether symmetry is an invariance of the
Hamiltonian action functional under the innitesimal
transformations.
Then the requirement of the the
Noether symmetry of the discrete dierence variational
Hamilton system gives
Dd f (zk ) =
(1)
Hd (φk )Dd (τk0 ) + X0,d [Hd (φk )] − 21 (pk+1 + pk )Dd (ξk0 )
+ 12 (qk+1 + qk )Dd (ηk0 ) + Dd (G0N k ) = 0,
where
(9)
is a discrete gauge function.
Criterion 1. If a discrete gauge function G0N k exists
such that the innitesimal generators τk0 , ξk0 and ηk0 satisfy
G0N k
=
G0N k (tk , qk , pk )
Eq. (9), the invariance is the Noether symmetry of the
discrete dierence variational Hamilton system (1), (2)
and (3).
Equation (9) is called the discrete the Noether identity
of the discrete dierence variational Hamilton system (1),
(2) and (3).
Theorem 1 [14]. For the discrete dierence varia-
tional Hamilton system (1), (2) and (3), if the innitesimal generators τk0 , ξk0 and ηk0 of the the Noether symmetry
and discrete gauge function G0N k satisfy the discrete the
Noether identity (9), the the Noether symmetry of the
system can induce the discrete the Noether exact invariant
IN0,d = τk0 (tk − tk−1 )D2 [R− Hd (φk )]
+ ξk0 (tk
(10)
− tk−1 )D4 [R− Hd (φk )]
+ ηk0 (tk − tk−1 )D6 [R− Hd (φk )] + τk0 R− Hd (φk )
− 21 (pk−1 + pk )ξk0 + 12 (qk−1 + qk )ηk0 + G0N k = const.
3. Perturbation to the Noether symmetry and
discrete the Noether adiabatic invariant
Small force acting on the discrete dierence variational
Hamilton system may result in a small change in its symmetries, i.e. perturbation to symmetries. The conserved
quantity associated with the symmetries, under a corresponding change, is an adiabatic invariant. In analytical mechanics, we study perturbation to symmetries and
adiabatic invariants of mechanical systems based on the
concept of high-order adiabatic invariant. According to
the concept of adiabatic invariant [17], for the discrete
dierence variational Hamilton system, we have
Denition. If Lz,d (tk , tk+1 , qk , qk+1 , pk , pk+1 , ε) is
a physical quantity including ε in which the highest
power is z in a discrete dierence variational Hamil-
ton system, and its derivative with respect to discrete
time tk is directly proportional to εz+1 , Iz,d is called
a z -th order adiabatic invariant of the discrete dierence variational Hamilton system. Denition. If
Lz,d (tk , tk+1 , qk , qk+1 , pk , pk+1 , ε) is a physical quantity
including ε in which the highest power is z in a discrete
dierence variational Hamilton system, and its derivative
with respect to discrete time tk is directly proportional to
εz+1 , Iz,d is called a z -th order adiabatic invariant of the
discrete dierence variational Hamilton system.
Suppose the discrete dierence variational Hamilton
system (1), (2) and (3) is perturbed by small quantity εWd (tk , tk+1 , qk , qk+1 , pk , pk+1 ) = εWd (φk ), then we
have the following total variation of corresponding discrete Hamiltonian action functional:
N
−1
∑
∆Sd +
εWd (φk )(tk+1 − tk )∆qk
−
k=0
N
−1
∑
εWd (φk )(qk+1 − qk )∆tk = 0,
k=0
where ∆ is a total variation symbol, and
(11)
Perturbation to Noether Symmetry and Noether Adiabatic Invariants . . .
Sd =
N
−1
∑
1
2
(pk+1 + pk )(qk+1 − qk )
(12)
k=0
−
N
−1
∑
Hd (tk , tk+1 , qk , qk+1 , pk , pk+1 )(tk+1 − tk ).
From (11), using the similar deduction in [14], we obtain the discrete equations of motion of the discrete difference variational Hamilton system perturbed by small
quantity εWd (φk ):
1
2
(pk−1 − pk+1 ) − D3 Hd (φk )(tk+1 − tk )
− D4 Hd (φk−1 )(tk − tk−1 )
+ εWd (φk )(tk+1 − tk ) = 0,
1
2
(14)
− D2 Hd (φk−1 )(tk − tk−1 )
− εWd (φk )(qk+1 − qk ) = 0.
(15)
Because of the action of εWd (φk ), the original symmetries and invariants of the system may vary. Assume that
the variation is a small perturbation based on the symmetrical transformations of the system without perturbation, and τk (tk , qk , pk ), ξk (tk , qk , pk ) and ηk (tk , qk , pk )
express the generators of innitesimal transformations after being perturbed, then
τk = τk0 + ετk1 + ε2 τk2 + . . . ,
ξk =
ηk =
ηk0
+
εξk1
+
εηk1
+
ε2 ξk2
+ ...,
+
ε2 ηk2
+ ...
(16)
The innitesimal transformations become
t∗k = tk + ∆tk = tk + ετk (tk , qk , pk ),
qk∗ = qk + ∆qk = qk + εξk (tk , qk , pk ),
p∗k = pk + ∆pk = pk + εηk (tk , qk , pk ).
(1)
Xd = εm Xm,d (m = 0, 1, . . . , z),
(21)
where
∂
∂
∂
∂
(1)
m
+ ξkm
+ ηkm
+ τk+1
Xm,d = τkm
∂tk
∂qk
∂pk
∂tk+1
m
+ ξk+1
∂
∂
m
+ ηk+1
.
∂qk+1
∂pk+1
(22)
Substituting (16) into (18), noticing (19)(22), and
making the coecients of εm equal, we obtain
(23)
Hd (φk )Dd (τkm ) + Xm,d [Hd (φk )]
− (pk+1 +
+ (qk+1 +
[
]
− Wd (φk ) ξkm−1 − Dd (qk )τkm−1 + Dd (Gm
N k ) = 0,
1
2
pk )Dd (ξkm )
1
2
qk )Dd (ηkm )
when m = 0, we note that τkm−1 = ξkm−1 = ηkm−1 = 0
holds, then Eq. (23) turns into Eq. (9).
Hd (φk ) − Hd (φk−1 ) − D1 Hd (φk )(tk+1 − tk )
ξk0
Substituting (16) into (19), we have
(1)
(13)
(qk+1 − qk−1 ) − D5 Hd (φk )(tk+1 − tk )
− D6 Hd (φk−1 )(tk − tk−1 ) = 0,
(20)
GN k = G0N k + εG1N k + ε2 G2N k + . . .
(1)
k=0
27
(17)
According to the the Noether symmetry theory, the
the Noether identity of the discrete dierence variational
Hamilton system perturbed by small quantity εWd (φk )
becomes
Criterion 2. For the discrete dierence variational
Hamilton system (1), (2) and (3) perturbed by small
quantity εWd (φk ), if the discrete gauge function Gm
N k exists such that the innitesimal generators τkm , ξkm and ηkm
satisfy (23), the corresponding variety of the the Noether
symmetry is called the perturbation to the the Noether
symmetry.
Equation (23) is the determining equation of the perturbation to the the Noether symmetry of the discrete
dierence variational Hamilton system (1), (2) and (3).
Theorem 2. For the discrete dierence variational Hamilton system (1), (2) and (3) perturbed by
small quantity εWd (φk ), if the innitesimal generators
τkm , ξkm and ηkm , and discrete gauge function Gm
N k satisfy
the determining of Eq. (23), the perturbation to the the
Noether symmetry of the system can induce a z -th order
discrete adiabatic invariant
{
INz,d = εm τkm (tk − tk−1 )D2 [R− Hd (φk )]
+ ξkm (tk − tk−1 )D4 [R− Hd (φk )]
+ ηkm (tk − tk−1 )D6 [R− Hd (φk )] + τkm R− Hd (φk )
}
m
m
m
1
1
− 2 (pk−1 + pk )ξk + 2 (qk−1 + qk )ηk + GN k
(1)
Hd (φk )Dd (τk ) + Xd [Hd (φk )] − 21 (pk+1 + pk )Dd (ξk )
+ 21 (qk+1 + qk )Dd (ηk ) − εWd (φk )[ξk − Dd (qk )τk ]
+ Dd (GN k ) = 0,
where
∂
∂
∂
∂
(1)
+ ξk
+ ηk
+ τk+1
Xd = τk
∂tk
∂qk
∂pk
∂tk+1
+ ξk+1
∂
∂
+ ηk+1
,
∂qk+1
∂pk+1
(18)
(24)
(m = 0, 1, . . . , z).
Proof: Using representations (7), (8) and (22), the
expansion of (23) gives
(1)
Hd (φk )Dd (τkm ) + Xm,d [Hd (φk )]
− 12 (pk+1 + pk )Dd (ξkm ) + 12 (qk+1 + qk )Dd (ηkm )
(19)
and GN k = GN k (tk , qk , pk ) is a gauge function. After
being perturbed, the gauge function comes into
[
]
− Wd (φk ) ξkm−1 − Dd (qk )τkm−1 + Dd (Gm
Nk)
= Hd (φk )
m
τk+1
− τkm
∂Hd (φk )
m ∂Hd (φk )
+ τkm
+ τk+1
tk+1 − tk
∂tk
∂tk+1
M.-J. Zhang, J.-H. Fang
28
+ ξkm
∂Hd (φk )
∂Hd (φk )
m ∂Hd (φk )
+ ξk+1
+ ηkm
∂qk
∂qk+1
∂pk
m
+ ηk+1
ξ m − ξkm
∂Hd (φk ) 1
− 2 (pk+1 + pk ) k+1
∂pk+1
tk+1 − tk
+ 12 (qk+1 + qk )
m
ηk+1
− ηkm
tk+1 − tk
]
[
− Wd (φk ) ξkm−1 − Dd (qk )τkm−1 + Dd (Gm
Nk)
[
∂Hd (φk ) tk − tk−1 ∂Hd (φk−1 )
+
∂pk
tk+1 − tk
∂pk
]
qk+1
qk−1
− 21
+ 21
tk+1 − tk
tk+1 − tk
[
∂Hd (φk−1 )
+ Dd τkm (tk − tk−1 )
∂tk
+ηkm
+ ξkm (tk − tk−1 )
∂Hd (φk−1 )
∂qk
∂Hd (φk−1 )
∂pk
= τkm
∂Hd (φk )
tk − tk−1 ∂Hd (φk−1 )
+ τkm
∂tk
tk+1 − tk
∂tk
+ ηkm (tk − tk−1 )
+ τkm
Hd (φk−1 )
Hd (φk )
− τkm
tk+1 − tk
tk+1 − tk
+ τkm Hd (φk−1 ) − 21 (pk−1 + pk )ξkm
+ ξkm
∂Hd (φk )
tk − tk−1 ∂Hd (φk−1 )
+ ξkm
∂qk
tk+1 − tk
∂qk
+ 21 (qk−1 + qk )ηkm + Gm
Nk
− 12
pk−1 ξkm
pk+1 ξkm
+ 12
tk+1 − tk
tk+1 − tk
+ ηkm
− 21
[
]
− Wd (φk ) ξkm−1 − Dd (qk )τkm−1 = 0.
∂Hd (φk )
tk − tk−1 ∂Hd (φk−1 )
+ ηkm
∂pk
tk+1 − tk
∂pk
qk+1 ηkm
qk−1 ηkm
+ 12
tk+1 − tk
tk+1 − tk
m
+ τk+1
∂Hd (φk )
tk − tk−1 ∂Hd (φk−1 )
− τkm
∂tk+1
tk+1 − tk
∂tk
m
+ ξk+1
∂Hd (φk )
tk − tk−1 ∂Hd (φk−1 )
− ξkm
∂qk+1
tk+1 − tk
∂qk
m
+ ηk+1
∂Hd (φk )
tk − tk−1 ∂Hd (φk−1 )
− ηkm
∂pk+1
tk+1 − tk
∂pk
m
+ τk+1
Hd (φk )
Hd (φk−1 )
− τkm
tk+1 − tk
tk+1 − tk
pk−1 ξkm
1
+2
+ 12
tk+1 − tk
pk ξkm
tk+1 − tk
−
−
p
ξm
1 k+1 k+1
2
tk+1 − tk
qk−1 ηkm
1
2
tk+1 − tk
−
1
2
+
1
2
m
pk ξk+1
tk+1 − tk
m
qk+1 ηk+1
tk+1 − tk
m
qk ηk+1
qk ηkm
− 12
+ Dd (Gm
Nk)
tk+1 − tk
tk+1 − tk
[
]
− Wd (φk ) ξkm−1 − Dd (qk )τkm−1
[
∂Hd (φk ) tk − tk−1 ∂Hd (φk−1 )
+
∂tk
tk+1 − tk
∂tk
]
Hd (φk )
Hd (φk−1 )
−
+
tk+1 − tk
tk+1 − tk
[
tk − tk−1 ∂Hd (φk−1 )
m ∂Hd (φk )
+
+ ξk
∂qk
tk+1 − tk
∂qk
]
pk−1
pk+1
+ 12
− 12
tk+1 − tk
tk+1 − tk
=
(25)
Using (13)(15), from (25), we obtain
{ [
∂Hd (φk−1 )
Dd εm τkm (tk − tk−1 )
∂tk
∂H
(φ
)
d
k−1
+ ξkm (tk − tk−1 )
∂qk
∂Hd (φk−1 )
+ ηkm (tk − tk−1 )
+ τkm Hd (φk−1 )
∂pk
]}
− 21 (pk−1 + pk )ξkm + 12 (qk−1 + qk )ηkm + Gm
Nk
{ [
= Dd εm τkm (tk − tk−1 )D2 (R− Hd (φk ))
+ ξkm (tk − tk−1 )D4 (R− Hd (φk ))
+ 12
τkm
]
+ ηkm (tk − tk−1 )D6 (R− Hd (φk )) + τkm R− Hd (φk )
]}
m
m
m
1
1
− 2 (pk−1 + pk )ξk + 2 (qk−1 + qk )ηk + GN k
= Dd INz,d
{
qk+1 − qk m
= εm εWd (φk )
τ − εWd (φk )ξkm
tk+1 − tk k
}
[ m−1
]
m−1
+ Wd (φk ) ξk
− Dd (qk )τk
{
= εm − εWd (φk )[ξkm − Dd (qk )τkm ]
[
]}
+ Wd (φk ) ξkm−1 − Dd (qk )τkm−1
= −εz+1 Wd (φk )[ξkz − Dd (qk )τkz ].
(26)
It shows that Dd INz,d is directly proportional to εz+1 ,
so INz,d is a z -th order adiabatic invariant of the discrete dierence variational Hamilton system, which can
be called the discrete the Noether adiabatic invariant.
When z = 0, which mean that εWd (φk ) vanish, the system becomes the one without perturbation. Then the
Perturbation to Noether Symmetry and Noether Adiabatic Invariants . . .
discrete the Noether adiabatic invariant (24) turns into
the discrete the Noether exact invariant (10).
τk0 (tk , qk , pk ) = 2tk ,
Take the Emden equation to illustrate the application
of the above results. The discrete Hamiltonian of the
system and its recursive operator are
R− Hd (φk ) = Hd (φk−1 ) =
(27)
1 p2k + p2k−1
2 t2k + t2k−1
1
6
+ (t2k qk6 + t2k−1 qk−1
),
(28)
12
and the system is perturbed by the small quantity
qk+1 pk+1 − qk pk
εWd (φk ) = −ε
.
(29)
3qk tk − qk tk+1 − 2qk+1 tk
Let us study the perturbation to the the Noether symmetry and the the Noether adiabatic invariant of the discrete
dierence variational Hamilton system.
We calculate the Dj Hd (φk ) (j = 1, 2, 3, 4, 5, 6) as
tk (p2 + p2 )
D1 Hd (φk ) = − 2 k+1 2 k2 + 16 tk qk6 ,
(tk+1 + tk )
Firstly, we seek the discrete the Noether exact invariant. Suppose that the generators τk0 , ξk0 and ηk0 are linear,
i.e.
τk0 (tk , qk , pk ) = C1 tk + C2 qk + C3 pk + C4 ,
ξk0 (tk , qk , pk ) = C5 tk + C6 qk + C7 pk + C8 ,
(31)
where C1 C12 are constants.
Substituting (30) and (31) into the the Noether identity (9) of the system without perturbation, we have
[ 2
]
1 pk+1 + p2k
1 2
6
2 6
0
(t
q
+
t
q
)
+
k k Dd (τk )
2 t2k+1 + t2k
12 k+1 k+1
[
]
tk (p2 + p2 )
+ 16 tk qk6 − 2 k+1 2 k2 τk0
(tk+1 + tk )
]
[
tk+1 (p2k+1 + p2k ) 0
6
τk+1
−
+ 16 tk+1 qk+1
(t2k+1 + t2k )2
5
0
+ 12 t2k qk5 ξk0 + 12 t2k+1 qk+1
ξk+1
+
+
0
pk+1 ηk+1
t2k+1 + t2k
pk ηk0
2
tk+1 + t2k
− 12 (pk+1 + pk )Dd (ξk0 )
(32)
ξk0 (tk , qk , pk ) = −qk ,
(33)
ηk0 (tk , qk , pk ) = pk ,
the function
G0N k = −qk pk
(34)
satises the the Noether identity (9). According to Theorem 1, the discrete dierence variational Hamilton system
has the following discrete the Noether exact invariant:
IN0,d =
+
tk (p2k + p2k−1 ) 2t2k (tk − tk−1 )(p2k + p2k−1 )
−
t2k + t2k−1
(t2k + t2k−1 )2
(tk − tk−1 )p2k
6
+ 61 tk tk−1 (tk qk6 + tk−1 qk−1
)
t2k + t2k−1
(35)
+ 21 (pk−1 qk + pk qk−1 ) = const.
Secondly, we seek the rst order discrete the Noether
adiabatic invariant. We also suppose that the generators
τk1 , ξk1 and ηk1 are linear, i.e.
τk1 (tk , qk , pk ) = C1′ tk + C2′ qk + C3′ pk + C4′ ,
ξk1 (tk , qk , pk ) = C5′ tk + C6′ qk + C7′ pk + C8′ ,
tk+1 (p2k+1 + p2k ) 1
6
,
D2 Hd (φk ) = −
+ 6 tk+1 qk+1
(t2k+1 + t2k )2
5
,
D3 Hd (φk ) = 12 t2k qk5 , D4 Hd (φk ) = 12 t2k+1 qk+1
pk
pk+1
D5 Hd (φk ) = 2
, D6 Hd (φk ) = 2
.(30)
tk+1 + t2k
tk+1 + t2k
ηk0 (tk , qk , pk ) = C9 tk + C10 qk + C11 pk + C12 ,
+ 12 (qk+1 + qk )Dd (ηk0 ) + Dd (G0N k ) = 0.
It follows that when
4. An example
1 p2k+1 + p2k
1
6
Hd (φk ) =
+ (t2k+1 qk+1
+ t2k qk6 ),
2 t2k+1 + t2k
12
29
′
′
′
ηk1 (tk , qk , pk ) = C9′ tk + C10
qk + C11
pk + C12
,
(36)
′
C1′ C12
are constants.
where
Substituting (30), (31) and (36) into the determining
Eq. (23) of the perturbation to the the Noether symmetry of the system, we have
[ 2
]
1 pk+1 + p2k
1 2
6
2 6
1
+
(t
q
+
t
q
)
k k Dd (τk )
2 t2k+1 + t2k
12 k+1 k+1
[
]
tk (p2 + p2 )
+ 16 tk qk6 − 2 k+1 2 k2 τk1
(tk+1 + tk )
[
]
tk+1 (p2k+1 + p2k ) 1
6
−
+ 16 tk+1 qk+1
τk+1 + 21 t2k qk5 ξk1
(t2k+1 + t2k )2
5
1
+ 12 t2k+1 qk+1
ξk+1
+
pk ηk1
2
tk+1 + t2k
+
1
pk+1 ηk+1
2
tk+1 + t2k
− 21 (pk+1 + pk )Dd (ξk1 ) + 12 (qk+1 + qk )Dd (ηk1 )
+
[ 0
]
qk+1 pk+1 − qk pk
ξk − Dd (qk )τk0
3qk tk − qk tk+1 − 2qk+1 tk
(37)
+ Dd (G1N k ) = 0.
It follows that when
τk1 (tk , qk , pk ) = 2tk ,
ηk1 (tk , qk , pk ) = pk ,
ξk1 (tk , qk , pk ) = −qk ,
(38)
the function
G1N k = −2qk pk
(39)
satises the determining Eq. (23) of the perturbation to
M.-J. Zhang, J.-H. Fang
30
the Noether symmetry. According to Theorem 2, the
discrete dierence variational Hamilton system has the
following rst order discrete the Noether adiabatic invariant:
[
tk (p2k + p2k−1 ) 2t2k (tk − tk−1 )(p2k + p2k−1 )
IN1,d =
−
t2k + t2k−1
(t2k + t2k−1 )2
(
)
(tk − tk−1 )p2k
6
+ 16 tk tk−1 tk qk6 + tk−1 qk−1
2
2
tk + tk−1
]
+ 12 (pk−1 qk + pk qk−1 ) (1 + ε) − εqk pk .
+
(40)
Further we can obtain more high-order discrete adiabatic
invariants. In addition, it is worth noting that there are
many calculations above, which can be also done in an
automatic way using a computer algebra system stated
in Refs. [26, 27].
[3] D.F.M. Torres, in:
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
The perturbation to the the Noether symmetry and
the the Noether adiabatic invariants of discrete dierence
variational systems are studied in this paper. We obtain the discrete the Noether adiabatic invariant induced
directly by the perturbation to the Noether symmetry
of the system. When z = 0, this means that εWd (φk )
vanish, the system becomes the one without perturbations, and the discrete the Noether adiabatic invariant
will turn into the discrete the Noether exact invariant
(i.e. the so-called discrete the Noether conserved quantity in Ref. [14]) naturally. The results of this paper
have important signicance for further study on the discrete mechanical systems and symmetrical perturbation
theory.
[14]
Acknowledgments
[22]
This research was supported by the National Natural
Science Foundation of China (grant No. 11103086) and
Fundamental Research Funds for the Central Universities
of China (grant No. 09CX04018A).
[23]
[1] F.X. Mei,
Symmetries and Conserved Quantities of
, Beijing Institute of
Technology Press, Beijing 2004 (in Chinese).
[2] S.K. Luo, Y.F. Zhang, et al., Advances in the Study of
Dynamics of Constrained Mechanics Systems, Science
Press, Beijing 2008 (in Chinese).
Constrained Mechanical Systems
, Eds. F. Colonius, L. Grüne, Vol. 273,
Springer, Berlin 2002, p. 287.
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5. Conclusion
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