## Abstract

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

## 1. Introduction

Integrating methods for differential equations are important in applied mathematics and mechanics. Lie symmetry analysis provides well-known instruments for integrating ordinary differential equations (ODEs) by using their symmetry properties (e.g. [1–6]). In particular, if an *n*-order ODE admits a sufficiently large symmetry group, one can integrate it by order reduction or using canonical variables. A classical reduction method is implied using the invariant differentiation operator and is applied to equations admitting solvable Lie algebras. The method of canonical variables is based on the classification of non-similar Lie algebras of operators and is applied to integrating second-order [1], third-order [7] and fourth-order [8,9] scalar ODEs.

Many works are related to the generalization of group analysis methods for integration of ODE systems of the form
*m* equations of order *n*.

For systems (1.1) consisting of first-order ODEs, as well as for scalar first-order ODEs, there are no constructive algorithms for constructing the symmetries of systems. And in this case Lie algebras of generators are used for investigating the nonlinear superposition [4,10–12].

If the order of equations of (1.1) is more than 1, then the symmetries of (1.1) are calculated by using constructive methods (e.g. [1–6,13,14]). Moreover, for system (1.1) with given functions *F*_{i} one can use any package of systems for computer algebra, by which one can find symmetries and invariants of systems. For example, the package GeM for Maple [15], the package SYM for Mathematica [16] and the package SYMMETRY for Reduce [17].

In [18], it is proved that if system (1.1) has *nm*+1 symmetries, then one can present the general integral of this equation by algebraic operations. This fact is the generalization of the classical results of integrating methods for scalar second-order ODEs with three symmetries [1,4].

However, for systems (1.1), which admit *nm* symmetries or less, the integration problem is not completely solved. On this subject systems of second-order ODEs are the most studied. In particular, group classification of systems of two second-order ODEs admitting three- and four-dimensional Lie algebras is provided in [19,20]. Group classification of systems of three second-order ODEs is provided in [21]. Lie symmetries of systems of second-order linear ODEs are investigated in [22–25].

The methods for reducing the order of systems of two second-order ODEs admitting three-dimensional solvable Lie algebras are described in [26]. In [27], the algorithm for the reduction of order for systems of two second-order ODEs admitting solvable four-dimensional Lie algebras of symmetries is suggested.

However, one can see that all integration methods for ODE systems are generalizations of the classical reduction method for scalar ODEs and are applied only for systems admitting solvable Lie algebras.

In this paper, we suggest a universal algorithm for constructing the first integrals of ODE systems. We show that any system of two second-order ODEs with four symmetries can be integrated by quadratures. The suggested algorithm can be extended to systems of higher-order ODEs, as well as to systems of more than two equations.

## 2. Main result

We consider systems of the form

Let us consider any four-dimensional Lie algebra *L*_{4} with generators (2.2). To construct system (2.1) admitting such an algebra, one can use the invariant representation of the system, i.e. the representation of equations (2.1) by invariants and differential invariants of the admitted transformation group (e.g. [1–4]). To obtain this representation, we construct invariants of Lie algebra *L*_{4} up to second order.

Let us introduce matrix *Λ*_{0} consisting of coordinates of infinitesimal generators (2.2) as well as matrices *Λ*_{1} and *Λ*_{2} consisting of coordinates of prolongations of (2.2) to the first and second derivatives, respectively:
*D*_{t} is the operator of total differentiation.

As is well known (e.g. [1–4,28]), any Lie algebra of operators in *R*^{n} has *k*=*n*−*r* invariants, where *n* is a number of variables and *r* is the rank of matrix, containing the coordinates of the operators. The *k*-order invariants are obtained from the system of linear equations
*k* is the order of prolongation and *X*_{i,k} is the prolongation of the operator *X*_{i} up to the *k*th derivatives.

The methods and examples of solving systems in the form (2.3) are well known; for details, see [29]. Also these methods are realized in modern computer systems such as Maple and Mathematica.

Table 1 contains admissible ranks of matrices *Λ*_{0}, *Λ*_{1} and *Λ*_{2} for arbitrary four-dimensional Lie algebras *L*_{4} of differential generators in *R*^{3} and forms of corresponding invariant systems (2.1).

One can see that, in cases 1, 4 and 7, algebras *L*_{4} have no realizations in *R*^{3}. In cases 3 and 6, algebras *L*_{4} have two second-order differential invariants and invariant systems are obtained by the theorem on invariant representation (e.g. [2]). According to this theorem, the rank of the coefficient matrix *Λ*_{2} on the solutions of (2.1) does not differ from its common rank. The obtained systems are called *regular*. Such systems are considered in [19,20]. In cases 2, 5 and 8, algebras *L*_{4} have only one second-order differential invariant. This invariant leads to one equation in (2.1). Another one in (2.1) can be obtained as a condition, when the rank of the coefficient matrix *Λ*_{2} is reduced. So obtained systems are called *singular*. In [26], singular systems admitting three-dimensional Lie algebras are considered.

Thus, one can split the set of systems (2.1) admitting four-dimensional Lie algebras into three classes. The first one consists of regular systems admitting *L*_{4} with rank(*Λ*_{0})=3. The second class contains regular systems admitting *L*_{4} with rank(*Λ*_{0})=2. Integration of these systems is based on their invariant representations and uses the operator of invariant differentiation. The last class consists of singular systems. It is approved that, if a system is singular, it can be reduced to a partially uncoupled form (in terms of [26]). Moreover, all systems with four symmetries can be integrated by quadratures. Both of these statements are checked for systems (2.1) admitting *canonical forms* of Lie algebras of operators (2.2) obtained in the classification [30]. Furthermore, such systems are also named *canonical*. For Lie algebras in canonical forms we construct 60 canonical systems of the first class, 14 systems of the second class and 18 systems of the third class. According to the group classification method any system (2.1) with four-point symmetries can be reduced to a canonical form and then integrated.

### (a) Integration of regular invariant systems

In the theory of differential invariants, the concept of the invariant differentiation operators is introduced. The actions of these operators to *k*-order invariants give (*k*+1)-order invariants (e.g. [1]). In the case of one independent variable and two dependent variables, the operator has the form λ(*t*,*x*,*y*,*x*′,*y*′,…)*D*_{t}, where *D*_{t} is the total differentiation operator.

In [2], it was shown that the operator of invariant differentiation λ*D*_{t} can be obtained from the condition of commutation
*D*_{t} with all infinitely prolonged generators *L*_{r}.

Let us calculate the commutator *r* is the dimension of the Lie algebra.

We use the invariant differentiation operator for the integration of systems (2.1). First, we investigate case 3 from table 1, where
*F* and *G*.

By the definition of the invariant differentiation operator, we obtain the representation
*Θ*. So, if the four-dimensional Lie algebras satisfy case 3 and is admitted by system (2.6), then
*Φ*(*I*_{1})=*Θ*(*I*_{1},*F*(*I*_{1}),*G*(*I*_{1})).

Equation (2.7) can be rewritten in the form
*L*_{4} of generators (2.2) satisfying case 3 and for the corresponding canonical systems, the right-hand side of (2.8) is integrated by quadratures. Mostly, one can choose the variables such that λ=1 and system (2.6) is easily integrated (see example 2.2). In other cases λ depends on derivatives, but (2.8) is still integrated by quadratures (see example 2.3). Now, using the solution of (2.8), one can integrate by quadratures all canonical systems arising in case 3.

An arbitrary change of variables keeps the ranks of matrices *Λ*_{0}, *Λ*_{1} and *Λ*_{2} and also keeps the integrability of the right-hand side of (2.8) (e.g. [31]), i.e. the following theorem is valid.

### Theorem 2.1.

*Let system* (*2.1*) *admit a four-dimensional Lie algebra with generators* (*2.2*). *If the operators satisfy the conditions rank*(*Λ*_{0})=3, *rank*(*Λ*_{1})= 4, *rank*(*Λ*_{2})=4, *then system* (*2.1*) *has a first integral. It is found from* (*2.8*). *Moreover, system* (*2.1*) *is integrated by quadratures*.

### Example 2.2.^{1}

Let system (2.1) admit Lie algebra *g*_{3,2}⊕*g*_{1} (see the classification provided in [30]) with generators
*Λ*_{0}, *Λ*_{1} and *Λ*_{2} are as follows:
*I*=*I*(*t*,*x*,*y*,*x*′,*y*′,*x*′′,*y*′′). And we obtain the next invariants
*t*,*x*,*y*,*x*′,*y*′,*x*′′,*y*′′). Here, we can choose λ=1, then equation (2.7) takes the form
*I*_{1}=*φ*(*t*+*C*_{1}) be the integral of this equation. Then system (2.9) can be rewritten in the form

### Example 2.3

Let system (2.1) admit Lie algebra *g*_{4,5} from [30] with generators
*y*′/*y*′′ and equation (2.7) is
*I*_{1}=*φ*(*C*_{1}*y*′) be the integral of this equation. Then system (2.6) can be rewritten in the form

and then, after simple algebraic transformations, in partially uncoupled form

Now let us consider the class of canonical systems, which are invariant with respect to Lie algebras satisfying case 6 from table 1. These systems have the form
*u*(*t*,*x*,*y*,*x*′,*y*′) such that
*Φ*.

On solutions of (2.10) this condition yields
*Ψ*(*t*)=*Φ*(*t*,*F*(*t*),*G*(*t*)). Hence, the first integral of system (2.10) has the form

An arbitrary change of variables keeps the integrability conditions of (2.10) (e.g. [31]), i.e. the following theorem is valid.

### Theorem 2.4.

*Let system* (*2.1*) *admit a four-dimensional Lie algebra of generators* (*2.2*) *and the generators satisfy the conditions rank*(*Λ*_{0})=2, *rank*(*Λ*_{1})= 4, *rank*(*Λ*_{2})=4. *Then the first integral of system* (*2.1*) *exists. Moreover, this system is integrable.*

### Example 2.5

Let system (2.1) admit Lie algebra *g*_{4,5} (see the classification result in [30]) with generators
*u* we can obtain the general solution of the system.

### Remark 2.6

If a system of form (2.1) admits a three-dimensional Lie symmetry algebra, then the operator of invariant differentiation can be used to reduce the order of the considered system. For example, if generators of Lie algebra *L*_{3} satisfy conditions
*F* and *G*.

### (b) Singular systems and their integrability

In cases 2, 5 and 8 from table 1, Lie algebras of operators have only one second-order differential invariant. Hence, invariant systems cannot be constructed by the theorem on invariant representation. However, the systems can be obtained as singular solutions of equations (2.3) on invariants. In other words, equations of the invariant systems may follow from the reduction conditions on the rank of matrices *Λ*_{2} [32]. Constructing singular systems for canonical forms of Lie algebras of generators [30] satisfying cases 2, 5 and 8, one can show that, for cases 2 and 8, the invariant systems reduce to the system

Singular invariant systems appear only for Lie algebras of operators satisfying the conditions of case 5. All such systems admitting canonical forms of Lie algebras are partially uncoupled and integrable. Hence, the following theorem is valid.

### Theorem 2.7.

*Let system* (*2.1*) *admit a four-dimensional Lie algebra of generators* (*2.2*) *and the algebra have only one independent second-order differential invariant. Then always one can find the change of variables such that system* (*2.1*) *is partially uncoupled. Moreover, the system is integrable.*

### Example 2.8

Let system (2.1) admit Lie algebra *g*_{2}⊕2*g*_{1} from [30] with generators
*y*′′=0 the rank of matrix *Λ*_{2} is reduced to three. So, the system admitting the given Lie algebra is partially uncoupled

### Remark 2.9

In this section, we used the canonical forms of four-dimensional Lie algebras. But the suggested algorithm can also be applied to systems which admit non-canonical Lie symmetries. We demonstrate this in the next section.

## 3. Applications

Here, we present a few physically important examples of systems (2.1) admitting four-dimensional Lie algebras and solve them by using the suggested algorithms.

### Example 3.1

The two-body problem [33] in terms of polar coordinates is

The first integral of the system is *C*_{1}=*r*^{2}*θ*′. Then system (3.1) takes the partially uncoupled form

### Example 3.2

The motion of a body in a resistant medium is described by the system [34]
*g* is a free-fall acceleration and *F*(*v*) is a resistance function.

If *F*(*v*)=*σ*=const., the system admits the Lie algebra *g*_{4,5} with generators
*y*′ from the second equation, we obtain

### Example 3.3

Consider the simple Newtonian system with velocity-dependent forces (another approach to solving this problem is provided by [27])

### Example 3.4

This example illustrates the application of the invariant differentiation operator to investigation of systems (2.1) admitting three-dimensional Lie algebras of generators. Consider the classical Kepler problem in polar coordinates (another approach to solving this problem is provided by [20])
*r*/*r*′ and obtain a new system on invariants

i.e. the system is reduced to the form

## 4. Conclusion

We have suggested algorithms for the integration of the systems of two second-order ODEs admitting four-dimensional Lie symmetry algebras by using the invariant differentiation operator. It is easy to see that these algorithms can be generalized for integration of the systems of *k* *m*-order ODEs, *km*≥4, as well as such systems with a small parameter, which is invariant with respect to approximate symmetries.

For example, let us consider the system
*rank*(*Λ*_{0})=4, *rank*(*Λ*_{1})=6 and *rank*(*Λ*_{2})=6. Then the algebra has one first-order and three second-order differential invariants. Consequently, by using the invariant form of (4.1) and the operator of invariant differentiation, we can obtain an equation for the first integral of the system. More detailed research in this direction will be carried out in future works.

## Authors' contributions

A.A.G. and R.K.G. conceived the theoretical base, found examples for application, interpreted the computational results and wrote the paper. A.A.G. performed most of the calculations in consultation with R.K.G. Both authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

The authors acknowledge the financial support of the Government of the Russian Federation through Resolution no. 220, agreement no. 11.G34.31.0042.

## Acknowledgements

We thank Prof. N.H. Ibragimov and Prof. S.V. Khabirov for useful discussions. We also thank the referees for helpful remarks which have improved this paper.

## Footnotes

↵1 In the examples in this section, we construct the systems by their admitted algebras.

- Received June 9, 2016.
- Accepted December 5, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.