#### Abstract

In this study we apply partial Noether and -symmetry approaches to a second-order nonlinear autonomous equation of the form , called Liénard equation corresponding to some important problems in classical mechanics field with respect to and functions. As a first approach we utilize partial Lagrangians and partial Noether operators to obtain conserved forms of Liénard equation. Then, as a second approach, based on the -symmetry method, we analyze -symmetries for the case that -function is in the form of . Finally, a classification problem for the conservation forms and invariant solutions are considered.

#### 1. Introduction

In classical mechanics, it is known that many important problems can be derived from Liénard equation of the form . For instance, in dynamics, the Van der Pol oscillator that is a nonconservative oscillator with nonlinear damping is a physically important example of Liénard equation. Liénard equation can also be considered as a model for a spring-mass system where the damping force corresponds to the position (e.g., the mass might be moving through a viscous medium of varying density), and the spring constant corresponds to how much the spring is stretched. It is possible to present other similar examples related to the Liénard equation. Thus, it can be said that Liénard equation has an important role in mathematical physics and mechanics fields.

In addition, with respect to the investigation of solution of differential equations, one of the most powerful methods for nonlinear differential equations is based on the study of Lie group of transformations. In the last century, the applications of Lie groups to the problems in mechanics, mathematics, physics, and so forth, have been carried out by many mathematicians, for example [1–12]. For the case of Liénard equation, the classical Lie point symmetries of Liénard equation are investigated in detail in the study [13]. However, the application of Noether theorem in the concept of theory of Lie groups to differential equations introduces that any Noether symmetry of the action of a physical system has a corresponding conservation law [1]. A conservation law means a quantity associated with a physical system that remains unchanged as the system evolves in time. In classical mechanics, the natural form of the Lagrangian is defined as the difference between the kinetic energy and potential energy of the system. An important property of the Lagrangian is that conservation laws can be easily derived from it. Furthermore, Noether theorem presents that variational symmetries are in one to one correspondence with conservation laws for the associated Euler-Lagrange equations. If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler-Lagrange equation. In addition, a Lagrangian of a system can be determined by making use of partial Lagrangian approach and the conservation laws can be obtained directly. The first study on this approach is carried out in [14]. In this study the Noether symmetries of Liénard equation are investigated for two different cases of arbitrary functions and . In fact, function is considered in two different forms for corresponding to two different mechanical problems, namely, linear undamped systems and linear damped systems. For both cases, the function can be chosen in a linear form for constant values of and . The first case for the linear undamped systems corresponds to the equation of motion of the free particle and simple harmonic oscillator, and the second case for the linear damped systems corresponds to the damped harmonic oscillator. As a result, for different choices of and , the conserved forms, partial Noether symmetries can be analyzed and classified for the Liénard equation.

In addition, in the literature, there is another approach called -symmetry, which is introduced by Muriel and Romero [15–18]. They introduce a new prolongation formula to investigate -symmetries for second order differential equations and they prove that although the equation has no Lie symmetries and by using the prolongation formula, then -symmetry of differential equation can be obtained and so the order of equation can be reduced. For convenience they consider a specific form (linear) for -function such as [17]. For example, for the specific choices functions and , the Liénard equations correspond to the general modified Emden-type equation and then its groups properties can be analyzed with linear term and constant external forcing via -symmetries. Moreover, it is possible to apply -symmetry approach to the Liénard equation by choosing other different forms of function and one can examine different functions corresponding to each function. So we can obtain different types of Liénard equations having physical meaning and highly nonlinear forms and by -symmetry concept new solutions, conservation laws and classification properties of the Liénard equation can be investigated.

The outline of paper is as follows. In Section 2 we introduce some preliminaries about partial Noether theorem. Section 3 is devoted to determine conservation laws and partial Noether symmetries of some important problems in classical mechanics derived from Liénard equation. In Section 4 we present some fundamental definitions about -symmetry approach and the calculation procedure of the integration factor and the conservation laws from -symmetries algorithmically. In Section 5 we investigate conservation laws, integration factors, and invariant solutions by using -symmetry concept, as an alternative approach, of some mechanics problems considered in Section 3. Some important results in the study are discussed in Section 6.

#### 2. Fundamental Definitions about Noether Theorem

Let be the independent variable with coordinates and the dependent variable with coordinates . The derivatives of with respect to are where is the operator of total differentiation. Additionally, if is a vector space of all differential functions of all finite orders, that is called the universal space and then the operators given below can be defined in the space .

*Definition 1. *For each the* Euler-Lagrange operator* is defined by

*Definition 2. *Generalized operator can be formulated as
and for convenience this operator can be written as
where the additional coefficients are examined by the formula
in which is the Lie characteristic function defined as
The generalized operator (5) can also be rewritten by using Lie characteristic function such as
and the Noether operator associated with a generalized operator is given by
where the Euler-Lagrange operators with respect to derivatives of are obtained from (3) by replacing by the corresponding derivatives

*Definition 3. *Suppose that a th-order ordinary differential equation system is given by
with maximal rank and is locally solvable.

*Definition 4. * which satisfies what is given below
is called a conservation law of (11). Moreover, (12) exists for all solutions of (11). In addition, (11) are assumed to be as the following form:
Let us consider differential equations of the form
where , , being the order of (14), is a Lagrangian and is the Euler-Lagrange operator defined by (3).

Suppose that if nonzero functions are defined such that (14) in which then is called a partial Lagrangian of (13).

*Definition 5. *If there exists a vector , , , are constants and . If is the characteristic of then
is called Noether-type symmetry operator corresponding to a partial Lagrangian . In addition, is called gauge function. If is a partial Noether operator corresponding to partial Lagrangian then the gauge function exists then the first integral is given by

#### 3. Partial Noether Symmetries of Liénard Equation

Let us consider the one-dimensional nonlinear Liénard-type differential equation such as
where and are smooth functions of . In this section we analyze the partial Noether symmetries of (18). For the Liénard equation (18), the* Euler-Lagrange* operator can be written as (3)
and the partial Lagrangian for the Liénard equation (18) is given by
Therefore (18) becomes
The partial Noether operators corresponding to are
where is equal to
By using the relation (16) we have
and analyzing (24) the coefficients of derivatives gives the following determining equations:
First, (25) gives
where is an arbitrary function. Then substituting into (26) gives
where is an arbitrary function. Differentiating (27) and (28), respectively, with respect to and we have the following equation:
which is a differential equation in terms of the unknown functions and . In the following subsections we consider some classical mechanics problems related with Liénard-type differential equation for different choices of and functions.

##### 3.1. Linear Undamped Systems:

If we consider , then (31) is equal to It is easy to see that a solution of (32) is where and are arbitrary constants. We now analyze the following subcases: (i),(ii) and ,(iii) and ,(iv) and .

*Case 1 (Free particle motion ). *If we take , then (33) is equal to zero and thus the Liénard equation (18) becomes
This form of (18) represents the equation of motion of a free particle. By using (32), (29), and (30) we obtain the partial Noether infinitesimals
where , , , , and are arbitrary constants. The corresponding partial Noether symmetry generators are
Additionally, the gauge function is obtained from (25) and (26)
The corresponding conservation laws by the (17) formula are obtained in the following forms:
It is important to mention that each conservation law satisfies the equality with respect to the characteristic .

*Case 2 (Free falling particle , ). *For this case of and (18) becomes
which corresponds to the equation of motion of a free falling particle. By solving (25)–(28) we get the , and gauge function
The corresponding partial Noether symmetry generators are
and the conservation laws corresponding to each partial Noether symmetries are

*Case 3 (Free linear harmonic oscillator ). *If we chose as an arbitrary and thus the Liénard equation (18) is equal to
which refers to the equation of a linear harmonic oscillator for . Then the solution of determining (25)–(28) for the and gives
The corresponding partial Noether symmetry generators are found as follows:
and the conservation laws are
in which it is easy to see that the relation is satisfied for each conserved form.

*Case 4 (Displaced linear harmonic oscillator ). *If the and are assumed to be arbitrary, then we obtain the Liénard equation (18) as the following form:
which represents a displaced simple harmonic motion. Equation (47) transforms the previous case with a change of variable such as . Therefore, this form of Liénard equation gives the similar partial Noether symmetries, generators, and conservation laws to Case 3.

##### 3.2. Linear Damped Systems:

The other case for is to be considered as a constant. By substituting into the (29) and (30) then we calculate the infinitesimals and rewrite (31), which includes the unknown functions and If we analyze the solutions of (49) then a solution of this equation for can be considered as linear function of , which is similar to the section ; that is, where and are arbitrary constants. To examine all possibilities of we again consider the following subcases: (i),(ii) and ,(iii) and ,(iv) and .

*Case 1 (Free particle in a viscous medium (**))*. For this case the Liénard equation (18) is equal to
which represents the equation of a free particle in a viscous medium. If we substitute into (49), then by using the determining (27)-(28) we find the partial Noether symmetries , and gauge function such that
and partial Noether symmetry generators are
By applying the definition of first integral (17) we obtain the conservation laws
which satisfy the relation .

*Case 2 (Falling particle in a viscous medium **, **)*. If the functions and are substituted into the Liénard equation (18) we obtain
which is the equation of a falling particle in a viscous medium. It is obvious that this form of Liénard equation (55) can be transformed into previous case by using the simple transformation such as . Therefore, if this transformation can be applied for (52)–(54) then partial Noether symmetries, partial Noether generators, and conservation laws are derived, respectively, in terms of for (55).

*Case 3 (Damped linear harmonic oscillator **)*. Suppose that is arbitrary and is equal to zero and thus Liénard equation (18) can be obtained as the form
which corresponds to the equation of a damped linear harmonic oscillator. By considering similar process to previous cases we get partial Noether symmetries, partial Noether generators, and gauge function
where , , , , and are arbitrary constants and ; is assumed to be positive. And the partial Noether generators are
and the corresponding conservation laws are

*Case 4 (Displaced damped harmonic oscillator **)*. For this case we get the Liénard equation (18)
which is the equation of displaced damped harmonic oscillator. By using the change of variable such as this case corresponds to the previous form of Liénard equation. So by applying this transformation to (57)–(59) partial Noether operators, partial Noether generators, and conservation laws can be obtained similarly.

#### 4. -Symmetry Approach for Differential Equations

In this section we consider -symmetry properties of Liénard equation and for this purpose we first present some fundamental definitions and theorems about -symmetries [15, 17]. Let us consider a th-order ordinary differential equation where variables are in some open set . For , denotes the corresponding -jet space and the elements of are denoted by . By applying implicit function theorem to (61), as a result, this equation can be written in the explicit form The vector field is called the vector field associated with (62). An integrating factor of (61) is assumed to be such as for some and . If the left-hand side of (61) is multiplied by integration factor , total derivative of some function is obtained as the form It is clear that -symmetries can be used for reduction of order of a differential equation. Namely, is an exact differential equation, which is the result of trivial reduction of order , [18]. If , , is any solution of the partial differential equation: when , then the vector field is a -symmetry of (61).

Theorem 6 (see [17]). *Assume that (61) is a th-order ordinary differential equation that admits an integrating factor such that . If is any particular solution of (65), then the vector field is -symmetry of (61).*

Now let us consider the second order differential equation (62) and a vector field (63) then a conservation law (first integral) of (66) is any function such as which satisfies the relation If is assumed to be a -symmetry of (66) and is a first-order invariant of and any particular solution of the equation then a first-order invariant reduced equation of the form can be obtained by using the reduction process associated with the -symmetry. Thus the general solution is found such as an equation of the implicit form It is clear that is an equivalent form of (66). Consequently, is an integrating factor of (66).

Theorem 7 (see [17]). *Let be a th-order ordinary differential equation, where is an analytic function of its arguments. There exists a function , for some , such that the vector field is a -symmetry of the equation.*

#### 5. -Symmetries, Conservation Laws, and Integrating Factors of Liénard Equation

In this section we investigate -symmetries of (18) for different cases of arbitrary functions of and . By considering theorem (62) the vector field is assumed to be as a -symmetry of (18). If we write explicit form of (18) corresponding to (66) we obtain By applying (65) to (72) we find in which is any particular solution of (73). For convenience a solution of of (73) can be assumed to be linear form such that Therefore, the expansion of (73) becomes The usual separation of powers of derivatives of (75) reduces to the system A particular solution of (76) is found And so (77) becomes To obtain the solution of (80) different cases of should be considered. Firstly in order to compare Noether and -symmetry approaches we investigate the similar cases of functions and , which are analyzed in Section 3.

*Case 1*. In this case we analyze and functions which correspond to the case of Section 3.1. Namely, and , which represent linear undamped systems. For this functions the determining equations (80) and (78) become
The solution of (81) is given by
*Case 1.1 ** and *. If we substitute (83) into (82), we obtain and so . Therefore, is found by using (79) as follows:
As a result, a first-order invariant of is applied to (69) then the conservation law and integration factor can be derived by using the -symmetry (84)
A solution of (85) is
In order to express the Liénard equation (18) in terms of , the terms and can be eliminated from (86) and so the reduced form becomes
and the general solution of (88) is
The integration factor corresponding to (71) can be written as
Here, which satisfies the relation (68) is equal to conserved form
*Case 1.2 (** and **)*. By applying same process with the previous case, it is clear that in order to satisfy (82), must be equal to zero. As a result, the results in this case are similar to the Case 1.1.

*Case 1.3 (** and **)*. For this form for (18), similar results can be obtained with (84)–(86). Therefore the reduced form of (18) can be written as
The general solution of (91) is
By using the relation (71) integration factor is given by
In (92), corresponds to conserved form
which is equal to original Liénard equation (18) for this case. This relation allows us to obtain new invariant solution of (18) which is
where and are constants.

*Case 2*. Now we analyze the case which is similar to the case in Section 3.2. As a reminder, in this section the Liénard equation represents linear damped systems for the choices of and . For this case of and the third determining equation (78) becomes
Now we evaluate different cases of and .

*Case 2.1 ** and *. By applying similar procedure and after determination of and , the reduced form of Liénard equation (18) is obtained in terms of
The general solution of (97) is
According to (71) the integration factor is
The relation related to the gives the corresponding conservation form
Then the new invariant solution of (18) can be written as
where and are constants.

*Case 2.2 ** and *. After same manipulations it is possible to show that this case corresponds to the Case 2.1.

*Case 2.3 ** and **)*. According to (84)–(86), the reduced form of (18) is given by
The general solution of (102) is
By using of the relation (71) the integration factor is obtained such that
which is equal to conserved form can be examined from (102) and it can be written as
which satisfies the original Liénard equation (18) for this case. And so we can derive new invariant solution of (18) in the following form:
where and are constants.

##### 5.1. Modified Emden Equations: ; Is a Constant

It is clear that a particular solution of (31) for this special choice of can be given by In this case by the choices of the functions of and , Liénard equation is considered in the nonlinear form [19]. We analyze four different case of and . These cases are(i),(ii) and ,(iii) and ,(iv) and .In this study we consider the first and the third cases since one can find only lambda function for the cases in which it is assumed in the linear form.

###### 5.1.1. Modified Emden Equation

Corresponding Liénard equation for is which is nonlinear ordinary differential equation. In order to obtain -symmetries the determining equations (76)–(78) should be evaluated together. So the results of these equations give -symmetry (79) such that and by substituting (109) into (69) we obtain It is clear that a solution of this (110) is By using (111) and can be derived in terms of and so Liénard equation can be written as which is the reduced form. The general solution of (112) is Integration factor corresponding to (71) is It is clear that is equivalent to conserved form and the reduced equation of (108) can be derived as The solution of (116) gives the new invariant solution of (108) where and are constants.

###### 5.1.2. Modified Emden Equation with Linear Term ( and )

Choosing as arbitrary Liénard equation is obtained as a nonlinear ordinary differential equation in the form By the same way and for this case the reduced form of (18) can be obtained The general solution of (119) is According to (71) the integration factor is examined It is clear that is equivalent to conserved form The reduced equation of (118) can be derived as and the solution of (123) gives the new invariant solution of (108) where and are constants.

##### 5.2. Alternative Approach for the Consideration of and

Alternatively, to get a classification, we first define function and then try to determine , , and functions. The corresponding subcases are given below.

##### 5.3.

where , , and are arbitrary constants. For this case of we get (80) of the form A particular solution of this equation gives the such that Hence, -symmetry can be written by using (79) If we substitute and into (78), then the new form of differential equation is obtained in terms of unknown function then the general solution of (128) is given by where is constant. Thus the Liénard equation for these special cases for and (18) becomes In order to obtain the conservation law and the integration factor by using -symmetry (127), a first-order invariant of is obtained in the following form from the relation (69): which has a solution of the form So in terms of the Liénard equation (18) can be written as And the general solution of (133) is given by According to (71) we obtain the integration factor as It is the fact that is equivalent to the following conserved form which gives the original Liénard equation given in the form (130). It is possible to say that the solution of reduced equation of (130) cannot be solved for the general . But, if we choose specifically then the reduced form of (130) is