Equivalence for Differential Equations

We shall study the equivalence problem for ordinary differential equations with respect to the affine transformations group.


Introduction
Two differential equations are called equivalent if one can be transformed into another by a certain change of variables. In particular this change of variables transforms solutions of one equation into solutions of another. In general, the problem of equivalence of differential equations consists in determining whether two equations are equivalent up to a given class of transformations. W. Kryńsky [12] and B. Dubrov [7] examine differential equations up to contact transformations. Sophus Lie was the first to use some approach to the problem of equivalence of differential equations (description of invariants, computation of symetry group).
In [9] Fels consider the problem of equivalence between two systems of secondorder differential equations under the pseudo-group of smooth invertible local point transformations ψ t, x j = s, y i = φ t, x j , ϕ i t, x j This notion of two system being equivalent defines an equivalence relation on the set of differential equations on the form (1.1). Fels was able to cast the question of equivalence between (1.1) and (1.2) into a question about the equivalence of exterior differential systems associated on jet space J 1 (R, R n ) , at wich point the Cartan's method [4,8,10,11,15,17] be may applied. The problem of equivalence for n = 1 was originally solved by Cartan in [4]. Chern in [6] investigated the two equivalence problems for systems under the restricted pseudo-groups of smooth invertible local transformations wich preserve the independent variable as given by ψ t, x j = t, ϕ i x j and ψ t, x j = t, ϕ i t, x j In the previous papers [1,2] the relationship between differential equations, Pfaffian systems and geometric structures are study. We have seen that every differential equation can be expressed as a Pfaffian system satisfying the structure equation and that the integration of a given equation is deeply related to the structure equation. We shall show it by means of interesting examples. My contribution here is the study of equivalence problem for the family of ordinary equations with respect to the affine transformation group A (2, R) (section 4). There exist 19 different types of first order ordinary differential equations which admit at least 1dimensional Lie groups in A (2, R). All the equations which belong to the above types can be integrated by quadrature.
In the paper, by the word differentiable we mean always differentiable of classe C ∞ .
Acknowledgments. Special thanks go to Witold Respondek and Yuri Sachkov for his encouragement and support.

Basic Definition, Examples
2.1. Pfaffian systems. In this subsection we will review some basic concepts and facts on Pfaffian systems theory [3]. Let M be an differentiable manifold. A F (M )-submodule Σ of Λ 1 (M ) is called a Pfaffian system of rank n on M if Σ is generated by n linearly independent Pfaffian forms θ 1 , . . . , θ n . A submanifold N of M is said an integral manifold of Σ if i * (θ) = 0 for all θ ∈ Σ, where i denotes the immersion N ֒→ M . A differentiable function is said a first integral of Σ if the exterior derivate df belongs to Σ. By the symbol Σ = θ 1 , . . . , θ n we mean that the Pfaffian system Σ is generated by linearly independent Pfaffian forms θ 1 . . . , θ n defined on M .
For each Pfaffian systems Σ on M we can construct the dual sytem, that is, the differentiable subbundle D = D(Σ) of the tangent bundle T (M ) on M such that the fiber dimension of D is equal to dim(M ) − n. Let Γ(D) be the sheaf of germs of local vector fields which belong to D and Γ(D) x , x ∈ M, the stalk of Γ(D) at x. For x ∈ M , we defined the subspaces Ch(D) x of T x (M ) by x the germ at x determined by X. We suppose that dim Ch(D) x is constant on M . Thus, we obtain the subbundle Ch(D) of T (M ) called Cauchy characteristic of D. The distrubution characteristic of D is the module spanned by all vector fields Y that belongs to D such that [Y, D] ⊆ D.
The dual system Ch(Σ) of Ch(D) is called the Cauchy characteristic system of Σ. The following theorem is due to E. Cartan.
By making use of the property 2, we can construct the Cauchy characteristic system Ch(Σ).
If we can find a solvable system of Σ, then m independents first integral of Ch(Σ) are given par quadrature.

Examples.
In the subsection we shall consider by means of examples [2] the relation between the differential equations, Pfaffian systems and structure equation.
a) Consider the Pfaffian system Σ = θ , θ = dz + pdx + p 2 dy, on R 4 = {(x, y, z, p)}. We have dθ = dp ∧ (dx + 2pdy) and w 1 = dp, w 2 = dx + 2pdy, w 3 = θ determine the Cauchy characteristic system of Σ. We can find by quadrature three independent first integrals as follows: We consider an absolute parallelism w 1 , w 2 , w 3 , w 4 , w 5 , w 6 on R 6 satisfying the equations Let x and y be two independent first integrales of the completely Pfaffian integrable system w 1 = w 2 = 0; the form w 3 is expressed as The functions x, y, z, p and q are independent first integrals of the completely integrale Pfaffian system w 1 = w 2 = w 3 = w 4 = w 5 = 0.Therefore w 4 and w 5 can be written by means of the exterior derivates dx, dy, dz, dp, dq and the formulas determine the fonctions r, s, s ′ , t and a ′ i s of the variables x, y, z, p, q and another u. From the equation , one can verify that the function s coincides with s ′ . Moreover, the equations dw 4 = 0, Therefore the functions r = r(x, y, z, p, q, u) , s = s(x, y, z, p, q, u) , t = t(x, y, z, p, q, u) determine a system of second-order partial differential equations. This family of systems of differential equations determined by an absolute parallelism satisfying (2.1) is the main subject of Cartan's researches in his paper [4,5]. For example, take the system of differential equations . The absolute parallelism satisfies the equations (2.1). It easy to see that the system w 2 , w 3 , w 4 , w 5 , w 6 forms a solvable system of Σ = w 3 , w 4 , w 5 . Five independent first integrals of the solvable system are given by quadrature as follows: By the expression, the general integral surface of (2.2) is given by the formulas: where f is a differentiable function and f ′ , f ′′ and f ′′′ denote its derivates.

Equivalence
In the paper [2] , we have seen that every differential equation can be expressed as a Pfaffian system satisfying the structure equation and the integration of a given equation is deeply related to the structure equation. In this section, we shall consider the equivalence problem for Pfaffian systems and hence for differential equations under the action of Lie groups.
Let M be a differentiable manifold, G be a Lie group acting on M on the left. For a Pfaffian system Σ on M we set where L g denotes the left action of g ∈ G on M ; g * Σ is a Pfaffian system on M and G(Σ) is a subgroup of G .
Two Pfaffian systems Σ 1 and Σ 2 on M are equivalent under the action of G if there is an element g ∈ G such that g * Σ 1 = Σ 2 .
Let F be a family of Pfaffian systems on M . The problems (Lie programme) to be solved are as follows: 1) Determine the condition for the equivalence of the elements of F .
2) Classify the Pfaffian systems in F under the action of G.
3) For each Σ ∈ F , determine the structure of the subgroup G(Σ).

4)
Research the relation between the integration of a Pfaffian system Σ and the structure of the subgroup G(Σ), i.e., reduce the integration of a Pfaffian system Σ to auxiliary systems obtained via the knowledge of the structure of G(Σ).
In section 4 we consider the problem 2) in the particular case where G = A(2, R).
For a Lie subgroup G ′ of G, we set For every Σ ∈ F (G ′ ), G ′ is the largest subgroup of G which leaves Σ invariant. It is possible that F (G ′ ) is an empty set.

The normalizer
and g * Σ 2 = Σ 1 for an element g ∈ G, then Σ 1 and Σ 2 lie in the same orbit determined by the action of N (G ′ : G) on F (G ′ ).
Proof. 1) and 2) may be clear. To prove 3), suppose that Σ ∈ F (G ′ ) and g ∈ N (G ′ : G). From (3.1) and 1) of the proposition, we have G(g * Σ) = g −1 G(Σ)g = g −1 G ′ g = G ′ and hence g * Σ ∈ F (G ′ ). 4) From (3.1) and 1) of proposition, we have By virtue of this proposition, the equivalence probleme and the classification are reduced to the following problems: (i) determine all conjugate classes of the subgroups of G.
(ii) For a representative G ′ of each conjugate class, determine the set F (G ′ ).
(iii) Observe the action of N (G ′ : G) on F (G ′ ).
Since there are, in general, many subgroups G ′ of G such that F (G ′ ) is empty set, this reduction of the problems is not always the best one. Moreover, the Pfaffian systems to be considered are not always defined globally on M . Therefore, instead of ordinary Lie groups, we have to consider Lie pseudogroups [4,13,14,16]. Then the subject of the study is invariants of a Pfaffian system with respect to a given Lie pseudogroup. At the rate, we can recognize that the subgroup G(Σ) plays an important role in the problems.

Equivalence with respect to the A(2,R)
Let G be a finite dimensional Lie group and let Σ be a left-invariant completely integrable Pfaffian system on G. We denote by I g (Σ) the maximal integral manifold through g ∈ G and we set G g (Σ) = {h ∈ G; L h (I g (Σ)) = I g (Σ)} , Since Σ is left-invariant, G g (Σ), g ∈ G, are mutually conjugate in G. A(2, R) be the affine transformation group on R 2 . By making use of the matrix representaion

Invariant forms of A(2,R). Let
we have a basis of invariant forms of A(2, R) where we put D = x 3 x 6 − x 4 x 5 . We have then the structure equation We remark that changing the basis of invariant forms by the formula where a ′ , b ′ and c ′ denote arbitrary constants with a ′ c ′ = 0, the structure equation (4.1) does not alter.

Classification under action of A(2,R)
. The systems to be considered are given by Σ = dx 2 − f (x 1 , x 2 )dx 1 where f denotes a differentiable function. In this paper I given some ideas for the classification. Since forms a characteristic invariant system. By using this invariant, we reduce the Cartan system to be submanifold M 0 defined by the equation The equation to be integrated is now given by w 2 = 0. On the submanilfold M 0 , we have where we put w 6 = w 6 − bw 1 and a, b, u 1 , u 2 , u 3 denote definite functions on M 0 . These functions are all invariants of induced Cartan system. 4.2.1 From now on, we shall determine all the equations wich admit at least 2-dimensional Lie subgroup of A(2, R) as an invariant group. Therefore we suppose always that the forms w 1 , w 2 are linearly independent.
I. The case a = 0. From (4.3) , we have u 1 = u 2 = 0. Moreover where c denotes a definite function on M .
1. b = 0. The manifold M 0 is given by the maximal integral manifold of w 5 = 0. Hence we obtain the first type: Integrating the system Σ, we have the result: 2.1. u 3 = 0. From the second equation of (4.6) we have c = 0. We have thus obtained the second type: Integrating the system Σ, we obtain the result: where r denotes a definite function on M 2 . The equations (4.7) does not determines a 3-dimensional Lie group. By using the invariant e, we can reduce M 2. Taking a ′ = 1, b ′ = 1, c ′ = (2−e) 3 in (4.2), we can reduce M 2 to the manifold M 3 = {g ∈ M 2 ; e(g) = 2} . From (4.7) we obtain on M 3 w 5 = w 2 , w 3 = w 1 − w 2 , w 6 = w 1 − 2w 2 , w 4 = 1 3 rw 2 , dr = 2rw 1 + r 0 w 2 (4.8) and Although the equations (4.9) do not contain any functions, the function r is an invariant of the group. Therefore (4.8) and (4.9) determine a 2-dimensional Lie group if and only if r is a constant on M 3 . In the case, we have r = 0 and Σ : w 3 = w 1 − w 2 , w 4 = 0, w 5 = w 2 , w 6 = 2w 1 − 2w 2 .
Integrating the system Σ, we obtain the result: Then y ′ = f (x, y) admits a 2-dimensional Lie group in A(2, R) and can be transformed to y ′ = x − (x 2 − 2y) 1 2 by an element A(2, R). II. The case a = 0. We go back to the manifold M 0 . Suppose that b = 0 on M 0 . Consider the submanifold N 0 defined by the equation b = const ( = 0). By setting we can assume that a = 1, b = 0 on M 0 . Therefore we have only to examine the case a = 0, b = 0 on M 0 . We can reduce M 0 to the submanifold on which we have By differentiating (4.9) we obtain 1. u 1 = u 2 = u 3 = 0. From (4.10), we have Σ : Integrating the system Σ, we obtain the result: If these equations determine a 2-dimensional Lie group, v 1 must be a constant. In this case, integrating the system Σ, we obtain the result: 3.1. The case u 2 0. We can assume always u 2 = 1. Integrating the system Σ, we obtain the result: 3.2. The case u 2 0. We can assume u 2 = −1. Integrating the system Σ we obtain the result: 4. u 1 = u 2 = 0, u 3 = 0. From (4.11) we have v i = 0, i = 1, 2, 3, 4. Since v 2 − v 3 + 3u 3 = 0, this contradicts the assumption u 3 = 0.
If 2u 2 2 = 3u 1 , then u 1 = 6, u 2 = 3, u 3 = 2, v 1 = 2 and v 2 = 1. We have Substituting this relation to the last equation of (4.10) we have a certain algebraic equation with respect to the quantities u 1 , u 2 . Hence we obtain Σ : Intregating the system Σ, we obtain the result: Theorem 4.11. The equations of this type are transformed by an element of A(2, R) to y ′ = x −1 y + x a (a const = 4, 1, 0, −1), which admits a 2-dimensional Lie group in A(2, R).

4.2.2.
As for the determination of the equations admitting a 1-dimensional Lie group in A(2, R), we can use the method developed in section 3. Here is the table of the standard forms and the invariant groups. We denote by a the parameter of a 1-dimensional Lie group.

Standard Forms
Invariant Groups 1-dimensional conformal transformation group