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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.

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. [

Noether's theorem [

The characteristic, multiplier, or integrating factor methods are also very powerful and elegant methods for construction of the first integrals. There are four different approaches based on the knowledge of the characteristics. The first method developed by Steudel [

The well-known Noether identity can be expressed in terms of Hamiltonian function and symmetry operators (see, e.g., [

Lie approach as described, for example, by Ibragimov and Nucci [

The paper is organized in the following manner. The fundamental relations are defined in Section

The following definitions are taken from the literature (see, e.g., [

Consider a

The total derivative with respect to

The Lie-B

The Euler operator is given by

The characteristic form of Lie-B

The

Now we present various approaches to construct first integrals taken from the literature.

The direct method was first used by Laplace [

Kara and Mahomed [

The first integrals are computed by the joint conditions (

In 1918, Noether developed a new approach to construct first integrals [

In the Noether approach we need to construct Lagrangian

The partial Noether approach for construction of first integrals was introduced by Kara et al. [

The first integrals of the system (

We can also use the partial Noether approach for equations arising from the variational principal and have the Lagrangian.

Then the first integrals are given from the formula

The dependence on the nonlocal variable

According to Steudel [

The variational approach was developed by Olver [

In this approach, the multiplier determining equation is obtained by taking the variational derivative of (

Consider the system (

On the solutions

The linearized system to (

The adjoint of the linearized system (

Moreover, the operators

We finish with the following definition.

The system (

We compute the first integrals of simple harmonic oscillator by utilizing different approaches. Consider

Equation (

Equation (

The first-order prolongation of the Lie point symmetry generators of (

The first integrals are computed by the joint conditions (

The second important aspect of this approach is that we can associate a symmetry with a first integral. The relationship (

Equation (

The Noether symmetry determining (

The separation of (

The solution of system (

Formula (

Equation (

The partial Noether operators

System (

Formula (

The adjoint equation for (

Let

The Lagrangian for the system consisting of (

For (

For the variational approach with multiplier of form

The solution of (

For (

Applying the results of Section

Substituting

Noticing that

The Noether symmetries associated with the first integrals can be utilized to derive the exact solutions of ordinary differential equations [

If

Suppose

In (

Now we will compute the exact solutions of (

A similar procedure is adapted to get the following exact solution of (

Now we show how one can find the exact solution of (

Similarly, the Noether symmetry

Suppose

The Euler operator, for each

Let

In [

The following important results which are analogs of Noether symmetries and the Noether theorem (see [

A Hamiltonian action

The canonical Hamilton system (

Let us transfer the preceding example into the Hamiltonian framework and define

One can separate (

The first integrals from formula (

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.

According to Gottlieb [

Let us look for first integrals for (

The Lagrangian for system

Using the self-adjoint condition

The only admitted Lie point symmetry generator of (

The Lie point symmetry generator (

Now we will derive first integrals for nonlinear jerk equation by multipliers approach. Assume multipliers of form

Equation (

Equation (

Equation (

Equation (

Equation (

The multipliers and first integrals for this case are

We now utilize first integrals to compute the exact solution of the jerk equation (

We firstly consider the first integrals

Now we obtain an implicit solution to

From (

Consider the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities [

Thus (

The only first integral for the two-degree-of-freedom gyroscopic system by formula (

The first integrals for simple harmonic oscillator were constructed using different approaches. All different approaches to compute first integrals for ordinary differential equations and systems of ordinary differential equations were explained with the ample example. The systematic way to compute the first integrals is by Noether's approach but it depends upon the existence of a standard Lagrangian. The Noether symmetries and the corresponding first integrals were constructed for simple harmonic oscillator. In the absence of a standard Lagrangian one can use the partial Noether approach which works with or without Lagrangian and the framework for this approach is similar to the Noether approach. The direct method and its use with the symmetry condition were explained in detail. We commented on some other approaches: the characteristic method, the multiplier approach for arbitrary functions as well as on the solution space, and the direct construction formula approach based on the knowledge of characteristics or multipliers which work without regard to a standard Lagrangian. The multipliers or characteristics can be easily constructed taking the variational derivative of

Furthermore, some solutions of ordinary differential equations using first integrals with its associated Noether symmetries were obtained. The first integrals are the reduced form of the given differential equation. Some of these reduced forms can be solved directly whereas the other form can be used to further reduce the order of a differential equation. The harmonic oscillator yielded five first integrals. Three first integrals were used to compute the solutions directly. Two first integrals were written as the first-order equations and the Noether symmetries were used to find the invariants which completely transform the reduced equation to quadrature.

First integrals for nonlinear jerk equation were derived by using Ibragimov's and multipliers' approach. Then using these first integrals, some exact solutions of jerk equation for different cases were also established. The partial Noether approach is used to derive the first integrals of the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Igor Leite Freire would like to thank FAPESP, Project no. 2011/19089-6, for financial support.