INTRODUCTION
In the study of differential equations, conservation laws have many significant
uses. An important problem is how to find conservation laws for given
differential equations. Emmy Noether, 1918 was the first to combine the
methods of variational calculus with the theory of Lie groups and to formulate
a general approach for constructing conservation laws for EulerLagrange
equations when their variational symmetries are known. "Noether proved
the remarkable result that for systems arising from a variational principle,
every conservation law of the system comes from a corresponding symmetry
property. For example, invariance of a variational principle under a group
of time translations implies the conservation of energy for the solutions
of the associated EulerLagrange equations and invariance under a group
of spatial translations implies conservation of momentum (Olver, 1993).
A thorough study of the Noether`s theorem with many references can be
found in Olver (1993). Numerous examples and applications of the Noether`s
theorem are presented in Ibragimov (1994). Algorithm of finding an admitted
Lie group is demonstrated in Meleshko (2005). Chatanin et al. (2008)
applied Lie group theory to the nonlinear equations of motion of inclined
unsagged cables to construct invariant solutions to the problem. In this
study, the Lie group theory is applied to the nonlinear equations of
motion of inclined sagged cables. By using the Noether`s theorem, conservation
of energy for the solutions of the associated EulerLagrange equations
is obtained.
MATERIALS AND METHODS
Physical model and equations of motion: The configuration
in the global coordinates (X,Y) describes the position of the inclined
sagged cable as shown in Fig. 1. The angle θ shows
the inclination of the cable with respect to X axis. After the disturbance
of an external excitation, the cable configuration is changed into the
dynamic configuration which is displaced from the initial configuration.
The displacement vectors in X and Y directions are represented by
respectively. The horizontal span X_{H} is fixed and the cable`s
vertical span Y_{H} is varied to attain specified values of θ.
In deriving the equation of motion by the virtual workenergy principle
the cable is considered to be perfectly flexible, homogeneous, linearly
elastic with negligible torsional, bending and shear rigidities. Therefore,
the strain energy is due only to stretching of the cable axis.

Fig. 1: 
Configurations of an inclined sagged cable 
The total
strain of the cable at the displaced state, with assumption of moderately
large vibration amplitudes, can be expressed as follows:
In this study, the initial strain e is assumed to have a small value
and is neglected. The prime (`) is used to denote differentiation with
respect to
From Srinil et al. (2004) and Rega et al. (2004), the governing
equations of motion are:
Where:
EA is the cable axial stiffness,
is the horizontal component of the cable static tension T_{s},
w_{c} is the cable weight per unit length, g is the gravity constant.
Here, the curve that defines shape of the cable is
This nonlinear system describes the coupled longitudinal and vertical displacement`s
dynamics and is valid also for slightly sagged horizontal cables if one assumes
T_{s} ≈ H rendering ρ = 1. It contains quadratic and cubic
nonlinear terms due to the cable`s axial stretching even in the absence of initial
sag (taut string case). A method for solving the nonlinear equations of motion
for horizontal and inclined elastic sagged cables by using a numerical technique
is demonstrated in Srinil et al. (2005).
Calculation of infinitesimal generators: An infinitesimal generator
X for the problem is written in the following form:
The main aim is to determine all functions α, β, γ and
η that correspond to the oneparameter symmetry groups of the Eq.
2 and 3. Since the governing equations form a system
of second order PDE, the second prolongation of the infinitesimal generator
is needed. The second prolongation of X is given by:
Where:
and
According to theorem 2.31 Olver (1993):
where, the notation _{F = 0} means that the second prolongation
of X is applied to the solutions of Eq. 2 and 3.
Substituting expressions (5,5`) and (23) into Eq. 6
and 7 and then equating the coefficients of all independent
monomials to zero, we obtain the determining equations. After solving
the determining equations, we get the functions α = 0, β = k_{1},
γ = k_{2}t+k_{3}, η = k_{4}t+k_{5},
where k_{1},...,k_{5} are arbitrary constants. Thus, the
infinitesimal generators have the form:
The infinitesimal generator contains five arbitrary constants. Consequently,
the Lie algebra derived from the governing equations is spanned by the
following five linearly independent generators:
Conservation laws: Consider a system of the second order partial
differential equations:
F_{i}(x, u, u_{(1)},
u_{(2)}) = 0 
(9) 
where, x = (x^{1}, x^{2},...,x^{n}) are the independent
variables, u = (u^{1}, u^{2},...,u^{m}) are the
dependent variables,
are the first and second order derivatives, respectively.
Definition: Equation
D_{i}[C^{i}(x,u,u_{(1)})]
= 0 
(10) 
is called a conservation law for Eq. 9 if it is satisfied
by all solutions u (x) of Eq. 9. The vector C = (C^{1},
C^{2},...,C^{n}) is called a conserved vector and D_{i}
is the total derivative with respect to x^{i}. Consider a variational
integral:
and its EulerLagrange equations:
where, the Lagrangian L of the system involves the independent variables
x = (x^{1}, x^{2},...,x^{n}), the dependent variables
u = (u^{1},u^{2},...,u^{m}) and the firstorder
derivatives
of u with respect to x^{i}.
Infinitesimal criterion of invariance: The following condition
will be necessary and sufficient for a connected group of transformations
to be a symmetry group of the variational problem.
Theorem: A connected group of transformations G is a variational
symmetry group of the variational integral (11) if and only if:
pr^{(1)}X(L)+LDi(ξ^{i})
= 0 
(13) 
for every infinitesimal generators:
Noether`s theorem: (Ibragimov, 2006) Let the variational integral
(11) be invariant under the group with the generator (14). In other words,
let the invariance test (13) be satisfied. Then the vector field C = (C^{1},C^{2},...,C^{n})
defined by:
is a conserved vector for Eq. 12, i.e., C satisfies
the conservation law D_{i}(C^{i}) = 0.
RESULTS AND DISCUSSION
Since the equations of motion of the inclined sagged cables are derived
by the variational method presented in Srinil et al. (2004), we
can apply Noether`s theorem to this system to obtain conservation laws
for the system as follows:
The Lagrangian of the equations of motion of the inclined sagged cables
is given by:
Where:
m is mass per unit length and ε_{0} is the initial static
strain of the cable centerline. The infinitesimal generators for the problem
have the form:
The Lie algebra corresponding to the governing equations is spanned by
the following five linearly independent generators:
The conservation law for the equations of motion of the inclined sagged
cables is:
Letthe generator X be a linear combination of five generators X_{1},X_{2},...X_{5}.
The first prolongation of the generator X is given by:
Applying the infinitesimal criterion of invariance (13) to X, we gets:
This gives:
It follows that λ_{2} = λ_{4} = λ_{5}
= 0 and we get that the family of all variational symmetries is given
by λ_{1}X_{1}+λ_{3}X_{3}, where,
λ_{1} and λ_{3} are arbitrary constants.
For the variational symmetries
the associated conserved vectors are given by:
If we let λ_{3} = 0 and λ_{1} = 1, the variational
symmetry of translation in time
are obtained. This gives the law of conservation of energy, D_{t}(C^{t})+D_{x}(C^{x})
= 0, where, the associated conserved vectors are:
CONCLUSION
Noether`s theorem is a powerful method for finding conservation laws
for complicated systems of differential equations arising from variational
principles. The equations of motion of inclined sagged cables were derived
by the variational method in Srinil et al.(2004). Thus, Noether`s
theorem can be applied to obtain the conservation laws, which play an
important role in the analysis of basic properties of the solutions of
this system. For each variational symmetry, there is a corresponding conservation
law. Applying the infinitesimal criterion of invariance, the researchers
find all variational symmetries. In particular, the variational symmetry
of translation in time is obtained and yields the conservation of energy
law.
ACKNOWLEDGMENTS
The research is supported by Development and Promotion for Science and
Technology Talents Project of Thailand (DPST). The first author would
like to express deep gratitude to Prof. Dr. Sergey V. Meleshko from Suranaree
University of Technology for his valuable suggestions.