ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 452689, 21 pagesdoi:10.5402/2011/452689

Research Article

Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

Sakka Sookmee and Sergey V. Meleshko

School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand

Received 1 July 2011; Accepted 4 August 2011

Academic Editor: M. Escobedo

Copyright © 2011 Sakka Sookmee and Sergey V. Meleshko. 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.

The necessary form of a linearizable system of two second-order ordinary differential equations (,,,,), (,,,,) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

1. Introduction

Almost all physical applications of differential equations are based on nonlinear equations, which in general are very difficult to solve explicitly. Ordinary differential equations play a significant role in the theory of differential equations. In the 19th century, one of the most important problems in analysis was the problem of classification of ordinary differential equations [1–4].

One type of the classification problem is the equivalence problem. Two systems of differential equations are said to be equivalent if there exists an invertible transformation which transforms any solution of one system to a solution of the other system. The linearization problem is a particular case of the equivalence problem, where one of the systems is a linear system. It is one of essential parts in the studies of nonlinear equations.

1.1. Linearization Problem

The analysis of the linearization problem for a single ordinary differential equation was started by Lie [1]. He gave criteria for linearization of a second-order ordinary differential equation by an invertible change of the independent and dependent variables (point transformation). Cartan [4] developed another approach for solving the linearization problem of a second-order ordinary differential equation.

Later the linearization problem was also considered with respect to other types of transformations, for example, contact and generalized Sundman transformations. These transformations have been applied to third-order and fourth-order ordinary differential equations, see [5–16] and references therein. It is worth to note that fiber preserving transformations, where the change of variables only depends on the independent variable, have played a special role: either only such transformations were studied [6], or one needs to study them separately during compatibility analysis [10, 11, 13, 17].

The linearization problem for a system of second-order ordinary differential equations was studied in [18–24]. In [18], necessary and sufficient conditions for a system of second-order ordinary differential equations to be equivalent to the free particle equations were given. Particular class of systems of two second-order ordinary differential equations were considered in [20]. In [19], criteria for linearization of a system of two second-order ordinary differential equations are related with the existence of an admitted four-dimensional Lie algebra. Some first-order and second-order relative invariants with respect to point transformations for a system of two ordinary differential equations were obtained in [21]. A new method of linearizing a system of equations is proposed in [22], where a given system of equations is reduced to a single equation to which the Lie theorem on linearization is applied.

1.2. Canonical Forms of a Linear System of Two Second-Order ODEs

The general form of a normal linear system of second-order ordinary differential equations is where and are vectors, and are square matrices. It can be shown [19] that there exists a change such that system (1.1) is reduced to one of the following forms: either or Here , , and are square matrices. Thus the problem of linearization via point transformations consists of solving the problem of reducibility of a system of second-order ordinary differential equations to one of these forms. In the present paper, the second canonical form (1.3) is used.

It is also worth to note that system (1.3) can be further simplified to one of three forms: where , , and are real valued (not complex) functions.

One of the main motivations for studying the linearization problem is the possibility of finding the general solution. Notice that even after finding the linearizing transformation one has to solve a linear system of second-order ordinary differential equations. The simplest case is where . More general and also not complicated is the case where the matrix is constant. For example, for , this case leads to solving either a simple linear fourth-order ordinary differential equation with constant coefficients or two simple linear second-order ordinary differential equations. Indeed, for , system (1.3) is where , are constant. If , then finding from the first equation of (1.5) and substituting it into the second equation of (1.5), one obtains a fourth-order ordinary differential equation Here, The general solution of the last equation depends on the roots of the characteristic equation The solution is similar for . On the other hand, if and , then system (1.1) is decoupled: Notice also that if in this case , then the last system of equations is equivalent to the system of two trivial equations .

1.3. The Problem Solved in the Present Paper

The present paper deals with a system of two second-order ordinary differential equations In the next section, the form of a linearizable system (1.10) is obtained. This form coincides with that obtained in [18]. Some invariants of this form with respect to the general set of point transformations related with a linearizable systems (1.10) were obtained in [21]. The main result of the paper consists of necessary and sufficient conditions for system (1.10) to be equivalent with respect to a fiber preserving point transformation to system (1.5).

The paper is organized as follows. In Section 2, the necessary form of a linearizable system of two second-order ordinary differential equations is presented. In Section 3, the main results of the paper are given: necessary and sufficient criteria for a system of two second-order ordinary differential equations to be equivalent via fiber preserving point transformations (1.12) to a system of the form (1.5). During the study presented in Section 3, we also obtained some necessary conditions for linearizability for the general case of point transformations (1.11) and for the general case of the matrix . These conditions are shown in Section 4. Examples demonstrating the procedure of using the linearization theorems are presented in Section 5.

2. Necessary Form of a Linearizable System (1.10)

For obtaining necessary conditions for system (1.10) to be linearizable via point transformations (1.11), one assumes that system (1.10) is obtained from the linear system of differential equations (1.3) by an invertible transformation (1.11). The derivatives are changed by the formulae where Replacing , , , and in system (1.3), it becomes where the coefficients are expressed through the functions , , , their partial derivatives, and the entries of the matrix as follows: where denotes , denotes , denotes , and is the Jacobian of the change of variables  (1.11)

Equation (2.3) presents the necessary form of a system of two second-order ordinary differential equations which can be mapped into a linear equation (1.3) via point transformations.

3. Sufficient Conditions for Equivalency to (1.5) via Fiber Preserving Transformations

For obtaining sufficient conditions of linearizability of system (2.3), one has to solve the compatibility problem of the system of (2.4), considering it as an overdetermined system of partial differential equations for the functions , , and with given coefficients of system (2.3). The compatibility analysis depends on the value of .

The next part of the present paper deals with a fiber preserving set of point transformations (1.12): and constant matrix

3.1. Case

Substitution of and into (2.4) gives where . Comparing the mixed derivatives and , one obtains Considering the conditions , one has Here, the functions are defined through and their derivatives (presented in the appendix).

Equating the mixed derivatives , , , , and using the conditions , , one obtains Note that is satisfied. Differentiating (3.16) and (3.17) with respect to and , one has

Equations , become Further analysis of the compatibility depends on the values of the coefficients , and of the last three equations (3.21)–(3.23).

3.1.1. Case

Substituting , found from (3.21), into (3.23) and (3.22), one obtains Differentiating (3.21) with respect to and , one has Equation (3.16) becomes Differentiating (3.26) with respect to , one gets the condition Notice that and are satisfied. Hence, there are no new conditions for the functions , , and . In summary, the criteria for linearization are conditions (3.3), (3.15), (3.19), (3.20), (3.24), (3.25), and (3.27).

3.1.2. Case and

Since and , (3.21) leads to the condition , and (3.22) becomes Substituting into (3.23) and (3.16), one gets Equation leads to the condition Note that and , are satisfied. Hence, there are no other conditions for the functions , , and . Summarizing, the linearization criteria in the case and are conditions (3.3), (3.15), (3.19), (3.20), (3.29), and (3.31).

3.1.3. Case , and

Substituting , found from (3.23), into (3.16), one has Differentiating (3.32) with respect to , one gets Note that the equations and , are satisfied. Hence, there are no more conditions for the compatibility, and the linearization criteria in the studied case are (3.3), (3.15), (3.19), (3.20) and (3.33).

Remark 3.1. In the case and , one has and . This case corresponds to (1.9).

Combining all derived results in the case , the following theorem is proven.

Theorem 3.2. Necessary and sufficient conditions for system (2.3) to be equivalent to a linear system (1.3) with constant matrix via fiber preserving transformations are as follows. (I)The conditions are (3.3), (3.15), (3.19), and (3.20), and the additional conditions are as(I.1)If , then the additional conditions are (3.24), (3.25), and (3.27).(I.2)If and , then the additional conditions are (3.29) and (3.31).(I.3)If , , and , then the additional condition is (3.33).(I.4) If , , and , then there are no additional conditions.

3.2. Case

If , then the change leads to the previous case. Hence, without loss of generality one can assume that . Equations (2.4) become and .

Comparing all mixed derivatives of third-order of the functions , , one finds where the coefficients are defined through and their derivatives (presented in the appendix).

Equations , , become Notice that are satisfied. Equation (3.39) generates a further analysis of the compatibility depending on the values of , , and .

3.2.1. Case

From the first equation of (3.39), one obtains that Substituting into the second and third equations of (3.39), one obtains Substitution of into (3.38) gives Note that and are satisfied. Hence, there are no new conditions. In summary, the linearization criteria are (3.34), (3.36), (3.40), (3.42), and (3.43).

3.2.2. Case and

From the second equation of (3.39), one obtains that Substituting into the third equation of (3.39), one obtains Substitution of into (3.38) gives Note that and are satisfied. Hence, there are no other conditions. Thus, the linearization criteria in this case are (3.34), (3.36), (3.40), (3.45), and (3.46).

3.2.3. Case , and

From the third equation of (3.39), one obtains Substitution of into (3.38) leads to the condition Note that and are satisfied. Hence, there are no more conditions. In brief, the linearization criteria are conditions (3.34), (3.36), (3.40), and (3.48). Notice also that

Remark 3.3. In the case , one has and . This case corresponds to (1.9).

Combining all obtained results in the case and , the following theorem is proven.

Theorem 3.4. Necessary and sufficient conditions for system (2.3) to be equivalent to a linear system (1.3) with constant matrix by fiber preserving transformations are as follows. (II)The conditions are (3.34), (3.36), and (3.40), and the additional conditions are as(II.1)If , then the additional conditions are (3.42) and (3.43).(II.2)If and , then the additional conditions are (3.45) and (3.46).(II.3)If , and , then the additional condition is (3.48).(II.4)If , and , then there are no additional conditions.

Remark 3.5. If one assumes that the conditions (II) of Theorem 3.4 are valid, then the conditions (I) of Theorem 3.2 vanish. Moreover, these conditions also imply that , , , and the following is valid: (a) the conditions (II.1) become a particular case of the conditions (I.1); (b) the conditions (II.3) are a particular case of the conditions (I.2); (c) the conditions (II.2) with and form particular cases of the conditions (I.2) and (I.3), respectively. This allows to propose the conjecture that Theorem 3.2 is valid independently of the values of and .Notice that this conjecture is to be expected. For example, for a linearizable single second-order equation via a point transformation, the linearizable criteria combine to only two conditions, whereas during compatibility analysis, one has to study two separable cases [17].

4. Necessary Conditions of Linearization under Point Transformations

During the study presented in the previous section, several relations for linearizability for the general case of point transformations (1.11) and for the general case of the matrix were noted. These relations are the necessary conditions for linearization, and they were obtained as follows. For example, assuming that , from (2.4), one obtains the derivatives , , , , , , , , . Comparing the mixed derivatives of the functions , , , and , one can find the expressions of the quantities where are expressed through and their derivatives (shown in the appendix). Excluding the functions , , , and from these expressions, one obtains the conditions where After obtaining these relations, one can directly check, by substituting (2.4) into (4.2), that they are satisfied for the general case of point transformations (1.11) and for the general case of the matrix . Notice also that using this substitution into the conditions obtained in [18, 23], one obtains that they are not satisfied unless the matrix .

Thus, the following theorem can be stated.

Theorem 4.1. The conditions (4.2) are necessary for system (2.3) to be linearizable under point transformations.

5. Examples

In this section, examples demonstrating the procedure of using the linearization theorems are presented.

Example 5.1. Let us consider a system of two second-order quadratically semilinear ordinary differential equations In [18, 20], it is shown that system (5.1) is equivalent via point transformations to the simplest equations if and only if where Application of fiber preserving transformation to system (5.1) also leads to the same conditions (5.2).

Example 5.2. Consider a nonlinear system where , are constant. Applying the linearization criteria obtained in [18, 23] to system (5.4), one obtains that system (5.4) is equivalent to the free particle equations via point transformations if and only if . Let us consider the case . Note that, for system (5.4), Since , then , and (3.21) becomes . Taking the simplest solution of this equation and solving the compatible system of (3.5)–(3.14) for the functions , and , one gets the solution and . Substituting , and into (3.18) and (3.26), one obtains and . Thus, Theorem 3.2 guarantees that system (5.4) can be transformed to the system of linear equations: and the linearizing transformation is

Example 5.3. A variety of applications in science and engineering such as the well-known oscillator system, the vibration of springs, and some types of conservative systems with two degrees of freedom are of the form For system (5.8), Theorem 3.2 provides conditions sufficiently for system (5.8) to be reduced to a linear system with constant coefficients via a fiber preserving transformation. For example, for the oscillator system , the condition is whereas the criteria of [18, 23] are only satisfied when .

6. Conclusion

The necessary form of a linearizable system of two second-order ordinary differential equations , via point transformations is obtained. Some other necessary conditions were also found. Necessary and sufficient conditions for a system of two second-order ordinary differential equations to be transformed to the general form of linear system with constant coefficients via fiber preserving transformations are obtained. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. On the way of establishing of main theorems, we also give an explicit procedure for constructing this linearizing transformation.

Appendix

Consider the following:

Acknowledgments

This research was supported by the Royal Golden Jubilee Ph.D. Program of Thailand (TRF). The authors are also thankful to E. Schulz for fruitful discussions.

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