On Conservation Forms and Invariant Solutions for Classical Mechanics Problems of Liénard Type

y 󸀠󸀠 + f (y) y 󸀠 + g(y) = 0, called Liénard equation corresponding to some important problems in classical mechanics field with respect to f(y) and g(y) 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 λ-symmetry method, we analyze λ-symmetries for the case that λ-function is in the form of λ(x, y, y󸀠) = λ 1 (x, y)y 󸀠 + λ 2 (x, y). Finally, a classification problem for the conservation forms and invariant solutions are considered.


Introduction
In classical mechanics, it is known that many important problems can be derived from Liénard equation of the form   + ()  + () = 0.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][2][3][4][5][6][7][8][9][10][11][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 Advances in Mathematical Physics 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 () =  1  +  2 can be chosen in a linear form for constant values of  1 and  2 .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  1 and  2 , 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][16][17][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 (, ,   ) =  1 (, )  +  2 (, ) [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.

Fundamental Definitions about Noether Theorem
Let  = ( 1 , . . .,   ) be the independent variable with coordinates   and  = ( 1 , . . .,   ) the dependent variable with coordinates   .The derivatives of  with respect to  are where is the operator of total differentiation.Additionally, if A 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 A.
Definition 1.For each  the Euler-Lagrange operator is defined by (3) 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 Advances in Mathematical Physics 3 Definition 3. Suppose that a th-order ordinary differential equation system is given by   (, ,  (1) ,  (2) , . . .,  () ) = 0,  = 1, 2, . . ., , (11) with maximal rank and is locally solvable.Definition 4.   =   (, ,  (1) ,  (2) , . . .,  (−1) ) ∈ A 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  = (, (, ,  (1) , . . .,  () )) ∈ A,  ≤ ,  being the order of ( 14), is a Lagrangian and /  is the Euler-Lagrange operator defined by (3).
Suppose that if nonzero functions    ∈ A are defined such that ( 14) in which  1  ̸ = 0 then  is called a partial Lagrangian of (13).

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  1 is equal to By using the relation (16) we have and analyzing (24) the coefficients of derivatives   gives the following determining equations: − () −   ∫  ()  −   = 0.
It is easy to see that a solution of (32) is where  1 and  2 are arbitrary constants.We now analyze the following subcases: Case 1 (Free particle motion ( 1 =  2 = 0)).If we take  1 =  2 = 0, 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  1 ,  2 ,  3 ,  4 , and  5 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    = (−  ) = 0 with respect to the characteristic .
Case 2 (Free falling particle ( 1 = 0,  2 ̸ = 0)).For this case of  1 and  2 (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 ( 1 ̸ = 0,  2 = 0)).If we chose  1 as an arbitrary and  2 = 0 thus the Liénard equation ( 18) is equal to which refers to the equation of a linear harmonic oscillator for  1 > 0. Then the solution of determining ( 25)-( 28) for the () = 0 and () =  1  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    = 0 is satisfied for each conserved form.
Case 4 (Displaced linear harmonic oscillator ( 1 ̸ =  2 ̸ = 0)).If the  1 and  2 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  → ỹ =  + ( 2 / 1 ).Therefore, this form of Liénard equation gives the similar partial Noether symmetries, generators, and conservation laws to Case 3.

Linear Damped
and rewrite (31), which includes the unknown functions () and () (49) 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  1 and  2 are arbitrary constants.To examine all possibilities of () we again consider the following subcases: Case 1 (Free particle in a viscous medium ( 1 =  2 = 0)).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 () = 0 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    = 0.
Case 2 (Falling particle in a viscous medium ( 2 ̸ = 0,  1 = 0)).If the functions () =  and () =  2 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  → ỹ =  + ( 2 /).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 ( 1 ̸ = 0,  2 = 0)).Suppose that  1 is arbitrary and  2 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  1 ,  2 ,  3 ,  4 , and  5 are arbitrary constants and  = √ 2 − 4 1 ;  2 − 4 1 is assumed to be positive.And the partial Noether generators are and the corresponding conservation laws are Advances in Mathematical Physics 7 Case 4 (Displaced damped harmonic oscillator ( 1 ̸ =  2 ̸ = 0)).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  → ỹ =  +  2 / 1 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.

𝜆-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  ⊂  ×  ≅ R 2 .
Theorem 6 (see [17]).Assume that (61) is a th-order ordinary differential equation that admits an integrating factor  such that | Δ=0 ̸ = 0.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) (67) 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 ] [, (1)] and any particular solution of the equation then a first-order invariant reduced equation of the form Δ  (, ,   ) = 0 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   ((, (, ,   )) = 0 is an equivalent form of (66).Consequently,  (, ,   ) =   (, ,  (, ,   )) ⋅    (, ,   ) (71) is an integrating factor of (66).
Theorem 7 (see [17]).Let   = (,  (−1) ) 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.

𝜆-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  (, ,   ) =  1 (, )   +  2 (, ) .
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, () = 0 and () =  1  +  2 , 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 (() = 0 and () = 0 ( 1 =  2 = 0)).If we substitute (83) into (82), we obtain () = 0 and so  2 (, ) = 0. Therefore, (, ,   ) is found by using (79) as follows: As a result, a first-order invariant (, ,   ) of ] [, (1)] 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 (() = 0 and () =  2 ( 1 = 0,  2 ̸ = 0)).By applying same process with the previous case, it is clear that in order to satisfy (82),  2 must be equal to zero.As a result, the results in this case are similar to the Case 1.1.
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 () =  = constant and () =  1  +  2 .For this case of () and () the third determining equation (78) becomes Now we evaluate different cases of  1 and  2 .
Case 2.1 (() =  and () = 0 ( 1 =  2 = 0)).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  2 are constants.

Case 2.3 (𝑓(𝑦) = 𝑘 and 𝑔
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  2 and  are constants.

Modified Emden Equations: 𝑓(𝑦)=𝑘𝑦; 𝑘 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  1 and  2 .These cases are (i)  1 =  2 = 0, (ii)  1 = 0 and  2 ̸ = 0, (iii)  1 ̸ = 0 and  2 = 0, 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.

Modified Emden Equation
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

Advances in Mathematical Physics
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  1 are constants.

Modified Emden Equation with
Linear Term ( 1 ̸ = 0 and  2 =0).Choosing  1 as arbitrary  2 = 0 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  1 are constants.

Alternative Approach for the Consideration of 𝑓(𝑦) and 𝑔(𝑦).
Alternatively, to get a classification, we first define () function and then try to determine  1 ,  2 , and () functions.The corresponding subcases are given below.

𝑓(𝑦)=𝑎𝑦
. where , , and  are arbitrary constants.For this case of () we get (80) of the form A particular solution of this equation gives the  2 (, ) such that Hence, -symmetry can be written by using ( 79) If we substitute  1 and  2 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 Advances in Mathematical Physics 11 where  1 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 ] [, (1)] 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 × ( ( ( 2 + 3 + 2)  + 2 (  ( + 2) + ( + 1) 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  = 1 then the reduced form of (130) is where  1 is a constant.It is a fact that this type of Liénard equation (130) for the case  = 1 cannot be solved by known methods in the literature.But in this analysis, the solution of reduced equation (127) as a new invariant solution of (130) can be obtained as where  2 is a constant.
In this case, it is possible to check that two specific cases for  should be analyzed separately such that ( = −2,  = −1).We investigate these, respectively, in the following subcases.

𝑓(𝑦)=(𝑎/𝑦
2 )+(/ 3 )+,  = −2.By using this form of (), (80) becomes A particular solution of  2 (, ) from differential equation ( 139) is Thus the -symmetry is obtained from the relation (79) as The differential equation ( 78) can be rewritten in terms of unknown function () by considering  1 and  2 in the following form: hence, the general solution of (142) is found as follows: where  1 is constant.So we can rewrite the Liénard equation (18) for this case of () and () If we substitute -symmetry into (69), the relation is obtained.A solution of this equation is In order to obtain Liénard equation (18) in terms of {, ,   }, the equalities   and   can be simplified (146).And so the reduced form is and the general solution of (147) is given by thus the integration factor corresponding to (71) is It is clear that (, (, ,   )) is equal to the conserved form which gives the original Liénard equation such as the form (143).

𝑓(𝑦)=(𝑎/𝑦)+(𝑏/𝑦
2 )+,  = −1.Equation ( 80) is found in the following form: A particular solution of (151) is equal to the general solution of (154) is where  1 is constant.For this case of () and () Liénard equation ( 18) corresponds to According to (69) we obtain A solution of (157) is By using (158) the reduced form of Liénard equation ( 18) in terms of {, ,   } can be written in the form and the general solution of (159) is According to (71) the integration factor is equal to (, (, ,   )) that is reduced to (160) corresponds to conserved form of ( 18) which is equal to the original Liénard equation of the form (156).
5.4.()= cosh +.According to (80) the differential equation in terms of  2 is given by The solution of above equation is found as follows: The -symmetry considered as in the linear form of functions of  1 and  2 is equal to Thus, (78) that includes the () function is obtained as the form The general solution of (166) is where  1 is a constant.Consequently, Liénard equation (18) for this case of () and () is found as below By consideration of the relation (69) the equation In order to find the reduced form of Liénard equation ( 18),   and   can be examined (170).Therefore we can write the following equation: and the general solution of (171) is equal to The integration factor corresponding to (71) is The conserved form (, (, ,   )) can be derived from (172) which gives the original Liénard equation of the form (168). ))) = 0,

Conclusion
In this study we apply partial Noether symmetry and symmetry approaches to Liénard equation, which has an important role in terms of physics and mechanics.Based on the studies of partial Noether operators, we derive partial Lagrangian of Liénard equation, which leads to obtain partial Noether symmetries.Since the Liénard equation includes two arbitrary functions such as () and (), we evaluate different cases of these functions for the classification.Partial Noether symmetries are investigated for two different problems related to the linear damped and undamped systems.And we derive new first integrals (conserved forms) corresponding to each partial Noether symmetry.Alternatively, in the literature the -symmetry concept is used to study integration factors, first integrals, and invariant solutions of ordinary differential equations.Hence, the other part of this study is based on the relation between integrating factors and -symmetries for Liénard equation related to some mechanics problems.In fact, in this approach the determining equation for the differential equation includes not only infinitesimal functions but also  function, which are unknown functions but it is not possible to get a solution of determining equations easily.In order to overcome this difficulty for determining equations, the  function is assumed to be in a linear form.In this way the system of determining equations which include partial differential equations can be solved easily.In the first part of the study we consider same functions () and () considered in the case of partial Noether symmetries.Based on these functions we study symmetries and first integrals.Then we conclude that it is possible to obtain new first integral by using -symmetry approach, which is different from partial Noether symmetry approach.
Furthermore, we analyze only some specific cases of arbitrary function () to determine -symmetries.As a result of determining equations we find different () functions corresponding to each of () functions and obtain new -symmetries.By using -symmetries we obtain new reduced forms, integration factors, and conserved forms of highly nonlinear Liénard type equation.We present that the conserved forms lead to new invariant solutions of corresponding classical mechanics problems, which are considered by different approaches in the literature.