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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 284653, 15 pages
Research Article

First Integrals, Integrating Factors, and Invariant Solutions of the Path Equation Based on Noether and -Symmetries

1Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey
2Division of Mechanics, Faculty of Civil Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey

Received 13 March 2013; Accepted 28 April 2013

Academic Editor: Nail Migranov

Copyright © 2013 Gülden Gün and Teoman Özer. 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.


We analyze Noether and -symmetries of the path equation describing the minimum drag work. First, the partial Lagrangian for the governing equation is constructed, and then the determining equations are obtained based on the partial Lagrangian approach. For specific altitude functions, Noether symmetry classification is carried out and the first integrals, conservation laws and group invariant solutions are obtained and classified. Then, secondly, by using the mathematical relationship with Lie point symmetries we investigate -symmetry properties and the corresponding reduction forms, integrating factors, and first integrals for specific altitude functions of the governing equation. Furthermore, we apply the Jacobi last multiplier method as a different approach to determine the new forms of -symmetries. Finally, we compare the results obtained from different classifications.

1. Introduction

In a fluid medium, drag forces are the major sources of energy loss for moving objects. Fuel consumption may have reduced to minimize the drag work. This can be achieved by the selection of optimum path. The drag force depends on the density of fluid, the drag coefficient, the cross-sectional area, and the velocity. These parameters are the combination of the altitude-dependent parameters which can be expressed as a single arbitrary function. If all parameters are assumed to be constants, then the minimum drag work path would be a linear path. But these parameters change during the motion. And all parameters can be defined as the function of altitude [1, 2].

The main purpose of the work is to study Noether and -symmetry classifications of the path equation for the different forms of arbitrary function of the governing equation [37]. Based on Noether’s theorem, if Noether symmetries of an ordinary differential equation are known, then the conservation laws of this equation can be obtained directly by using Euler-Lagrange equations [8]. However, in order to apply this theorem, a differential equation should have standard Lagrangian. Thus, an important problem in such studies is to determine the standard Lagrangian of the differential equation. In fact, for many problems in the literature, it may not be possible to determine the Lagrangian function of the equation. To overcome this problem, partial Lagrangian method can be used alternatively and the Noether symmetries and first integrals can be obtained in spite of the fact that the differential equation does not have a standard Lagrangian [9]. Here, we examine the partial Lagrangian of path equation and classify the Noether symmetries and first integrals corresponding to special forms of arbitrary function in the governing equation.

The second type of classification that is called -symmetries is carried out by using the relation with Lie point symmetries as a direct method. For second-order ordinary differential equation, the method of finding -symmetries has been investigated extensively by Muriel and Romero [10, 11]. They have demonstrated that integrating factors and the integrals from -symmetries for a second-order ordinary differential equation can be determined algorithmically [12]. In their studies, for the sake of simplicity, the -symmetry is assumed to be a linear form as . However, it is possible to show that the -symmetry cannot be chosen generally in this linear form. Therefore, we propose in this study to use the relation between Lie point symmetries and -symmetries for the classification.

The other classification that we discuss in our study is how to obtain -symmetries with the Jacobi last multiplier approach. Recently, Nucci and Levi [13] have shown that -symmetries and corresponding invariant solutions can be algorithmically obtained by using the Jacobi last multiplier. This new approach includes the new determining equation including -function that can be obtained from the divergence of the ordinary differential equation. In the -symmetries approach based on a new form of the prolongation formula, the determining equations are difficult to solve since they include three unknown variables to determine and then the determining equation cannot be reduced to a simpler form. However, by considering the Jacobi last multiplier approach, first we determine the -function, which reduces to two the number of unknown functions, and then the other functions called infinitesimals functions can be calculated easily. Taking into account these ideas we analyze -symmetries of the path equation for different cases of the altitude function.

The outline of this work is as follows. In the next section, we present the necessary preliminaries. In Section 3, Noether symmetries, first integrals, and some invariant solutions of path equation are obtained. In Section 4, firstly we introduce some fundamental information about -symmetries, integration factors, and first integrals, and then -symmetries corresponding to different choice of the arbitrary function are investigated. Also for some cases the reduced forms of path equation are found and the new solutions of path equation are established. Section 5 is devoted to introduce another approach that is called Jacobi last multiplier to investigate the -symmetries. The conclusions and results are discussed in Section 6.

2. Preliminaries

Let us assume that be the independent variable and be the dependent variable with functions . The derivatives of with respect to are given by where is the total derivative operator [1418] with respect to , which can be defined as

Definition 1. For each we can define the operator which is called the Euler-Lagrange operator.

Definition 2. Generalized operator can be formulated as where in which is the Lie characteristic function For convenience the generalized operator (4) can be rewritten by using characteristic function such as and the Noether operator associated with a generalized operator can be defined

Definition 3. Let us consider an th-order ordinary differential equation system then the first integral of this system is a differential function (9) , the universal space and the vector space of all differential functions of all finite orders, which is given by the following formula: and this equality is valid for every solution of (9). The first integral is also referred to as the local conservation law.

Definition 4. Let (9) be in the following form and , and then nonzero functions satisfy the relations , , in which is called partial Lagrangian of (11). Otherwise, is a standard Lagrangian.
On the other hand the Euler-Lagrange equations can be defined as following form and similarly the form of partial Euler-Lagrange equations is

Definition 5. Let be a vector that satisfies , where is a constant. Then represents prolongation of the generalized operator (7), and partial Noether operator corresponding to a partial Lagrangian is formulated as in which , , is the characteristic of . Also is called the gauge function.

Definition 6. If is a partial Noether operator corresponding to partial Lagrangian , then the gauge function exists. Hence, the first integral is given by

3. Noether Symmetries of Path Equation

The differential equation describing the path of the minimum drag work is given in the form where is the altitude function. In this section we use partial Lagrangian approach to analyze Noether symmetries. Firstly, we can determine the Euler-Lagrange operator (3) for the path equation (16) such as and the partial Lagrangian for the path equation (16) is Then the application of (18) to (14) and separation with respect to powers of and arranging yield the set of determining equations, the over-system of partial differential equations To find the infinitesimals and , (19)–(22) should be solved together. First, (19) is integrated as and then substituting (23) into (20) and solving for yield Differentiating (21)-(22) with respect to and , respectively, gives Using (25) and eliminating , we find that If the infinitesimals (23) and (24) are inserted into (26) then one can find the following classification relationship in terms of : Here several cases should be examined separately for different forms of .

3.1. Constant

For this case the solution of (27) gives to the following infinitesimals: where are constants . Integrating (21) with respect to gives

The associated infinitesimal generators turn out to be

Thus, the first integrals by Definition 6 are given as follows:


For the linear case of , we obtain where is a constant. The partial Noether operator is and the first integral is


The solution of determining equations for the form of gives the following infinitesimals where are constants , and the gauge function is

The associated five-parameter symmetry generators take the form and the corresponding first integrals are


For this case, the infinitesimal functions read where are constants , and the gauge function is

The corresponding Noether symmetry generators are And the conservation laws are


For this choice of , we find the infinitesimals where is constant, and we have the first integral For convenience all Noether symmetries and first integrals are presented in Table 1.

Table 1: Noether symmetry classification table of path equation.
3.6. Invariant Solutions

Invariant solutions that satisfy the original path equation can be obtained by first integrals according to the relation . We here determine some special cases and investigate the corresponding invariant solutions.

Case  1. (a) For the case of , the conservation law is by using the relation , then the invariant solution of path equation (16) is where , are constants.

(b) For the same function, the conservation law is and the invariant solution similar to previous one is where , are constants.

Case  2. Let us consider , then the first integral yields and the solution of this equation gives where , are constants, in which it is obvious that the invariant solution (50) satisfies the original path equation.

4. -Symmetries of Path Equation

The relationship between -symmetries, integration factors and first integrals of second-order ordinary differential equation is very important from the mathematical point of view [1012]. Let us consider first the second-order differential equation of the form and let vector field of (51) be in the form of

In terms of , a first integral of (51) is any function in the form of providing equality of . An integrating factor of (51) is any function satisfying the following equation: where is total derivative operator in the form of Thus -symmetries of second-order differential equation (51) can be obtained directly by using Lie symmetries of this same equation. Secondly, let be a Lie point symmetry of (51), and then the characteristic of is and for the path equation (16) the total derivative operator can be written as thus the vector field is called -symmetry of (16) if the following equality is satisfied. The following four steps can be defined for finding -symmetries and first integrals.(1)Find a first integral of , that is, a particular solution of the equation where is the first-order -prolongation of the vector field .(2)The solution of (59) will be in terms of first order derivative of . To write equation of (51) in terms of the reduced equation of , we can obtain the first-order derivative the solution of (59) and we can write (51) equation in terms of .(3)Let be an arbitrary constant of the solution of the reduced equation written in terms of . Therefore, is an integrating factor of (51).(4)The solution of is the first integral of .

4.1. -Symmetries Using Lie Symmetries of Path Equation

Let us consider an th-order ODE as follows:

Thus the invariance criterion of (61) is The expansion of relation (62) gives the determining equation related to path equation, which is the system of partial differential equations. In this system there are three unknowns, namely, , , and , which are difficult to solve because they are highly nonlinear. In the literature [1012], for the convenience the function are chosen generally in the form In addition, for solving the remaining determining equations, the infinitesimal functions and are chosen specifically as and [1012]. Therefore, the number of unknowns in the equation is reduced to find and functions, and finally, -symmetries can be determined explicitly.

However, for the path equation (16), it is possible to check that -symmetries of this equation cannot be determined by taking the form of in (63). Thus, we study -symmetries of path equation by using the relation with the Lie point symmetries of the same equation [2, 19]. Here Lie point symmetries of path equation are examined by considering four different cases of function .

4.1.1. Arbitrary

For arbitrary the one-parameter Lie group of transformations is and the generator is Applying this generator (56), we obtain the characteristic

Using (58), the -symmetry is obtained in the following form:

If we substitute -symmetry (67) in (59), then we have

It is clear that a solution of (68) is

To write (16) in terms of , we can express the following equality using (69):

Taking derivative of (70) with respect to gives and by using and , (16) becomes

It is easy to see that the general solution of this equation is

According to (60),we find the integration factor to be of the form

Then the conserved form satisfies the following equality: which gives the original path equation. Thus the reduced equation is where is a constant, and the solution of (76) is determined for two different cases of arbitrary function.(i)For , where is a constant, is the solution of original path equation (16).(ii)For , is the other solution of the same equation.


For another case , the infinitesimal generators are

Thus, we can calculate -symmetry of path equation using, for example, Lie symmetry generator. For this generator the infinitesimals are

Therefore, the characteristic is written as

By using (58) we obtain the -symmetry

A solution of (59) for this case is and we can write , then to obtain path equation in terms of one can have

By using these equalities (84) we find the following equation: in which the general solution is

To find the integration factor one can write above equation in terms of as and then the integration factor becomes

If we substitute in (87), then the reduced equation in terms of is and the solution of (89) is where and are constants. It is clear that this solution satisfies the original path equation (16). Also, one can write which is the first integral of equation that provides the path equation (16).


For this case the eight-parameter symmetry generators are obtained as follows:

Now let us consider operator, and then the corresponding infinitesimals and are

Using these infinitesimals we find the characteristic and the -symmetry is

By using (95) the equation (59) becomes

A solution of (96) is

This equation can be written as

By differentiation of (98) we have and if we substitute (98) and (99) into the path equation, we obtain and the solution of (100) is

To define , one can write

Therefore, by using the relation (60) we find the integration factor

If we rewrite (102) in terms of and then we substitute this expression into integration factor, the reduced equation of path equation becomes where is a constant. By the solution of (104), we obtain the solution that satisfies the original path equation (16) as where is a constant, and the corresponding conservation law is


For this case the infinitesimal generators of path equation are

If we consider, for example, symmetry generator and then and are then the characteristic by (56) is

If we apply the operator (52) to this characteristic (109), we obtain , and the -symmetry is equal to zero. For symmetry generator we find also similar to previous one. Hence, we can use another symmetry generator, for example, to obtain -symmetry. For this case, are infinitesimals, and the corresponding characteristic is

We find the -symmetry from (58) as in the following form:

By applying (112) to (59) we obtain the solution And we write this expression (113) in terms of as

By differentiating (114) with respect to one can write and by substituting and to the original path equation we obtain where the solution of (116) is

To define this equality in terms of variable then is defined as follows: so we obtain the integration factor using (60)

Finally one can write the conservation law which gives the original path equation. And thus we can express the first integral, which is reduced form of the path equation where is a constant. Integrating (121) we obtain the solution that satisfies the original equation where is a constant.


If is assumed in the polynomial form and then Lie symmetry generators are , for example, can be used to obtain -symmetry, and for this generator the infinitesimals are

By using , the characteristic function is written as

By considering (58), the -symmetry becomes

The solution of (59) is

To write (16) in terms of , we can express the following equality:

By taking derivative (128) with respect to , then we have

If we substitute and into the path equation, then one can find and a solution of this equation (130) is

By using (60) we find the integration factor of the form

It is easy to see that the conserved form satisfies the following equality: and this equality gives the original path equation. Thus the reduced form of path equation is where is a constant. And all results are summarized in Table 2.

Table 2: Table of -symmetry classification with Lie symmetry of path equation.

5. -Symmetries and Jacobi Last Multiplier Approach

Definition of -Symmetry. Let be a vector field on which is open subset, and has the property of . For , denotes the corresponding -jet space, and their elements are , where, for , denotes the derivative of order of with respect to . In addition let be a vector field defined on , and let be an arbitrary function. Then the -prolongation of is with where is total derivative operator with respect to such that

In this section we analyze -symmetries of path equation by using Jacobi last multiplier as another approach. First (61) can be written by using system of first-order equations, which is equivalent to the expression and by solving the following differential equation, the Jacobi last multiplier of (138) is found: where, namely, is

The nonlocal approach [13, 20] to -symmetries is analyzed to seek -symmetries such that With this idea always can be considered to be of the form such as . But this relation cannot be considered if the divergence of (138) is equal to zero. So is chosen like this form because any Jacobi last multiplier is a first integral of (138). In this section we again consider different choices of for -symmetry classification.

5.1. Constant

For this case the divergence of the path equation yields Substituting into (135) then from the solution of the determining equations (62) we obtain eight-parameter -infinitesimals and the generators are which corresponds to the classical Lie point symmetries since is equal to zero.


Another special form we consider here is . For this case we obtain the divergence of (16) in the form and by substituting into the prolongation formula, the -infinitesimals can be found as follows: and the corresponding generator is which is a new -symmetry.


For this case of the divergence of (16) gives and the corresponding -infinitesimals are while the corresponding new -symmetries are found to be as follows


For this case we find that and it is clear that -infinitesimal functions are

Therefore, we find new -symmetries as follows:

5.5. ,

The divergence of the path equation yields

If we substitute (154) into (135), we obtain -infinitesimals and the corresponding one-parameter -generator

It is clear that we should analyze two specific values for .

Case  1  . The divergence of path equation for this value of is the -infinitesimals can be written as and the -generator is

Case  2  . For another specific value of the divergence is the -infinitesimals are found as follows: and the -generator is

In summary all new -symmetries are presented in Table 3.

Table 3: -Symmetry classification table of path equation with Jacobi last multiplier.
5.6. Invariant Solutions

In this section we present some invariant solutions based on Jacobi multiplier approach.

Case  1. For the case