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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 831647, 14 pages
http://dx.doi.org/10.1155/2011/831647
Research Article

First Integrals for Two Linearly Coupled Nonlinear Duffing Oscillators

1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
2Department of Mathematics, LUMS, School of Science and Engineering, Lahore Cantt 54792, Pakistan
3Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

Received 6 September 2010; Revised 1 December 2010; Accepted 3 January 2011

Academic Editor: G. Rega

Copyright © 2011 R. Naz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate Noether and partial Noether operators of point type corresponding to a Lagrangian and a partial Lagrangian for a system of two linearly coupled nonlinear Duffing oscillators. Then, the first integrals with respect to Noether and partial Noether operators of point type are obtained explicitly by utilizing Noether and partial Noether theorems for the system under consideration. Moreover, if the partial Euler-Lagrange equations are independent of derivatives, then the partial Noether operators become Noether point symmetry generators for such equations. The difference arises in the gauge terms due to Lagrangians being different for respective approaches. This study points to new ways of constructing first integrals for nonlinear equations without regard to a Lagrangian. We have illustrated it here for nonlinear Duffing oscillators.

1. Introduction

In this paper, we study a system of two linearly coupled nonlinear Duffing oscillators𝑦+Ω2𝑦=𝛽(𝑦𝑧)𝛼𝑦3,𝑧+Ω2𝑧=𝛽(𝑧𝑦)𝛼𝑧3,(1.1) where 𝛼 and 𝛽 are nonlinearity linear coupling parameters, and prime denotes differentiation with respect to 𝑥. The system of two second-order ordinary differential equations (ODEs) frequently arises in nonlinear oscillations, nonlinear dynamics, relativity, fluid mechanics, and so forth. These oscillators, in general, describe different mechanical systems of practical importance and have two types of characteristics (see, e.g., [1, 2]). The hardening characteristic approaches to linear for small amplitudes and it raises towards infinity when the amplitude tends to certain limits. The softening characteristic performs in a non-monotonic way, and for large amplitude the vibration shape changes to the rectangular shape.

There is great interest in studying the system of two linearly coupled nonlinear Duffing oscillators, especially for first integrals, which are important from a physical point of view and for reduction purposes as well. Of course, a Lagrangian exists for the system of two equations under study. Our purpose is two fold: one is how first integrals can be constructed for ODEs; secondly, we want to investigate the effectiveness of the partial Lagrangian approach.

The Noether symmetries and first integrals are important due to their wide range of applications for Euler-Lagrange equations (see, e.g., Noether [3] and later works [4, 5]). This theorem is based on the existence of a Lagrangian and variational symmetries. It gives the relationship between equivalence classes of point symmetries and first integrals. There are equations that arise in application that do not admit standard Lagrangians, for example, the simple system of two second-order equations 𝑦=𝑦2+𝑧2,𝑧=𝑦; the curve family is nonextremal for this system. The scalar evolution equations also do not admit Lagrangians. Similarly, for the system of two coupled Van der pol oscillators with linear diffusive coupling𝑦𝑦+𝜖2𝑦1𝑧+𝑦=𝐴(𝑧𝑦)+𝐵𝑦,𝑧𝑧+𝜖2𝑧1𝑦+𝑧=𝐴(𝑦𝑧)+𝐵𝑧,(1.2) where 𝐴 and 𝐵 are constants and 𝜖 is a small parameter and no variational problem exists which can be verified from Douglas [5]. The interested reader is referred to the interesting paper cited above for the classification of Lagrangians in which Douglas [5] has provided the complete solution to the inverse problem for three-dimensional space (system of two second-order ODEs). Now, we raise the question as to how one can construct first integrals for equations which are not variational as mentioned above. It is well known that there are other methods which provide first integrals without making use of a Lagrangian. The most elementary method is the direct method [6, 7] which is often used for constructing first integrals without the variational principle. There are some other approaches as well (see, e.g., [4, 8, 9]) in which the equations can be expressed in characteristic form.

The Noether symmetries and the corresponding first integrals have been subjects of rigorous investigations and play an important role in the reduction of differential equations. Some important works have been done relating to first integrals. In [10, 11], the authors deduce a relation between “symmetries” and first integrals without a variational principle. The classification of Noether point symmetries for a three degrees of freedom Lagrangian system was given by Damianou and Sophocleous in [12], and the results for one and two degrees of freedom were also reported in their paper. The first integrals associated with Noether and partial Noether operators of point type for a linear system of two second-order ODEs with variable coefficients were constructed by the authors in [13]. They showed that the first integrals resulting from Noether and partial Noether approaches are equivalent. The difference occurs in the gauge terms due to Lagrangians being different for the respective approaches. The classification of partial Noether operators and first integrals for a system with two degrees of freedom was also discussed by Naeem and Mahomed in [14]. Furthermore, a new perturbation method based on integrating vectors and multiple scales for perturbed systems of ODEs is presented in [15]. By using this perturbation method a strongly nonlinear forced oscillator based on integrating factors is studied which yields the approximate first integrals [16]. For an account of this theory, the reader is referred to an interesting book [17].

In this paper, we show that the first integrals corresponding to Noether and partial Noether operators of point type for a system of two linearly coupled nonlinear Duffing oscillators are equivalent, and the algebras for both cases are isomorphic. We give an elegant way to construct first integrals for such equations with respect to point symmetries. Firstly, we obtain the partial Noether operators of point type and then first integrals are constructed by utilizing a partial Noether's theorem. This approach gives new ways of constructing first integrals for the equations without regard to a Lagrangian as partial Lagrangians do exist for such equations. It would definitely be of interest to obtain the first integrals of the Duffing oscillator system with respect to first-order Lie-Bäcklund operators. This will provide a more complete picture for such systems. Further work can then be done on reductions using these symmetries.

The outline of the work is as follows. In Section 2, the basic definitions are adapted from the literature. The Noether point symmetries and the corresponding first integrals are presented in Section 3. In Section 4, we construct the partial Noether operators of point type and the relating first integrals. Concluding remarks are summarized in Section 5.

2. Preliminaries

Suppose 𝑥 is considered to be the independent variable and 𝑢=(𝑢1,𝑢2)=(𝑦,𝑧), the dependent variable with coordinates 𝑦 and 𝑧. The derivatives of 𝑢𝛼 with respect to 𝑥 are denoted by𝑢𝛼𝑥=𝐷𝑥(𝑢𝛼),𝛼=1,2,(2.1) in which𝐷𝑥=𝜕𝜕𝑥+𝑢𝛼𝑥𝜕𝜕𝑢𝛼+𝑢𝛼𝑥𝑥𝜕𝜕𝑢𝛼𝑥+,(2.2) is the total derivative operator with respect to 𝑥. Note that the summation convention is assumed for repeated indices throughout. The collection of first derivative 𝑢𝛼𝑥 is denoted by 𝑢=(𝑢1,𝑢2)=(𝑦,𝑧), and all higher-order derivatives are denoted by 𝑢,𝑢,𝑢,.

The following definitions and results are easily adaptable from the literature (see, e.g., [1012, 1820]).(1)The Lie-Bäcklund operator is 𝜕𝑋=𝜉𝜕𝑥+𝜂𝛼𝜕𝜕𝑢𝛼+𝜁𝛼𝑥𝜕𝜕𝑢𝛼𝑥,(2.3) where 𝜁𝛼𝑥 are defined by 𝜁𝛼𝑥=𝐷𝑥(𝜂𝛼)𝑢𝛼𝑥𝐷𝑥(𝜉),𝛼=1,2.(2.4)(2)A Lie-Bäcklund operator 𝑋 defined in (2.3) is a Noether symmetry generator corresponding to a Lagrangian 𝐿 if there exists a function 𝐵(𝑥,𝑢) such that 𝐷𝑋𝐿+𝑥𝜉𝐿=𝐷𝑥(𝐵),(2.5) where 𝐷𝑥 is the total derivative operator given in (2.2). The function 𝐵(𝑥,𝑢) is known as the gauge term.(3)The Lie-Bäcklund operator 𝑋 is known to be a partial Noether operator corresponding to a partial Lagrangian 𝐿 if it is determined from 𝐷𝑋𝐿+𝑥𝜉𝜂𝐿=𝛼𝜉𝑢𝛼𝑥𝛿𝐿𝛿𝑢𝛼+𝐷𝑥(𝐵),(2.6) for a suitable function 𝐵(𝑥,𝑢), where 𝛿𝛿𝑢𝛼=𝜕𝜕𝑢𝛼𝐷𝑥𝜕𝜕𝑢𝛼+𝐷2𝑥𝜕𝜕𝑢𝛼𝑥𝐷3𝑥𝜕𝜕𝑢𝛼𝑥𝑥+,𝛼=1,2(2.7) is the Euler operator.(4)If the Lie-Bäcklund operator 𝑋 is a Noether symmetry generator corresponding to a Lagrangian 𝐿𝒜 of Euler-Lagrange differential equations, then 𝜂𝐼=𝐵𝜉𝐿+𝛼𝜉𝑢𝛼𝑥𝜕𝐿𝜕𝑢𝛼𝑥,𝛼=1,2(2.8) is known as the first integral of Euler-Lagrange differential equations.(5)If the Lie-Bäcklund operator 𝑋 is a partial Noether operator with respect to a partial Lagrangian 𝐿(𝑥,𝑢,𝑢) of partial Euler-Lagrange equations, then the constant of motion or first integral 𝐼 can be constructed from (2.8).

3. Noether's Approach

In this section, we derive the Noether point symmetry operators for the coupled nonlinear Duffing oscillators (1.1) corresponding to a standard Lagrangian and construct the first integrals by Noether's theorem.

3.1. Noether Point Symmetry Generators for Coupled Nonlinear Duffing Oscillators

A Lagrangian for system (1.1) satisfying the Euler-Lagrange equations 𝛿𝐿/𝛿𝑦=0 and 𝛿𝐿/𝛿𝑧=0 is1𝐿=2𝑦2+12𝑧2Ω22𝑦2+𝑧2𝛽2𝑦2+𝑧2𝛼4𝑦4+𝑧4+𝛽𝑦𝑧.(3.1) The Noether point symmetry operator 𝑋 corresponding to a Lagrangian for a system of two linearly coupled nonlinear Duffing oscillators (1.1) is calculated from the formula (2.5) with respect to some function 𝐵(𝑥,𝑦,𝑧). The Noether point symmetry determining equations are𝜉𝑦=0,𝜉𝑧𝜂=0,(3.2)1𝑦12𝜉𝑥=0,𝜂2𝑧12𝜉𝑥=0,𝜂1𝑧+𝜂2𝑦𝜂=0,(3.3)1𝑥=𝐵𝑦,𝜂2𝑥=𝐵𝑧𝜂,(3.4)1Ω2𝑦𝛽𝑦𝛼𝑦3+𝛽𝑧+𝜂2Ω2𝑧𝛽𝑧𝛼𝑧3+𝛽𝑦(3.5)+𝜉𝑥12𝑦2+𝑧2𝛽+Ω2𝛼4𝑦4+𝑧4+𝛽𝑦𝑧=𝐵𝑥.(3.6) After some simple calculations, (3.2)–(3.4) give rise to𝜂𝜉=𝑓(𝑥),1=12𝑓𝑦𝑧𝑏1𝜂+𝑔(𝑥),2=12𝑓𝑧+𝑦𝑏11+(𝑥),𝐵=4𝑓𝑦2+𝑧2+𝑦𝑔(𝑥)+𝑧(𝑥)+𝑇(𝑥),(3.7) where 𝑏1 is an arbitrary constant and 𝑓, 𝑔, , and 𝑇 are arbitrary functions of 𝑥. The insertion of (3.7) in (3.6) and separation with respect to powers of 𝑦 and 𝑧 reduce to the following system:𝛼𝑏11=0,𝛽𝑓=0,(3.8)𝛼𝑓(𝑥)=0,𝛼𝑔(𝑥)=0,𝛼(𝑥)=0,(3.9)4𝑓+𝛽𝑓+Ω2𝑓𝛽𝑏11=0,(3.10)4𝑓+𝛽𝑓+Ω2𝑓+𝛽𝑏1𝑔=0,(3.11)(𝑥)+𝛽+Ω2𝑔(𝑥)𝛽(𝑥)=0,(3.12)(𝑥)+𝛽+Ω2(𝑥)𝛽𝑔(𝑥)=0,(3.13)𝑇(𝑥)=0.(3.14) Equation (3.14) results in𝑇(𝑥)=𝑐0,(3.15) where 𝑐𝑜 is a constant.

In order to solve system (3.8)–(3.13), the following cases arise.

Case 1. 𝛼=0, 𝛽=0.
The solution of system (3.8)–(3.13) with 𝛼=0 and 𝛽=0 yields 𝑓(𝑥)=𝑏2+𝑏3cos2Ω𝑥+𝑏4𝑔sin2Ω𝑥,(𝑥)=𝑏5cosΩ𝑥+𝑏6sinΩ𝑥,(𝑥)=𝑏7cosΩ𝑥+𝑏8sinΩ𝑥,(3.16) where 𝑏1,,𝑏8 are constants.
Now, (3.7) together with (3.16) gives 𝜉=𝑏2+𝑏3cos2Ω𝑥+𝑏4𝜂sin2Ω𝑥,(3.17)1=𝑏3sin2Ω𝑥+𝑏4cos2Ω𝑥Ω𝑦𝑏1𝑧+𝑏5cosΩ𝑥+𝑏6𝜂sinΩ𝑥,(3.18)2=𝑏3sin2Ω𝑥+𝑏4cos2Ω𝑥Ω𝑧+𝑏1𝑦+𝑏7cosΩ𝑥+𝑏8sinΩ𝑥,(3.19)𝐵=Ω2𝑏3cos2Ω𝑥+𝑏4𝑦sin2Ω𝑥2+𝑧2+𝑦Ω𝑏5sinΩ𝑥+𝑏6cosΩ𝑥+𝑧Ω𝑏7sinΩ𝑥+𝑏8cosΩ𝑥+𝑐𝑜.(3.20)
Setting one of the constants equal to one and the rest to zero, we find the following Noether point symmetry operators and gauge terms: 𝑋1𝜕=𝑧𝜕𝜕𝑦+𝑦𝑋𝜕𝑧,𝐵=0,2=𝜕𝑋𝜕𝑥,𝐵=0,3𝜕=cos2Ω𝑥𝜕𝜕𝑥𝑦Ωsin2Ω𝑥𝜕𝜕𝑦𝑧Ωsin2Ω𝑥𝜕𝑧,𝐵=Ω2𝑦2+𝑧2𝑋cos2Ω𝑥,4𝜕=sin2Ω𝑥𝜕𝜕𝑥+𝑦Ωcos2Ω𝑥𝜕𝜕𝑦+𝑧Ωcos2Ω𝑥𝜕𝑧,𝐵=Ω2𝑦2+𝑧2𝑋sin2Ω𝑥,5𝜕=cosΩ𝑥𝑋𝜕𝑦,𝐵=Ω𝑦sinΩ𝑥,6𝜕=sinΩ𝑥𝑋𝜕𝑦,𝐵=Ω𝑦cosΩ𝑥,7𝜕=cosΩ𝑥𝑋𝜕𝑧,𝐵=Ω𝑧sinΩ𝑥,8𝜕=sinΩ𝑥𝜕𝑧,𝐵=Ω𝑧cosΩ𝑥.(3.21)

Case 2. 𝛼=0, 𝛽0.
The utilization of solution of (3.8)–(3.13) together with (3.7) results in 𝑏1=0,𝜉=𝑓(𝑥)=𝑐1,𝜂(3.22)1𝑐=𝑔(𝑥)=22𝑐cosΩ𝑥+32sinΩ𝑥+𝑐4cos2𝛽+Ω2𝑥+𝑐5sin2𝛽+Ω2𝜂𝑥,(3.23)2𝑐=(𝑥)=22𝑐cosΩ𝑥+32sinΩ𝑥𝑐4cos2𝛽+Ω2𝑥𝑐5sin2𝛽+Ω2Ω𝑥,(3.24)𝐵=2(𝑦+𝑧)𝑐2sinΩ𝑥+𝑐3cosΩ𝑥+(𝑦𝑧)2𝛽+Ω2×𝑐4sin2𝛽+Ω2𝑥+𝑐5cos2𝛽+Ω2𝑥+𝑐𝑜,(3.25) where 𝑐1,,𝑐5 are constants and 2𝛽+Ω2>0. If 2𝛽+Ω2<0 then cos(2𝛽+Ω2)𝑥 is replaced with cosh(2𝛽+Ω2)𝑥 and sin(2𝛽+Ω2)𝑥 with sinh(2𝛽+Ω2)𝑥.
The Noether point symmetries for 𝛼=0, 𝛽0 are 𝑋1=𝜕𝑋𝜕𝑥,𝐵=0,(3.26)2=12𝜕cosΩ𝑥+𝜕𝜕𝑦Ω𝜕𝑧,𝐵=2𝑋(𝑦+𝑧)sinΩ𝑥,(3.27)3=12𝜕sinΩ𝑥+𝜕𝜕𝑦Ω𝜕𝑧,𝐵=2𝑋(𝑦+𝑧)cosΩ𝑥,(3.28)4=cos2𝛽+Ω2𝑥𝜕𝜕𝜕𝑦𝜕𝑧,𝐵=2𝛽+Ω2(𝑦𝑧)sin2𝛽+Ω2𝑋𝑥,(3.29)5=sin2𝛽+Ω2𝑥𝜕𝜕𝜕𝑦𝜕𝑧,𝐵=2𝛽+Ω2(𝑦𝑧)cos2𝛽+Ω2𝑥.(3.30)

Case 3. If 𝛼0, 𝛽=0, then from system (3.8)–(3.13), we obtain 𝑏1=0,𝑓(𝑥)=𝑑1,(𝑥)=0,𝑔(𝑥)=0,(3.31) where 𝑑1 is a constant. Hence, 𝜉=𝑑1,𝜂1=0,𝜂2=0,𝐵=0.(3.32)
We get only one generator for this case as 𝑋1=𝜕𝜕𝑥,𝐵=0.(3.33)

Case 4. 𝛼0, 𝛽0.
System (3.8)–(3.13) with 𝛼0 and 𝛽0 yields the following results: 𝑏1=0,𝑓(𝑥)=𝑒1,(𝑥)=0,𝑔(𝑥)=0,(3.34) where 𝑒1 is a constant. Thus, 𝜉=𝑒1,𝜂1=0,𝜂2=0,𝐵=0.(3.35)
For this case, the symmetry generator is 𝑋1=𝜕𝜕𝑥,𝐵=0.(3.36)

The interpretation of the results in case of Noether point symmetries are given below.

Noether point symmetries
Case 1: in this case, we obtain an eight-dimensional Lie algebra.
Case 2: the Lie algebra is five dimensional.
Case 3: we deduce a one-dimensional Lie algebra.
Case 4: the Lie algebra for this case is also one dimensional.

3.2. First Integrals

The first integrals for Noether point symmetry generators for system (1.1) corresponding to the Lagrangian (3.1) are (by (2.8))𝜂𝐼=𝐵𝜉𝐿+1𝜉𝑦𝑦+𝜂2𝜉𝑧𝑧,(3.37) where we have used 𝜕𝐿/𝜕𝑦=𝑦 and 𝜕𝐿/𝜕𝑧=𝑧 which hold for all the cases. The first integrals for each case computed with the help of (3.37) are summarized below.

Case 1. 𝛼=0, 𝛽=0.
If we invoke the Noether point symmetry generators and gauge terms from system (3.21) in (3.37), we obtain the following first integrals: 𝐼1=𝑦𝑧𝑦𝑧,𝐼2=12𝑦2+𝑧2+Ω22𝑦2+𝑧2,𝐼3=Ω2𝑦2+𝑧21cos2Ω𝑥2𝑦cos2Ω𝑥2+𝑧2Ω2𝑦2+𝑧2+𝑦𝑦+𝑧𝑧𝑦Ωsin2Ω𝑥+2+𝑧2𝐼cos2Ω𝑥,4=Ω2𝑦2+𝑧21sin2Ω𝑥2𝑦sin2Ω𝑥2+𝑧2Ω2𝑦2+𝑧2𝑦𝑦+𝑧𝑧𝑦Ωcos2Ω𝑥+2+𝑧2𝐼sin2Ω𝑥,5=Ω𝑦sinΩ𝑥𝑦𝐼cosΩ𝑥,6=Ω𝑦cosΩ𝑥𝑦𝐼sinΩ𝑥,7=Ω𝑧sinΩ𝑥𝑧𝐼cosΩ𝑥,8=Ω𝑧cosΩ𝑥𝑧sinΩ𝑥.(3.38) For a system of two second-order ODEs, there are exactly four functionally independent first integrals. What we obtained are seven first integrals. Out of these, there are only four functionally independent first integrals.

Case 2. 𝛼=0, 𝛽0.
The first integrals for this case are 𝐼1=12𝑦2+𝑧2+Ω22𝑦2+𝑧2+𝛽2𝑦2+𝑧2𝐼𝛽𝑦𝑧,2Ω=2(1𝑦+𝑧)sinΩ𝑥2𝐼cosΩ𝑥(𝑦+𝑧),3=Ω21(𝑦+𝑧)cosΩ𝑥2𝐼sinΩ𝑥(𝑦+𝑧),4=(𝑦𝑧)2𝛽+Ω2sin2𝛽+Ω2𝑥(𝑦𝑧)cos2𝛽+Ω2𝐼𝑥,5=(𝑦𝑧)2𝛽+Ω2cos2𝛽+Ω2𝑥(𝑦𝑧)sin2𝛽+Ω2𝑥.(3.39)

Case 3. 𝛼0,𝛽=0.
The simple calculations lead to the following integral: 𝐼1=12𝑦2+𝑧2+Ω22𝑦2+𝑧2+𝛼4𝑦4+𝑧4.(3.40)

Case 4. 𝛼0, 𝛽0.
Straightforward manipulations result in the following first integral: 𝐼1=12𝑦2+𝑧2+12𝑦2+𝑧2𝛽+Ω2+𝛼4𝑦4+𝑧4𝛽𝑦𝑧.(3.41) Note that the partial Lagrangian approach is similar to the Lagrangian approach; when one uses point-type operators one may not obtain all the first integrals. A familiar example is the Laplace-Runge-Lenz vector for the well-known Kepler plroblem which comes from a higher symmetry and not a geometric sysmmetric of point type. Here, too, we expect that the existence of an elliptic type integral will arise from a derivative dependent operator which will be of interest to look at in a future work. We only obtain the kinetic plus potential energy integrals here. Presumably, a higher derivative dependent operator could yield further integrals if they exist. These need to be further investigated.

4. Partial Noether's Approach

In this section, we derive the partial Noether operators of point type associated with a partial Lagrangian of (1.1) and then construct the first integrals for (1.1).

4.1. Partial Noether Operators of Point Type for Coupled Nonlinear Duffing Oscillators

A partial Lagrangian for system (1.1) is1𝐿=2𝑦2+12𝑧2,(4.1) so that system (1.1) can be written as𝛿𝐿𝛿𝑦=Ω2𝑦+𝛽(𝑦𝑧)+𝛼𝑦3,𝛿𝐿𝛿𝑧=Ω2𝑧+𝛽(𝑧𝑦)+𝛼𝑧3.(4.2) The partial Noether operator 𝑋 corresponding to a partial Lagrangian for the system (1.1) can be determined from (2.6) with respect to some function 𝐵(𝑥,𝑦,𝑧). The partial Noether operator of point-type determining equations are𝜉𝑦=0,𝜉𝑧𝜂=0,(4.3)1𝑦12𝜉𝑥=0,𝜂2𝑧12𝜉𝑥=0,𝜂1𝑧+𝜂2𝑦𝜂=0,(4.4)1𝑥=𝐵𝑦Ω𝜉2𝑦+𝛽𝑦+𝛼𝑦3𝜂𝛽𝑧,(4.5)2𝑥=𝐵𝑧Ω𝜉2𝑧+𝛽𝑧+𝛼𝑧3𝐵𝛽𝑦,(4.6)𝑥+𝜂1Ω2𝑦+𝛽𝑦+𝛼𝑦3𝛽𝑧+𝜂2Ω2𝑧+𝛽𝑧+𝛼𝑧3𝛽𝑦=0.(4.7) The solution of (4.3)–(4.6) yields𝜂𝜉=𝑓(𝑥),1=12𝑓𝑦𝑧𝑏1𝜂+𝑔(𝑥),2=12𝑓𝑧+𝑦𝑏11+(𝑥),𝐵=4𝑓𝑦2+𝑧2+𝑦𝑔(𝑥)+𝑧1(𝑥)+𝑓2Ω2𝑦+𝛽2+𝑧2+𝛼4𝑦4+𝑧4𝑦𝑧𝛽+𝑇(𝑥),(4.8) where 𝑏1 is an arbitrary constant and 𝑓, 𝑔, , and 𝑇 are arbitrary functions of 𝑥. The substitution of (4.8) in (4.7) and separation with respect to powers of 𝑦 and 𝑧 reduce to the system of (3.8)–(3.13) that we have obtained in the case of Noether point symmetries and 𝑇(𝑥)=𝑐𝑜. In case of partial Noether operators of point type the same cases need to be considered as we have for the Noether point symmetry operators.

Case 1. 𝛼=0, 𝛽=0.
For this case 𝜉, 𝜂1, and 𝜂2 are the same as given in (3.17), (3.18), and (3.19) and, therefore, the partial Noether operators of point type are identical to the Noether point symmetry operators but the gauge terms are different. The gauge term for the partial Noether operators of point type is Ω𝐵=2𝑏22𝑦2+𝑧212Ω2𝑏3cos2Ω𝑥+𝑏4𝑦sin2Ω𝑥2+𝑧2+𝑦Ω𝑏5sinΩ𝑥+𝑏6cosΩ𝑥+𝑧Ω𝑏7sinΩ𝑥+𝑏8cosΩ𝑥+𝑐𝑜.(4.9) Notice from (3.20) and (4.9) that only the 𝐵's relating to 𝑋2, 𝑋3, and 𝑋4 for the partial Noether's case are different from that for the Noether's approach. The 𝐵's relating to 𝑋2, 𝑋3, and 𝑋4 for the partial Noether case are Ω𝐵=22𝑦2+𝑧2,Ω𝐵=22𝑦2+𝑧2Ωcos2Ω𝑥,𝐵=22𝑦2+𝑧2sin2Ω𝑥.(4.10)

Case 2. 𝛼=0, 𝛽0.
For this case, 𝜉,𝜂1, and 𝜂2 are the same as given in (3.22), (3.23), and (3.24) and therefore the partial Noether operators of point type are similar to the case of Noether point symmetries (3.30) but the gauge terms are different. The gauge term for partial Noether case is Ω𝐵=2(𝑦+𝑧)𝑐2sinΩ𝑥+𝑐3cosΩ𝑥+(𝑦𝑧)2𝛽+Ω2𝑐4sin2𝛽+Ω2𝑥+𝑐5cos2𝛽+Ω2𝑥+𝑐112Ω2𝑦+𝛽2+𝑧2𝑦𝑧𝛽+𝑐𝑜.(4.11) Note that in (4.11) all the gauge terms are similar as in the case of Noether symmetries (see (3.25)) except one gauge term relating to 𝑋1 is different which is given as follows: 1𝐵=2Ω2𝑦+𝛽2+𝑧2𝛽𝑦𝑧.(4.12)

Case 3. 𝛼0, 𝛽=0.
The partial Noether operator of point type when 𝛼0, 𝛽=0 is 𝑋1=𝜕/𝜕𝑥 which is identical to the case of Noether point symmetries but the difference occurs in the gauge term as Ω𝐵=22𝑦2+𝑧2+𝛼4𝑦4+𝑧4.(4.13)

Case 4. 𝛼0, 𝛽0.
The operator for this case is also similar to the case of Noether point symmetries except 𝐵 in (3.36) is different as follows 1𝐵=2Ω2𝑦+𝛽2+𝑧2+𝛼4𝑦4+𝑧4𝛽𝑦𝑧.(4.14)

4.2. First Integrals

For the partial Lagrangian (4.1) 𝜕𝐿/𝜕𝑦=𝑦 and 𝜕𝐿/𝜕𝑧=𝑧 also hold and therefore the first integrals can be computed from (3.37). Moreover, since Noether and partial Noether operator of point type are the same for each case, only the term 𝐵𝜉𝐿 in formula (3.37) will be different from the Noether case.

Case 1. 𝛼=0, 𝛽=0.
For operators 𝑋1 and 𝑋5 to 𝑋8, we have 𝜉=0 and the gauge terms are the same as for the case of Noether point symmetries. Therefore, we obtain the same first integrals. The gauge term 𝐵's relating to 𝑋2, 𝑋3, and 𝑋4 for the partial Noether case are different from that for the Noether's approach and 𝜉0 for these operators; hence, formula (3.37) yields the same first integrals as we have constructed for the Noether's case.

Similarly, for other cases, the first integrals for the partial Noether operators of point type are identical to the case of Noether point symmetry generators.

The interpretation of the results in case of partial Noether operators of point type is as follows.

4.3. Partial Noether Operators of Point Type

The partial Noether operators of point type for the coupled nonlinear Duffing oscillators are the same as the Noether point symmetry operators. The difference arises in the “gauge” terms due to the Lagrangians being different. The algebras for both cases are isomorphic due to the fact that 𝛿𝐿/𝛿𝑦 and 𝛿𝐿/𝛿𝑧 are independent of derivatives. The first integrals due to the partial Noether operators of point type associated with a partial Lagrangian are equivalent to the case of Noether point symmetries.

5. Concluding Remarks

The Noether point symmetry operators corresponding to a Lagrangian of a system of two linearly coupled nonlinear Duffing oscillators have been constructed. We have computed the first integrals associated with Noether point operators by utilization of Noether's theorem. Then the partial Noether operators of point type and first integrals for the system under study are constructed corresponding to a partial Lagrangian. The partial Euler-Lagrange equations obtained herein are independent of derivatives, and therefore the partial Noether operators become point symmetry operators and the algebras are isomorphic for both the Noether and partial Noether cases. The first integrals corresponding to the Noether and partial Noether operators of point type are equivalent and these form one-, five-, and eight-dimensional Lie algebras. It would be of value to extend the results here to more general symmetries of Lie-Bäcklund type. Moreover, it is of interest to obtain reductions for the system using point or Lie-Bäcklund symmetries.

This study also points to new ways of constructing first integrals for systems without use of a Lagrangian. However, a partial Lagrangian can exist for such equations and one can invoke a formula for the construction of the first integrals. Since, in general, no variational problem exists for equations, one can utilize a partial Lagrangian approach which we have shown here to be effective for the coupled Duffing oscillators. The first integrals still can be constructed by using a Noether-like theorem.

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