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

and Applied Analysis 3 Definition 6. IfX is a partial Noether operator corresponding to partial Lagrangian L, then the gauge function B(x, y) exists. Hence, the first integral is given by I = ξL + (η − y 󸀠 ξ) L y 󸀠 − B. (15) 3. Noether Symmetries of Path Equation Thedifferential equation describing the path of theminimum drag work is given in the form y 󸀠󸀠 − f 󸀠 (y) f (y) − y 󸀠2 f 󸀠 (y) f (y) = 0, (16) where y = y(x) 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 δ αy = ∂ ∂y − D x ∂ ∂y x + D 2 x ∂ ∂y xx , (17) and the partial Lagrangian L for the path equation (16) is L = 1 2 y 󸀠2 + lnf (y) . (18) Then the application of (18) to (14) and separation with respect to powers of y and arranging yield the set of determining equations, the over-system of partial differential equations 1 2 ξ y + ξ f 󸀠 (y) f (y) = 0, (19) η y − 1 2 ξ x + η f 󸀠 (y) f (y) = 0, (20) η x + ξ y lnf (y) − B y = 0, (21) ξ x lnf (y) − B x + η f 󸀠 (y) f (y) = 0. (22) To find the infinitesimals ξ and η, (19)–(22) should be solved together. First, (19) is integrated as


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 [3][4][5][6][7].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 (, ) =  1 (, )  + 2 (, ).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.
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)  ∈ A, 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.
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  ∈ A be a vector that satisfies  ̸ =  + , where  is a constant.Then  () represents th prolongation of the generalized operator (7), and partial Noether operator corresponding to a partial Lagrangian is formulated as in which  = (

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 Here several cases should be examined separately for different forms of ().
3.1.() =  = .For this case the solution of (27) gives to the following infinitesimals: where   are constants  = 1, . . ., 5. 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: (31) 3.2.() = .For the linear case of (), we obtain where  1 is a constant.The partial Noether operator is and the first integral is (34) 3.3.() =   .The solution of determining equations for the form of () =    gives the following infinitesimals where   are constants  = 1, . . ., 5, and the gauge function is The associated five-parameter symmetry generators take the form and the corresponding first integrals are 3.4.() = 1/( + ).For this case, the infinitesimal functions read where   are constants  = 1, . . ., 5, and the gauge function is The corresponding Noether symmetry generators are Abstract and Applied Analysis 5 And the conservation laws are 3.5.() =   .For this choice of (), we find the infinitesimals where  1 is constant, and we have the first integral For convenience all Noether symmetries and first integrals are presented in Table 1.

Invariant Solutions.
Invariant solutions that satisfy the original path equation can be obtained by first integrals according to the relation    = 0. 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    = 0, then the invariant solution of path equation ( 16) is where  1 ,  are constants.
(b) For the same () function, the conservation law is and the invariant solution similar to previous one is where  1 ,  are constants.
Case 2. Let us consider () = 1/( + ), then the first integral yields and the solution of this equation gives where  1 ,  are constants, in which it is obvious that the invariant solution (50) satisfies the original path equation.

𝜆-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 [10][11][12].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 () = 0.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 (57) 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  [, (1)] , that is, a particular solution of the equation where  [, (1)] 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 firstorder 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  [, (1)] .

𝜆-Symmetries Using Lie Symmetries of Path Equation.
Let us consider an th-order ODE as follows: Thus the invariance criterion of (61) is pr ( () −  (, ,   ,   , . . .,  (−1) )       () = = 0. (62) 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 [10][11][12], 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  = 0 and  = 1 [10][11][12].Therefore, the number of unknowns in the equation is reduced to find  1 (, ) and  2 (, ) 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 ().

Arbitrary 𝑓(𝑦).
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  1 is a constant, is the solution of original path equation ( 16).(ii) For () =   , is the other solution of the same equation.(79) Thus, we can calculate -symmetry of path equation using, for example,  1 Lie symmetry generator.For this generator  1 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  3 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).

𝑓(𝑦) = 1/(𝑚𝑦 + 𝑛).
For this case the eight-parameter symmetry generators are obtained as follows: Using these infinitesimals we find the characteristic and the -symmetry is Abstract and Applied Analysis 9 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  3 is a constant, and the corresponding conservation law is If we apply the operator  (52) to this characteristic (109), we obtain (O ¸) = 0, and the -symmetry is equal to zero.For  2 symmetry generator we find also  = 0 similar to previous one.Hence, we can use another symmetry generator, for example,  7 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  1 is a constant.
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 ) ⋅ (129) 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.

𝜆-Symmetries and Jacobi Last Multiplier Approach
Definition of  ∈  ∞ ( (1) )-Symmetry.Let V be a vector field on  which is open subset, and has the property of  ⊂  × .For  ∈ N,  () ⊂  ×  () denotes the corresponding jet space, and their elements are (,  () ) = (, ,  1 , . . .,   ), where, for  = 1, . . ., ,   denotes the derivative of order  of  with respect to .In addition let  = (, )  + (, )  be a vector field defined on , and let  ∈  ∞ ( (1) ) be an arbitrary function.Then the -prolongation of  is pr =  (, )   +  (, )   +  (1) (, ,   ,   , . . .,  (−1) )    +  (2) (, ,   ,   , . . .,  (−1) )    , 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  = log (1/).But this relation cannot be considered if the divergence of (138) Div ≡ ∑  =1 (  /  ) 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.

𝑓(𝑦) = 𝑘 = 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.
5.2.() = .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.
Case 1 ( = 1/3).The divergence of path equation for this value of  is the -infinitesimals can be written as and the -generator is Case 2 ( = 1/2).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.

Invariant Solutions.
In this section we present some invariant solutions based on Jacobi multiplier approach.
Case 1.For the case () =   we can investigate   1 to find the invariant solution of path equation.The first prolongation of   1 is Pr and the Lagrange equations are gives the first order invariants that replaced into path equation generate the first-order equation the solution of this equation yields and the first integral is this equality gives the original path equation (16).The reduced form of path equation is in which the solution of (169) is where  1 and  are constants.It is clear that (170) is similar to the solutions (48) and ( 122).If we apply similar process for the  2 symmetry generator, we obtain first-order invariants for this case as and the first integral is another reduced form of path equation ( 16) is The solution of ( 173) is given by The solution of (178) is where  1 and c are constants, and it is clear that (182) is similar to the solution (105).

Conclusion
The aim of this study is to classify Noether and -symmetries of path equation describing the minimum drag work.The symmetry classification of the equation is analyzed with respect to different choices of altitude-dependent arbitrary function () of the governing equation, which represents a combination of the density, the drag coefficient, the cross sectional area, and the velocity.It is a fact that an ordinary differential equation should have a Lagrangian function to obtain Noether symmetries.In this study we consider the partial Lagrangian approach for obtaining Noether symmetries and constructing a classification in the problem.Thus, new first integrals (conserved forms) are obtained directly by using each Noether symmetry given by symmetry of the equation.With this point of view we find and classify the new forms of first integrals, and then the invariant solutions of path equation are constructed for specific forms of ().
In the literature, as a different and a new concept, symmetries of the second order ordinary differential equations are analyzed by assuming -function in the linear form.However, in our study, we prove that it is not possible to obtain -symmetries of the drag equation by selecting function in a linear form.So we study another approach to obtain -symmetries based on using Lie point symmetries of the path equation.Thus, we have derived -symmetries, integrating factors, first integrals, and the reduced form of the original path equation.Based on using these new symmetries, we present some new different invariant solutions by calculating new reduced forms and first integrals.
In our study, additionally, the Jacobi last multiplier concept is presented as a new and an alternative approach to construct -symmetries of the path equation algorithmically.In this method, first, -function is determined by taking divergence of the governing equation and then the infinitesimals functions  and  are determined from the determining equations, then we calculate new -symmetries.In this study we generate first-order equations by using these new symmetries, which provide invariant solutions of path equation.After all calculations we present that all methods discussed in this study have their own important properties to find first integrals and invariant solutions of ordinary differential equations, and the advantages of these approaches are given for specific cases.Furthermore, all symmetry classifications are presented in tables.

Table 1 :
Noether symmetry classification table of path equation.
4.1.5.() =   .If () is assumed in the polynomial form and then Lie symmetry generators are

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