Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.


Introduction
The study of conserved quantities plays a great role in mathematical physics and in applied mathematics.For instance, a considerable number of phenomena have some kind of "conservation." Examples can be easily found from the hydrodynamics, electrodynamics, shallow water phenomena, and so forth.One can also mention the celebrated Kepler's third law or the conservation of energy in the classical mechanics, particularly the one-dimensional harmonic oscillator.In regard to these last two phenomena, the conserved quantity is called first integral, which is the analogous of conservation laws for ordinary differential equations models.
In a recent paper, Naz et al. [1] studied different approaches to construct conservation laws for partial differential equations.The purpose of this paper is to analyze all different approaches for construction of first integrals for ordinary differential equations.In fact, different approaches to derive first integrals can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches.In 1798, Laplace [2] developed a method known as the direct method for the construction of first integrals.Although such a method does not originally require any symmetry of the considered equation, Kara and Mahomed [3] developed a relationship between symmetries and conservation laws.The joint conditions of symmetry and direct method are used to construct the first integrals.
Noether's theorem [4] is a powerful technique to derive first integrals for the differential equations having Lagrangian formulations using its symmetries, although it requires a suitable Lagrangian.Kara et al. [5] developed the partial Noether approach.The partial Noether approach is applicable to differential equations with or without a Lagrangian.The interested readers are referred to [6][7][8][9][10][11] for discussions on first integrals by the Noether approach and partial Noether the main ideas behind the mentioned methods in Section 3.Then, in Section 4, we apply these different approaches to the harmonic oscillator equation.In Section 5 some solutions are obtained via first integrals.Relations between Hamiltonian functions and first integrals are discussed in Section 6.In Section 7, the first integrals for nonlinear jerk equation are computed by the Noether approach for the equation and its adjoint as a system and by the multiplier approach.The exact solutions of jerk equation for different cases are also established via first integrals.The first integrals for the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are also derived.Finally, concluding remarks are presented in Section 8.
Consider a th-order ordinary differential equation system   (, ,  (1) ,  (2) , . . .,  () ) = 0,  = 1, . . ., , where  is the independent variable and   ,  = 1, 2, . . ., , are the  dependent variables.We will adopt the summation convention and there is summation over repeated upper and lower indices.The total derivative with respect to  is ( The following are the basic operators defined in A, the vector space of differential functions.The Lie-B ä cklund operator  is defined as where in which   0 ≡   .The Euler operator is given by The characteristic form of Lie-B ä cklund operator (3) is in which   is the Lie characteristic function defined by The Noether operator associated with a Lie-Bäcklund operator  is where First Integral.A first integral of system (1) is a differential function  ∈ A, such that for every solution of (1).

Approaches to Construct First Integrals
Now we present various approaches to construct first integrals taken from the literature.

Direct Method.
The direct method was first used by Laplace [2] in 1798 to construct all local first integrals.The determining equations for the first integrals for the direct method are 3.2.Symmetry and First Integral Relation.Kara and Mahomed [3] added a symmetry condition to the direct method.The Lie-Bäcklund symmetry generator  and the first integral  are associated with the following equation: The first integrals are computed by the joint conditions (11) and (12).

Noether's Approach.
In 1918, Noether developed a new approach to construct first integrals [4] and it is currently known as Noether's approach.
In the Noether approach we need to construct Lagrangian (, , . . .,  (−1) ).The Noether symmetries are then computed from (14) and finally (16) provides the first integrals corresponding to each Noether symmetry.The reader is guided to [46] for further discussions about this technique and its relations with the so-called Noether symmetries.

Partial Noether
Approach.The partial Noether approach for construction of first integrals was introduced by Kara et al. [5] and it can be useful for constructing first integrals when the differential equation does not have a known Lagrangian.
The first integrals of the system (1) associated with a partial Noether operator  corresponding to the partial Lagrangian  are determined from (16).
We can also use the partial Noether approach for equations arising from the variational principal and have the Lagrangian.

Noether Approach for a System and Its Adjoint
Adjoint Equations.Let V = (V 1 , V 2 , . . ., V  ) be the new dependent variables.The system of adjoint equations to the system of th-order differential equations (1) is defined by (Atherton and Homsy [13], Ibragimov [12]) where Symmetries of Adjoint Equations.Suppose system (1) has the generator Ibragimov [12] showed that the following operator is a Lie point symmetry for the system (1) and ( 20): The operator ( 23) is an extension of (22) to the variable V  and yields    .
Conservation Theorem.Every Lie point, Lie B ä cklund, and nonlocal symmetry of the system of kth-order differential equations (1) yields a first integral for the system consisting of (1) and the adjoint equations (21).Let  be the Lagrangian defined by Then the first integrals are given from the formula where   ,   are the coefficient functions of the generator (22).The first integrals constructed from (26) contain the arbitrary solutions V of adjoint equation (21) and, thus, for each solution V one has first integrals.The dependence on the nonlocal variable V provides a nonlocal first integral.One can eliminate such variable if the original system of ODEs is nonlinearly self-adjoint [14,17] and to the equations admitting this remarkable property one can establish a first integral for the original system.[23] and Olver [24], the first integral can be expressed in the characteristic form as

Characteristic Method. According to Steudel
where   are the characteristics or multipliers.

Variational Approach.
The variational approach was developed by Olver [24].The variational derivative of ( 27) yields all the multipliers for which the equation can be expressed as a local first integral.The multipliers determining equation is and it holds for arbitrary functions ().

Variational Approach on Solution Space of the Differential Equation.
In this approach, the multiplier determining equation is obtained by taking the variational derivative of ( 27) on the solution space of the differential equation; that is, Equations ( 29) are less overdetermined than (28).Sometimes the characteristics constructed from ( 29) may correspond to adjoint symmetries (see [28]) and not to a first integral.
3.9.Integrating Factor Method for First Integrals.Consider the system (1) and let On the solutions  of system (1), it is concluded that [] = const, which means that [] is a first integral of (1) and the functions Λ  are integrating factors; see [28] for further details and deeper discussion.
The linearized system to ( 1) is given by where The adjoint of the linearized system (31) is given by Moreover, the operators   and  *  satisfy the identity where If Λ  satisfy the condition where then the first integral  is In ( 38) and ũ() = (ũ 1 (), . . ., ũ ()) are any fixed functions such that the function () is finite, while We finish with the following definition.
Definition 1.The system (1) is said to be self-adjoint if and only if

First Integrals of Simple Harmonic Oscillator
We compute the first integrals of simple harmonic oscillator by utilizing different approaches.Consider 4.1.Direct Method.Equation (11) with (, ,   ) becomes where Equation ( 43) after expansion results in or If we further restrict  to be then (46) becomes Splitting (47) according to derivatives of , we obtain The system of ( 49) is solved for , , and  to obtain where The first integrals are computed by the joint conditions (11) and (12).
The second important aspect of this approach is that we can associate a symmetry with a first integral.The relationship (12) holds for symmetry  3 and first integral  2 and, thus, symmetry  3 is associated with  2 .Similarly  4 is associated with  3 .This association of symmetries with a first integral helps in finding a solution via double reduction theory [38][39][40][41][42][43].

Noether's Approach. Equation (42) admits the standard Lagrangian
which satisfies the Euler Lagrange equation / = 0. Now we show how to compute the Noether symmetries corresponding to a Lagrangian (53).The Noether symmetry determining ( 14) results in where  = (, ),  = (, ), lie symmetry operators and  = (,) is the gauge terms.Expansion of (54) gives Noether symmetry determining equation The separation of (55) with respect to powers of derivatives of  gives rise to The solution of system (56) is Formula ( 16) with , , and  from (57) yields the first integrals (50). where The partial Noether operators  = / + / corresponding to  satisfy (19); that is, The usual separation by derivatives of  gives System (61) yields Formula ( 16) with , , and  from (62) yields the first integrals (50).Hence the first integrals in each case are    = (−  − ) = 0 with respective characteristic .Here the partial Noether's approach yields all nontrivial first integrals as obtained by Noether's approach.The difference here lies in the forms of  and  which are distinct from the ones used in the Noether approach.

Noether Approach for a
System and Its Adjoint.The adjoint equation for (42) is and this yields Let V = (, ), then V  =   +     , and 2 +     .Substituting these expressions of V and V  into (64) and equating the coefficients of    and   to zero, one obtains V =  +  1 cos  +  2 sin .Thus, ( 42) is nonlinearly self-adjoint.
The Lagrangian for the system consisting of ( 42) and ( 64) is The Lagrangian  satisfies The formula for first integrals from ( 26) is Equation ( 67) with  from (65) results in According to the conservation theorem, every Lie point, Lie-Bäcklund, and nonlocal symmetry of the system of secondorder differential equation ( 42) yields a first integral for the system consisting of ( 42) and the adjoint equation (64).For the Lie symmetry  1 = /, the first integrals are where V is the solution of adjoint equation (64).Note that V =  yields  1 and  2 ,  3 can be obtained from the substitution V = sin  and V = cos , respectively.One can use the other Lie symmetries given in (51) to derive the first integrals but one requires the solution of adjoint equation to construct first integrals.In order to illustrate this fact, let us consider the generator It is easy to check that (70) is not a Noether symmetry generator.Substituting  = 0 and  =  into (68), one arrives at The substitution V =  in (71) provides the trivial first integral  = 0.However, setting V = cos  and V = sin , respectively, into (71), then the first integrals  2 and  3 are obtained again.
where  1 , . . .,  6 are constants.The multipliers with respect to constants  1 , . . .,  5 are the same as obtained in Section 4.7 and yield the first integrals obtained in (50).The multiplier associated with  6 is  which does not correspond to any first integral.It might correspond to an adjoint symmetry.

Exact Solutions via First Integrals
The Noether symmetries associated with the first integrals can be utilized to derive the exact solutions of ordinary differential equations [34].
If  is a Noether symmetry and  is a first integral of (1) corresponding to a first-order Lagrangian  = (,  [1] ), then the following properties are satisfied [34]: [1] where [1] ( Proposition 2. Suppose  is a symmetry of   =   , where   =   (, ); then it satisfies Proposition 3. In (87) if  = 1 and  [1] () = 0, then  is a point symmetry of reduced equation (, ,   ) = , in which  is an arbitrary constant.Now we will compute the exact solutions of (42) using its first integrals which are reduced forms of the equation under consideration.The first integrals reduce an th-order ODE to ( − 1)th-order ODE.For scalar first-order ODE, the first integrals transform to quadrature whereas for scalar second-order ODE the first integrals result in the first-order ODEs.Some of these reduced forms (first integrals) can be solved directly.The other reduced form can be transformed to quadrature by using the Noether symmetries with its associated first integrals which yield the exact solutions.The first three integrals  1 ,  2 , and  3 of (42) yield a solution directly.Since    = 0 which implies  = , the first integral  1 in (50) can be written as which can also be expressed as Equation ( 153) is a variable separable and yields and this comprises the exact solution of ODE (42).
A similar procedure is adapted to get the following exact solution of (42) using  2 or  3 : Now we show how one can find the exact solution of (42) using Noether symmetries associated with the first integral.The Noether symmetry is associated with the first integral  4 in (50).The induced equation  4 =  1 can be expressed as Using (92) one can easily find the invariant and it reduces (93) to Equation ( 95) is expressible as a variable separable and it finally yields or The solution in ( 94) is an exact solution of (42) with  which can be determined from ( 96) or (97).

Hamiltonian Functions and First Integrals
Suppose  is the independent variable and (, ) = ( 1 , . . .,   ,  1 , . . .,   ) are the phase space coordinates.The derivatives of   ,   with respect to  are given by where is known as the total derivative operator with respect to .
Here we present the basic operators needed in the sequel after introducing the necessary notations.The Euler operator, for each , is and the associated variational operator is Applying operators ( 103) and (104) on equated to zero results in the following canonical Hamilton equations: Equations ( 106) are obtained using /  = 0 and /  = 0. Equation ( 105) is the well-known Legendre transformation which relates the Hamiltonian and Lagrangian, where   = / q  and q  = / ṗ  .

Let
on the system (106).
In [24], the authors have studied the Hamiltonian symmetries in evolutionary or canonical form.The symmetry properties of the Hamiltonian action have been considered in the space (, , ) in [30,31].They presented the Hamiltonian version of Noether's theorem considering the general form of the symmetries (107).
The following important results which are analogs of Noether symmetries and the Noether theorem (see [24,30,44,47] for a discussion) were established.

Theorem 4 (Hamilton action symmetries). A Hamiltonian action
is said to be invariant up to gauge (, , ) associated with a group generated by (107) if Theorem 5 (Hamiltonian version of Noether's theorem).The canonical Hamilton system (106) which is invariant has the first integral for some gauge function  = (, , ) if and only if the Hamiltonian action is invariant up to divergence with respect to the operator  given in (107) on the solutions to (106).

First Integrals of Harmonic Oscillator in Hamiltonian
Framework.Let us transfer the preceding example into the Hamiltonian framework and define The Hamiltonian function for this problem is The canonical Hamiltonian equations (106) for Hamiltonian function (113) result in The Hamiltonian operator determining equation (110), after expansion, yields in which we assume that  = (, ),  = (, ), and  = (, ).One can also assume these functions to be dependent on .We have chosen (, ) dependence to simplify the calculations here and this leads to at least one Hamiltonian Noether operator.Equation (115) with the help of (114) can be written as One can separate (116) with respect to powers of derivatives of q and finally arrive at the following Hamiltonian Noether operators and the gauge terms: It is worthy to notice here that no integration is required to derive solutions of (114).

Applications to Some Models from Real World
In this section we apply the considered techniques to some equations arising from concrete problems, namely, the jerk equation and free oscillations with two-degree-of-freedom gyroscopic system with quadratic nonlinearities.

Jerk Equation.
According to Gottlieb [48,49], the most general nonlinear jerk equation is where the prime denotes differentiation with respect to  and , , , ,  are constants.In (119), at least one of , ,  should be different from zero and if  = 0, then  ̸ = 2 so that the jerk equation is not a derivative of a second-order ODE [50].

First Integrals for Nonlinear Jerk Equation by Noether
Approach for a System and Its Adjoint.Let us look for first integrals for (119) using Noether approach for a system and its adjoint.The adjoint equation for (119) is The Lagrangian for system  1 = 0 and  * 1 = 0 is  = V 1 .The system  1 = 0 and  * 1 = 0 possesses a first integral where is any Lie point symmetry of (119).Now we use strictly selfadjointness for eliminating the nonlocal variable V in the first integral (121).
Using the self-adjoint condition we conclude that  = −1 and  =  =  = 0,  = 1,  = 1.Then we conclude that the strictly self-adjoint subclass of (119) is given by the family The only admitted Lie point symmetry generator of ( 124) is If we take  =  =  = 0,  = 1,  = 0, (119) admits not only (125), but also The Lie point symmetry generator (125) provides the trivial first integral  = 0.However, from (126) one can construct a nontrivial one.In fact, using (126), we obtain  = − −   .Substituting this expression for ,  = , and  = − into (121) and setting V = , after reckoning, we have (127) After expansion of (128), the multipliers and first integrals are computed for specific values of parameters.