The characterization of third order ordinary differential equations admitting fiber-preserving point symmetries

Guy Grebot^{1}

Departamento de Matemática Universidade de Brasília Brasil.

Email:guy@mat.unb.br

Abstract

Third order ordinary differential equations are characterized with respect to the dimension of the symmetry group of fiber preserving point transformations they admit. For the symmetry groups of dimension greater or equal to 4, we obtain parametric families of equations admitting those groups. The equivalence classes of linear equations are also explicitly given. The results presented are based on the solution of the local equivalence problem for third order ordinary differential equations under fiber preserving point transformations.

1 Introduction

The local equivalence problem for third order ordinary differential equations (ODE’s), consists in determining the conditions for a coordinate transformation, taking one third order ODE into another one, to exist. In [4], Élie Cartan’s method of equivalence was used to solve this problem under the group G of point transformations of the form X = X(x), Y = Y (x, y), where y satisfies

y′′′ = F (x, y, y′, y′′),

the ‘′’denoting total differentiation with respect to x.

The method of equivalence directly gives the different equivalence classes which are characterized by a set of functions, invariant under a subgroup of a given prolongation of G.

For self-equivalence, the group formed by the solutions of the problem is the symmetry group of the equation under study. Its dimension is related to the number of invariants determining the class of the considered equation (see [8]).

The aim of this work is to give a characterization of third order ODE’s admitting a symmetry group of fiber preserving point transformations. This characterization is based on the possible values assumed by the invariants returned by the application of the equivalence method. The complexity of these expressions forbids us to give, as for second order ODE’s [7], the complete list of possible symmetry groups of third order ODE’s. But when the invariants are assumed to be constant such a description is accessible, and we obtain a characterization of the third order ODE’s admitting symmetry groups of dimension four to seven, as well as a characterization of those groups.

1Supported by CNPq, Brazil.

The reason for not obtaining a characterization of the equations admitting lower dimensional symmetry groups is due to the high number of invariants. In fact, by the method of equivalence, a maximal functionally independent set must be selected among those invariants and their derivatives. This is always possible for particular cases with the help of the symbolic program clasode3 [5]. But in the general case it is not possible: many possibilities appear as one must select at most three functionally independent invariants among, say, nine.

In the next section we briefly resume the solution of the equivalence problem for third order ODE’s under fiber preserving point transformations. The linear equations are presented in Section 3. In the other sections we give the characterization of the equations admitting symmetry groups of dimension 6 and 4.

In what follows, we have used the nomenclature of [4]; the basic invariants, obtained by the application of the equivalence method, are named I_{i} where i ≤ 10.

2 The solution of the equivalence problem

We briefly present the solution, obtained in [4], to the problem of finding necessary and sufficient conditions for the existence of a transformation

\begin{aligned}X & = & ϕ(x) \\ Y & = & ψ(x, y) \end{aligned}  (1)

taking one third order ODE

y′′′ = F (x, y, y′, y′′),  (2)

into another one

Y′′′ = G(X, Y, Y′, Y′′).  (3)

Such a problem is called an equivalence problem, and the method to solve it is E. Cartan’s method of equivalence (see [3]). We start with a coframe {ω}

\underline{ω^{0}} = dx
\underline{ω^{1}} = dy − p dx
\underline{ω^{2}} = dp − q dx
\underline{ω^{3}} = dq − F dx

defined on the second order jet bundle J^{2}(R, R), with coordinates (x, y, p, q), and a similar coframe \underline{Ω} determining the solutions of (3). The problem to be solved is to find the transformations Φ : J^{2}(R, R) → J^{2}(R, R), which are given by the second prolongation of (1) such that

Φ^{∗}\underline{Ω}^{i} = Γ \underline{ω}^{i},  (4)

where Γ are defined on J^{2}(R, R) with values in the group G of matrices of the form

Γ = ( \begin{matrix}A & 0 & 0 & 0 \\ 0 & B & 0 & 0 \\ 0 & C & B/A & 0 \\ 0 & D & E & B/A^{2} \end{matrix} ) .  (5)

The functions A, B, C, D and E are given by

A = ϕ_{,x}
B = ψ_{,y}
C = (ψ_{,xy} + p ψ_{,yy})/ϕ_{,x}
D = ϕ_{,x}^{−3} (q ϕ_{,x} ψ_{,yy} + p (2 ψ_{,xyy} ϕ_{,x} − ψ_{,yy} ϕ_{,xx})
+ p^{2} ψ_{,yyy} ϕ_{,x} + ψ_{,xxy} ϕ_{,x} − ϕ_{,xx} ψ_{,xy})
E = ϕ_{,x}^{−3} (2 ψ_{,xy} ϕ_{,x} − ψ_{,y} ϕ_{,xx} + 2 p ψ_{,yy} ϕ_{,x}).

The idea of the method is to reformulate the problem such that it becomes

Φ̃^{∗}Σ^{i} = σ^{i}.  (6)

The problem is completely solved in this form (see [8]). The solution is given in terms of a set F of functions defined on J^{2}(R, R). Among the elements of F, a maximal set of functionally independent functions must be selected. The dependence of the elements of F with respect to these independent functions will characterize the equivalence class of a given equation.

For self-equivalence, the solutions of the problem form the group of symmetries of the equation. The dimension r of this group is related to the number p of functionally independent invariants in F and the dimension n of the space on which the coframe {σ} is defined. We have

r = n − p  (7)

(see [3] [8] [12]).

For the solution of the equivalence problem of third order ODE’s, the algorithm of the method splits depending on whether A: F_{,qq} = 0 or B: F_{,qq} ≠ 0.

In the first case, F_{,qq} = 0, we are led to consider two more cases A-a: T D_{0} = 0 and A-b: T D_{0} ≠ 0, where

T D_{0} = (18 F_{,p} F_{,q} + 4 F_{,q}^{3} − 18 F_{,q} \frac{d}{dx}F_{,q} + 54 F_{,y} − 27 \frac{d}{dx}F_{,p}
+ 9 \frac{d^{2}}{dx^{2}} F_{,q})/(54 A^{3}).  (8)

If T D_{0} = 0 then we obtain two more cases. These cases lead to the class of the trivial equation and to equivalence classes of equations admitting symmetry groups of dimension 3, 1 or 0. If T D_{0} ≠ 0, then we must consider the cases

A-ba) T A_{1}, T A_{2}, T A_{3} not null,

A-bb) T A_{1} = 0 and T A_{2}, T A_{3} not null,

A-bc) T A_{2} = 0 and T A_{1}, T A_{3} not null,

A-bd) T A_{3} = 0 and T A_{1}, T A_{2} not null,

A-be) T A_{1} = T A_{2} = 0 and T A_{3} not null,

A-bf) T A_{1} = T A_{3} = 0 and T A_{2} not null, A-bg) T A_{2} = T A_{3} = 0 and T A_{1} not null,

A-bh) T A_{1}, T A_{2}, T A_{3} all null,

where

T A_{1} = (−6 A_{,p} F_{,q} + A_{,q} (9 F_{,p} + F_{,q}^{2} − 3 \frac{d}{dx}F_{,q}) + 18 A_{,y})/(18 B A)
+ (−18 B A_{,p} A^{2} E + 9 A_{,q} A^{4} E^{2})/(18 B^{3} A)  (9)
T A_{2} = (B A_{,p} − A_{,q} A^{2} E)/B^{2}  (10)
T A_{3} = A_{,q} A/B,  (11)

and

A^{3} = (18 F_{,p} F_{,q} + 4 F_{,q}^{3} − 18 F_{,q} \frac{d}{dx}F_{,q} + 54 F_{,y} − 27 \frac{d}{dx}F_{,p} + 9 \frac{d^{2}}{dx^{2}} F_{,q})/54.

(12)

All these cases, apart from A-be and A-bf, furnish a set of invariants that are used to determine classes of equivalence. The case A-bh leads to the classes of the non trivial linear equations, and to other classes of equations admitting a 2-dimensional symmetry group. Apart from this case, the case A-ba is the only one leading to possible classes of equations admitting a 4-dimensional group of symmetries. For that case, the possible 4-dimensional symmetry groups would appear by letting the invariants constant. In doing so, we observe that these constant do not satisfy the Jacobi identities, and define no Lie algebra.

In case A, only the linear equations admit a maximal symmetry group of fiber preserving point transformations, of dimension 4,5 or 7.

In case B: F_{,qq} ≠ 0, we have to consider the following cases

B-a) T B_{3} = T B_{0} = T D_{0} = 0,

B-b) T B_{3} = T B_{0} = 0, T D_{0} non null,

B-c) T B_{3} = 0, T B_{0} non null, T B_{2} = −1,

B-d) T B_{3} = 0, T B_{0} non null, T B_{2} ≠ −1,

B-e) T B_{3} non null,

where

T B_{0} = (F_{,qq} F_{,q} + 3 \frac{d}{dx} F_{,qq})/(3 A F_{,qq})  (13)
T B_{2} = (−9 A E F_{,qqq} + 3 F_{,pqq} F_{,qq})/F_{,qq}^{3}  (14)
T B_{3} = (3 A F_{,qqq})/F_{,qq}^{2}  (15)
T D_{0} = (18 F_{,p} F_{,q} + 4 F_{,q}^{3} − 18 F_{,q} \frac{d}{dx} F_{,q} + 54 F_{,y} + 9 \frac{d^{2}}{dx^{2}} F_{,q}
− 27 \frac{d}{dx} F_{,p})/(54 A^{3}),  (16)

A and E being group parameters. Each of these cases will furnish a set of invariants which must be used to characterize the equivalence class of a given equation. Only case B-e will allow equivalence classes of equations admitting a 4-dimensional symmetry group. The case B-a leads to the equivalence class of the equations admitting a 6-dimensional symmetry group.

3 The Linear equations

In this section we give the charactrerization of the linear equations which admit fiber preserving symmetry groups of dimensions 4, 5 and 7.

Replacing F in the expressions of the preceding section, by the expression of a canonical linear equation (see [6])

F = −2 M(x) y′ − (N(x) + M′(x))y  (17)

we obtain the conditions determining the equivalence classes of the linear equations. There are two possibilities for the so-called (see [2]) pseudo invariant T D_{0} which assumes the form

T D_{0} = −\frac{N}{A^{3}} .  (18)

¿From this expression we directly see that the linear equations that can be equivalent to the trivial equation must satisfy N = 0. This is exactly the condition found in the example 1 of [1].

The equivalence classes of the linear equations are then determined by the value of the so called Laguerre invariant N.

The conditions characterizing the class of the trivial equation are given by T D_{0} = 0 and we obtain that a third order ordinary differential equation

y′′′ = F (x, y, y′, y′′)  (19)

can be transformed into the trivial equation

y′′′ = 0,  (20)

by a transformation X = ϕ(x), Y = ψ(x, y), if, and only if,

\{ \begin{matrix}F & = & p^{3} (1/3 C_{0,y} − (C_{0}/3)^{2}) + p^{2} (C_{0,x} − C_{0}/3 C_{1}) \\ & + & p C_{2} + p q C_{0} + q C_{1} + C_{3} \\ 0 & = & \frac{∂}{∂ y}C_{3} − C_{3} C_{0}/3 + T X/54 \\ T X & = & 9 ∂_{x}(C_{1,x} − C_{1}^{2} − 3 C_{2}) + C_{1} (18 C_{2} + 4 C_{1}^{2}), \end{matrix}  (21)

where C_{i} = C_{i}(x, y).

The equations belonging to this class admit the maximal 7-dimensional group of symmetries.

If T D_{0} ≠ 0, then we obtain two other classes of linear equations which are characterized by the invariant

I_{1} = (A^{2} (−3 F_{6,xx} + F_{6,x}^{2} + 3 F_{1}) + 6 A A_{,xx} − 9 A_{,x}^{2})/(6 A^{4}),  (22)

with F_{i} = F_{i}(x, y) for i = 0, 1, 6 and the expression for A is given in (12). The Maurer-Cartan equations for the symmetry group are given by

\{ \begin{matrix}dω^{0} & = & 0 \\ dω^{1} & = & β^{4} ∧ ω^{1} + ω^{0} ∧ ω^{2} \\ dω^{2} & = & β^{4} ∧ ω^{2} + ω^{0} ∧ ω^{1} I_{1} + ω^{0} ∧ ω^{3} \\ dω^{3} & = & β^{4} ∧ ω^{3} + ω^{0} ∧ ω^{1} + ω^{0} ∧ ω^{2} I_{1} \\ dβ^{4} & = & 0. \end{matrix}  (23)

As I_{1} depends only on x, there can be at most only one functionally independent invariant, meaning that we can either have a 5-dimensional or a 4-dimensional symmetry group.

Result 1 There is a transformation X = ϕ(x), Y = ψ(x, y), taking a third order ordinary differential equation

y′′′ = F (x, y, y′, y′′)  (24)

into the linear equation

Y′′′ + 2 M Y′ + N Y = 0,  (25)

where M and N are constant, if, and only if,

F = q (p F_{6,y} + F_{6,x}) + p^{3} (3 F_{6,yy} − F_{6,y}^{2})/9
+ p^{2} (3 F_{6,xy} − F_{6,x} F_{6,y})/3 + p F_{1} + F_{0},  (26)

where F_{i} = F_{i}(x, y), and

F_{6,xxy} = (3 F_{1,y} + 2 F_{6,xy} F_{4})/3  (27)
0 = 27 F_{0,yy} − 9 F_{0,y} F_{6,y} − 9 F_{1,xy} + 3 F_{1,y} F_{6,x}
− 6 F_{6,xx} F_{6,xy} + 2 F_{6,xy} F_{6,x}^{2} + 9 F_{6,xy} F_{1} − 9 F_{6,yy} F_{0}  (28)
M/N^{2/3} = (A^{2} (−3 F_{6,xx} + F_{6,x}^{2} + 3 F_{1}) + 6 A A_{,xx} − 9 A_{,x}^{2})/(6 A^{4}).  (29)

We see that the linear equations with constant coefficients are parametrized by the constant M/N^{2/3}, which is the value of the invariant I_{1} of case A-bha. These equations admit a 5-dimensional symmetry group of fiber preserving point transformations.

The characterization of the linear equations with non constant coefficients is given by the following result

Result 2 There is a transformation X = ϕ(x), Y = ψ(x, y), taking a third order ordinary differential equation

y′′′ = F (x, y, y′, y′′)  (30)

into the linear equation

Y′′′ + 2 M Y′ + (M′ + N) Y = 0,  (31)

where M and N are functions of the independent variable X only, if, and only if,

F = q (p F_{6,y} + F_{6,x}) + p^{3} (3 F_{6,yy} − F_{6,y}^{2})/9
+ p^{2} (3 F_{6,xy} − F_{6,x} F_{6,y})/3 + p F_{1} + F_{0},  (32)

where F_{i} = F_{i}(x, y), and

F_{6,xxy} = (3 F_{1,y} + 2 F_{6,xy} F_{4})/3  (33)
0 = 27 F_{0,yy} − 9 F_{0,y} F_{6,y} − 9 F_{1,xy} + 3 F_{1,y} F_{6,x}
− 6 F_{6,xx} F_{6,xy} + 2 F_{6,xy} F_{6,x}^{2} + 9 F_{6,xy} F_{1} − 9 F_{6,yy} F_{0},  (34)

and the equation

\begin{aligned}& −1/6 N^{−8/3} (6 M N^{2} + 2 N N_{,XX} − N_{,X}^{2}) \\ = & (A^{2} (−3 F_{6,xx} + F_{6,x}^{2} + 3 F_{1}) + 6 A A_{,xx} − 9 A_{,x}^{2})/(6 A^{4}). \end{aligned}  (35)

is satisfied.

Linear equations with non-constant coefficients will admit a 4-dimensional group of fiber-preserving point symmetries.

As a final remark, the equations given in the last two results furnish the sufficiency of the conditions for the existence of an equivalence, and can, in the case of the previous results, be used to determine the coordinate function X of the transformation.

3.1 6-dimensional symmetry groups

The only equations admitting a 6-dimensional symmetry group are of the form

F = (12 q^{2} + 36 q F_{21,x} + 64 p^{4} F_{01} + 384 p^{3} F_{21} F_{01}
+ 4 p^{2} (216 F_{21}^{2} F_{01} − F_{22}) + 12 p (−F_{21,xx} + 72 F_{21}^{3} F_{01} − F_{21} F_{22})
+ 9 (−2 F_{21,xx} F_{21} + 3 F_{21,x}^{2} + 36 F_{21}^{4} F_{01} − F_{21}^{2} F_{22}))/(8 p + 12 F_{21}),

where F_{21} = F_{21}(x), F_{22} = F_{22}(x) and F_{01} = F_{01}(x, y), and the further condition

F_{01,x} = 3/2 F_{01,y} F_{21},

has to be satisfied. All these equations are equivalent to

y′′′ = \frac{3}{2} \frac{q^{2}}{p} .  (36)

The symmetry group of fiber preserving point transformations for such equations is determined by the Maurer-Cartan equations

\{ \begin{matrix}dω^{0} & = & α^{2} ∧ ω^{0} \\ dω^{1} & = & ω^{0} ∧ ω^{2} + ω^{1} ∧ ω^{2} + α^{2} ∧ ω^{1} \\ dω^{2} & = & ω^{0} ∧ ω^{3} − ω^{1} ∧ ω^{3} + μ^{2} ∧ ω^{1} \\ dω^{3} & = & −ω^{2} ∧ ω^{3} + μ^{2} ∧ ω^{2} − α^{2} ∧ ω^{3} \\ dα^{2} & = & 2 ω^{0} ∧ ω^{3} − μ^{2} ∧ ω^{0} \\ dμ^{2} & = & 2 ω^{2} ∧ ω^{3} + μ^{2} ∧ α^{2} − 2 μ^{2} ∧ ω^{2}. \end{matrix}  (37)

The transformations X = ϕ(x) and Y = ψ(x, y), relating (36) to another equation of the class are determined by

ψ_{,x} = \frac{3}{2} ψ_{,y} F_{21}
ψ_{,yyy} = \frac{3}{2} \frac{(ψ_{,yy})^{2}}{ψ_{,y}} − 8 F_{01} ψ_{,y}
ϕ′′′ = \frac{3}{2} \frac{ϕ^{′′2}}{ϕ′} − \frac{ϕ′}{2} F_{22}.

3.2 4-dimensional symmetry groups

The determination of the equations admitting a four dimensional symmetry group of fiber preserving point transformations is not as direct as it was for higher dimensions. In some cases, the result is direct, but in general, we have to assume all the invariants to be constant and look at the conditions imposed by this constraint. In doing so it is possible to show that the only equations admitting such groups are characterized as below.

As mentioned above, the only possible case for which a 4-dimensional symmetry group is possible is the case B-e. Let us assume that all the invariants given in equations (62)- (71) in appendix A, are constant, so that we can determine which equations have a 4-dimensional symmetry group of fiber preserving point transformations. I_{4}, I_{3} lead to

\frac{d}{dx} F_{,qqq} = (−3 F_{,pqq} − 4 F_{,qqq} F_{,q} − F_{,qq}^{2} I_{3} I_{6} + F_{,qq}^{2} I_{6} − F_{,qq}^{2})/3  (38)
F_{,qqqq} = F_{,qqq}^{2} (I_{3} + 2)/F_{,qq}.  (39)

I_{6} and I_{5} give

\frac{d}{dx} F_{,qq} = F_{,qq} (−F_{,qqq} F_{,q} + F_{,qq}^{2} I_{6})/(3 F_{,qqq})  (40)
\frac{d}{dx} F_{,q} = (−9 F_{,pqq}^{2} + 6 F_{,qqq} (F_{,pq} F_{,qq} + 3 F_{,qqy} − F_{,pqq} F_{,q})
+ F_{,qqq}^{2} (9 F_{,p} + F_{,q}^{2}) − 6 F_{,pqq} F_{,qq}^{2} + 2 F_{,qq}^{4} I_{5})/(3 F_{,qqq}^{2}),  (41)

and from ∂_{q} \frac{d}{dx} F_{,q} − \frac{d}{dx} F_{,qq} − F_{,pq} − F_{,q} F_{,qq} = 0 we get

F_{,yqqq} = (−6 F_{,pqq} F_{,qq}^{2} I_{3} + 6 F_{,pq} F_{,qqq} F_{,qq} I_{3} + 18 F_{,qqq} F_{,qqy} I_{3} + 36 F_{,qqq} F_{,qqy}
+ 4 F_{,qq}^{4} I_{3} I_{5} + F_{,qq}^{4} I_{6})/(18 F_{,qq}).  (42)

We can compute \frac{d^{2}}{dx^{2}} F_{,q} and substitute it into the I_{i}. I_{2} and I_{7} respectively give

F_{,pqqq} = (F_{,pqq} F_{,qqq} (I_{3} + 2))/F_{,qq}  (43)
F_{,ppqq} = (9 F_{,pqq}^{2} I_{3} + 18 F_{,pqq}^{2} + 6 F_{,pqq} F_{,qq}^{2} − 6 F_{,pq} F_{,qqq} F_{,qq} − F_{,qq}^{4} I_{7}
− F_{,qq}^{4} I_{5})/(9 F_{,qq}).  (44)

¿From I_{9} we obtain

F_{,ppq} = −((2 I_{3} I_{6}^{2} + 8 I_{3} I_{6} I_{7} + 4 I_{3} I_{8} − I_{6}^{2} − 4 I_{6} I_{5}
− 8 I_{6} I_{7} + 2 I_{6} + 4 I_{5} − 2 I_{8}) F_{,qq}^{4}
− 18 F_{,pqq}^{2} − 12 F_{,pqq} F_{,qq}^{2} + 12 F_{,pq} F_{,qqq} F_{,qq})/(18 F_{,qqq}).  (45)

I_{8} furnishes

F_{,ypqq} = (−18 F_{,pqq}^{2} F_{,qq}^{2} (2 I_{3} − 3) + 6 F_{,pqq} (6 (3 (I_{3} + 2) F_{,qqy} + F_{,qq}^{2} F_{,q}) F_{,qqq}
− (4 I_{3} I_{7} − I_{6} − 2 I_{5} − 2 I_{7} − 2) F_{,qq}^{4} + 6 F_{,pq} F_{,qqq} F_{,qq} I_{3})
+ F_{,qq}^{6} (4 I_{3} I_{6}^{2} + 16 I_{3} I_{6} I_{7} + 8 I_{3} I_{8} − I_{6}^{2} − 12 I_{6} I_{5}
− 16 I_{6} I_{7} + 4 I_{6} − 4 I_{7} − 2 I_{8})
− 4 F_{,qq}^{2} ((I_{5} + I_{7}) F_{,qq}^{2} F_{,q} + 9 F_{,qqy}) F_{,qqq}
+ 6 F_{,qq} F_{,qqq} (−2 (3 F_{,qqq} F_{,q} + F_{,qq}^{2}) F_{,pq}
− 9 F_{,pp} F_{,qqq} + 6 F_{,qqq} F_{,yq}))/(108 F_{,qq} F_{,qqq}).  (46)

The compatibility conditions between these expressions lead to

0 = (I_{3} − 1) (I_{5} + I_{7})  (47)
0 = (I_{3} − 2) I_{9}  (48)
0 = (I_{3} − 1) (I_{6} + 4 I_{3} I_{5}),  (49)

and to

I_{1} = −(2 I_{3} I_{7} − I_{6} − 2 I_{5} − 2 I_{7})/2  (50)
I_{2} = 0  (51)
I_{4} = I_{3} I_{6} + 2 I_{6} + 1  (52)
I_{9} = (2 I_{3} I_{6}^{2} + 8 I_{3} I_{6} I_{7} + 4 I_{3} I_{8} − I_{6}^{2} − 4 I_{6} I_{5}
− 8 I_{6} I_{7} + 2 I_{6} + 2 I_{5} − 2 I_{7} − 2 I_{8})/2  (53)
I_{10} = −(I_{3} − 1) I_{7}.  (54)

For I_{3} = 1, we obtain

\{ \begin{matrix}I_{1} & = & (I_{6} + 2 I_{5})/2 \\ I_{2} & = & 0 \\ I_{4} & = & 3 I_{6} + 1 \\ I_{7} & = & 1/2 I_{6}^{2} + I_{8} − I_{6} (I_{5} − 1) − I_{5} (I_{6} − 1) \\ I_{9} & = & 0 \\ I_{10} & = & 0, \end{matrix}  (55)

and I_{5}, I_{6}, I_{8} are free constants. If I_{3} = 2, then

\{ \begin{matrix}I_{1} & = & −2 I_{5} \\ I_{2} & = & 0 \\ I_{4} & = & −32 I_{5} + 1 \\ I_{6} & = & −8 I_{5} \\ I_{7} & = & −I_{5} \\ I_{9} & = & 3 (48 I_{5}^{2} − 2 I_{5} + I_{8}) \\ I_{10} & = & I_{5}, \end{matrix}  (56)

and I_{5} and I_{8} are free. For the other cases, i.e I_{3} ≠ 2 and I_{3} ≠ 1, we are left with

\{ \begin{matrix}I_{7} & = & −I_{5} \\ I_{6} & = & −4 I_{3} I_{5} \\ 0 & = & (2 I_{3} − 1) (8 I_{3} I_{5}^{2} (I_{3} + 1) − 2 I_{5} + I_{8}). \end{matrix}  (57)

For I_{3} = 1/2, I_{5} and I_{8} are free, and

\{ \begin{matrix}I_{1} & = & (−I_{5})/2 \\ I_{2} & = & 0 \\ I_{4} & = & −(5 I_{5} − 1) \\ I_{6} & = & −2 I_{5} \\ I_{7} & = & −I_{5} \\ I_{9} & = & 0 \\ I_{10} & = & (−I_{5})/2. \end{matrix}  (58)

For I_{3} ≠ 1/2, I_{3} and I_{5} are free and

\{ \begin{matrix}I_{1} & = & −I_{3} I_{5} \\ I_{2} & = & 0 \\ I_{4} & = & −(4 I_{3}^{2} I_{5} + 8 I_{3} I_{5} − 1) \\ I_{6} & = & −4 I_{3} I_{5} \\ I_{7} & = & −I_{5} \\ I_{8} & = & 2 I_{5} − 8 I_{3} I_{5}^{2} (I_{3} + 1) \\ I_{9} & = & 0 \\ I_{10} & = & (I_{3} − 1) I_{5}. \end{matrix}  (59)

Replacing these values into the Maurer-Cartan equations

\{ \begin{matrix}dω^{0} & = & ω^{0} ∧ ω^{1} I_{1} + ω^{0} ∧ ω^{2} I_{2} + ω^{0} ∧ ω^{3} I_{3} \\ dω^{1} & = & ω^{0} ∧ ω^{1} I_{4} + ω^{0} ∧ ω^{2} + ω^{1} ∧ ω^{2} I_{2} + ω^{1} ∧ ω^{3} (I_{3} − 1) \\ dω^{2} & = & ω^{0} ∧ ω^{1} I_{5} + ω^{0} ∧ ω^{2} I_{6} + ω^{0} ∧ ω^{3} + ω^{1} ∧ ω^{2} I_{7} \\ & − & ω^{1} ∧ ω^{3} (I_{2} + 1) − ω^{2} ∧ ω^{3} \\ dω^{3} & = & ω^{0} ∧ ω^{1} I_{8} + ω^{0} ∧ ω^{2} I_{5} + ω^{0} ∧ ω^{3} (2 I_{6} − I_{4}) \\ & + & ω^{1} ∧ ω^{2} I_{9} + ω^{1} ∧ ω^{3} I_{10} \end{matrix}  (60)

we obtain the characterization of the groups admitted by the different classes of equations, if the I_{i} satisfy the Jacobi identities

\{ \begin{matrix}0 & = & I_{3} I_{9} + I_{1} I_{2} + I_{2} I_{7} \\ 0 & = & −I_{3} I_{1} − I_{3} I_{10} + I_{1} + I_{2}^{2} + I_{2} \\ 0 & = & I_{2} \\ 0 & = & I_{3} I_{5} + I_{1} − I_{2} I_{4} + I_{2} I_{6} − I_{5} − I_{7} \\ 0 & = & −2 I_{3} I_{4} + 2 I_{3} I_{6} + 2 I_{2} + I_{4} − 2 I_{6} + 1 \\ 0 & = & I_{2} \\ 0 & = & I_{1} I_{6} − 3 I_{2} I_{5} + I_{4} I_{7} − I_{5} + I_{8} − I_{9} \\ 0 & = & −2 I_{3} I_{5} + I_{1} − I_{2} I_{6} − I_{6} + I_{7} − I_{10} \\ 0 & = & −I_{3} I_{6} + I_{4} − 2 I_{6} − 1 \\ 0 & = & −I_{3} I_{7} − I_{2}^{2} − I_{2} + I_{7} − I_{10} \\ 0 & = & I_{1} I_{5} − 2 I_{2} I_{8} + 2 I_{4} I_{9} − I_{5} I_{7} + I_{5} I_{10} − I_{6} I_{9} \\ 0 & = & −2 I_{3} I_{8} − I_{1} I_{4} + 2 I_{1} I_{6} + I_{2} I_{5} + I_{4} I_{10} + I_{5} + I_{8} + I_{9} \\ 0 & = & −I_{3} I_{5} − I_{2} I_{4} + 2 I_{2} I_{6} + I_{5} + I_{10} \\ 0 & = & −I_{3} I_{9} + I_{2} I_{10} + 2 I_{9}. \end{matrix}  (61)

Replacing, for each of the cases above, the values obtained for the I_{i}, in (61), we obtain the results shown in table 3.2. In the last column of this table, we have put the normal form of the commutators as furnished by the computer algebra program presented in [11]. With the admissible values for I_{i}, we can go back to equations (62)-(71) and get the conditions for the equations to admit a 4-dimensional symmetry group.

4 Conclusion

The solution of the local equivalence problem for third order ODE’s gives a complete characterization of the equations admitting symmetry groups of fiber preserving point transformations of dimension ≥ 4.

The different possible 4-dimensional groups are given as parametric families. For those groups, the use of a computer algebra program presented in [11] allows us to classify their Lie algebras. In principal, it should be possible to obtain the symmetry generators from the information obtained. This avoids the solution of a system of equations. But in practice the author was not able to obtain these generators on a concrete example, as has been done in [10].

It is interesting to note that, unlike the case of second order ODE’s, it is not possible to characterize the equations admitting symmetry groups of dimension less than 4, in the general case. Nonetheless, with the help of the computer algebra program clasode3, several equations admitting such groups, such as some of the Painlevé equations of the third order, were clasified.

References

[1] Bocharov A. V., Sokolov V. V., Svinolupov S. I. (1993) On Some Equivalence Problems for Differential Equations, preprint.

[2] Chern S.S. (1940) The geometry of a differential equation Y′′′ = F (X, Y, Y′, Y′′), Tensor, N.S., Vol. 28, 173-176.

[3] Gardner, R. (1989) The Method of Equivalence and its Applications, Siam, Philadelphia, Pensylvania.

[4] Grebot G. (1994) On Killing tensors in vacuum Space-times and The classification of third order ordinary differential equations, PhD Thesis, Queen Mary College, London.

[5] Grebot G. (1995) The classification of third order ordinary differential equations, Proceedings of the CATHODE workshop, Nijmegen, 9-12 January 1995, p. 23.

[6] Gregus, M. (1987) Third Order Linear Differential Equations, Mathematics and its Applications, D. Reidel Publishing Company, Dordrecht, Holland.

[7] Hsu L., Kamran N. (1989) Classification of Second-Order Differential Equations admitting Lie Groups of Fiber-preserving point Symmetries, Proc. Lond. Math. Soc. (3), 58, No 2, 387-416.

[8] Kamran, N. (1989) Contributions to the Study of the Equivalence problem of Elie Cartan and its Applications to partial and Ordinary Differential Equations, Memoires Cl. Sci. Acad. Roy. Belg., 45, Fac.7.

[9] Kamran N., Shadwick W.F. (1986) The solution of the Cartan Equivalence Problem for \frac{d^{2} y}{dx^{2}} = F (x, y, \frac{d y}{dx}) under the Pseudo-group \overline{x} = ϕ(x), \overline{y} = ψ(x, y), Lecture Notes in Physics 246, 320-336.

[10] Karlhede A. and MacCallum M.A.H. (1982) On Determining the Isometry Group of a Riemannian Space, Gen. Rel. Grav., Vol 14, No 7, 673.

[11] F. Schöbel (1992) The Symbolic Classification of Real Four Dimensional Lie Algebras, Universität Leipzig Preprint.

[12] Sternberg, S. (1964) Lectures on differential geometry, Prentice Hall, Englewood Cliffs, NJ.

A Invariants for the case B-e

The invariants obtained for the case B-e are given below.

I_{1} = −(18 F_{,pqqq} F_{,pqq} F_{,qqq} F_{,qq} + 6 F_{,pqqq} F_{,qqq}^{2} F_{,qq} F_{,q}
− 9 F_{,pqq}^{2} F_{,qqqq} F_{,qq} − 18 F_{,pqq}^{2} F_{,qqq}^{2} − 12 F_{,pqq} F_{,qqq}^{3} F_{,q}
− 9 F_{,p} F_{,qqqq} F_{,qqq}^{2} F_{,qq} + 18 F_{,p} F_{,qqq}^{4} − F_{,qqqq} F_{,qqq}^{2} F_{,qq} F_{,q}^{2}
+ 3 F_{,qqqq} F_{,qqq}^{2} F_{,qq} \frac{d}{dx} F_{,q} − 18 F_{,yqqq} F_{,qqq}^{2} F_{,qq} + 2 F_{,qqq}^{4} F_{,q}^{2}
− 6 F_{,qqq}^{4} \frac{d}{dx} F_{,q} + 36 F_{,qqq}^{3} F_{,yqq})/(2 F_{,qqq}^{2} F_{,qq}^{4})  (62)
I_{2} = 3 (F_{,pqqq} F_{,qqq} − F_{,pqq} F_{,qqqq})/(F_{,qqq}^{2} F_{,qq})  (63)
I_{3} = (F_{,qqqq} F_{,qq} − 2 F_{,qqq}^{2})/F_{,qqq}^{2}  (64)
I_{4} = (−3 F_{,pqq} F_{,qq} − F_{,qqq} F_{,qq} F_{,q} + 9 F_{,qqq} \frac{d}{dx} F_{,qq}
− 3 F_{,qq} \frac{d}{dx} F_{,qqq})/F_{,qq}^{3}  (65)
I_{5} = (−9 F_{,pqq}^{2} − 12 F_{,pqq} F_{,qqq} F_{,q} − 18 F_{,pqq} \frac{d}{dx} F_{,qqq}
− 9 F_{,p} F_{,qqq}^{2} − F_{,qqq}^{2} F_{,q}^{2} + 3 F_{,qqq}^{2} \frac{d}{dx} F_{,q}
− 18 F_{,qqq} F_{,yqq} + 18 F_{,qqq} ∂_{p} \frac{d}{dx} F_{,qq})/(2 F_{,qq}^{4})  (66)
I_{6} = F_{,qqq} (F_{,qq} F_{,q} + 3 \frac{d}{dx} F_{,qq})/F_{,qq}^{3}  (67)
I_{7} = −(18 F_{,ppqq} F_{,qqq}^{2} F_{,qq} − 36 F_{,pqqq} F_{,pqq} F_{,qqq} F_{,qq}
+ 18 F_{,pqq}^{2} F_{,qqqq} F_{,qq} + 9 F_{,pqq}^{2} F_{,qqq}^{2} + 6 F_{,pqq} F_{,qqq}^{3} F_{,q}
− 6 F_{,pqq} F_{,qqq}^{2} F_{,qq}^{2} + 6 F_{,pq} F_{,qqq}^{3} F_{,qq} − 9 F_{,p} F_{,qqq}^{4}
− F_{,qqq}^{4} F_{,q}^{2} + 3 F_{,qqq}^{4} \frac{d}{dx} F_{,q} − 18 F_{,qqq}^{3} F_{,yqq})/(2 F_{,qqq}^{2} F_{,qq}^{4})  (68)
I_{8} = F_{,qqq}^{3} (18 F_{,p} F_{,q} + 4 F_{,q}^{3} − 18 F_{,q} \frac{d}{dx} F_{,q}
+ 54 F_{,y} + 9 \frac{d^{2}}{dx^{2}} F_{,q} − 27 \frac{d}{dx} F_{,p})/(2 F_{,qq}^{6})  (69)
I_{9} = 3 (−18 F_{,ppqq} F_{,pqq} F_{,qqq}^{2} − 6 F_{,ppqq} F_{,qqq}^{3} F_{,q} + 9 F_{,pp} F_{,qqq}^{4}
+ 27 F_{,pqqq} F_{,pqq}^{2} F_{,qqq} + 6 F_{,pqqq} F_{,pqq} F_{,qqq}^{2} F_{,q}
+ 9 F_{,pqqq} F_{,p} F_{,qqq}^{3} + F_{,pqqq} F_{,qqq}^{3} F_{,q}^{2} − 3 F_{,pqqq} F_{,qqq}^{3} \frac{d}{dx} F_{,q}
+ 18 F_{,ypqq} F_{,qqq}^{3} − 9 F_{,pqq}^{3} F_{,qqqq} + 6 F_{,pqq}^{2} F_{,qqq}^{2} F_{,qq}
− 12 F_{,pqq} F_{,pq} F_{,qqq}^{3} − 9 F_{,pqq} F_{,p} F_{,qqqq} F_{,qqq}^{2}

− F_{,pqq} F_{,qqqq} F_{,qqq}^{2} F_{,q}^{2} + 3 F_{,pqq} F_{,qqqq} F_{,qqq}^{2} \frac{d}{dx} F_{,q}
− 18 F_{,pqq} F_{,yqqq} F_{,qqq}^{2} + F_{,pqq} F_{,qqq}^{3} F_{,qq} F_{,q}
+ 3 F_{,pqq} F_{,qqq}^{3} \frac{d}{dx} F_{,qq} + 2 F_{,pq} F_{,qqq}^{4} F_{,q} − 3 F_{,qqq}^{4} ∂_{p} \frac{d}{dx} F_{,q})
/ (2 F_{,qqq}^{2} F_{,qq}^{5})  (70)
I_{10} = (−9 F_{,pqq}^{2} − 6 F_{,pqq} F_{,qqq} F_{,q} − 6 F_{,pqq} F_{,qq}^{2}
+ 6 F_{,pq} F_{,qqq} F_{,qq} + 9 F_{,p} F_{,qqq}^{2} + F_{,qqq}^{2} F_{,q}^{2}
− 3 F_{,qqq}^{2} \frac{d}{dx} F_{,q} + 18 F_{,qqq} F_{,yqq} − F_{,qqq} F_{,qq}^{2} F_{,q}
− 3 F_{,qqq} F_{,qq} \frac{d}{dx} F_{,qq})/(2 F_{,qq}^{4}) + I_{1}.  (71)

Table 1: 4-dimensional Lie algebras for the case B-e

structure constants  normal form for the commutators

and classification

For I_{5} = 0:

I_{3} = 1,  [X, Y] = W, [X, Z] = Y,
I_{1} = I_{7} = I_{5},  [W, X] = [W, Y] = [W, Z] = 0
I_{4} = 1,  [Y, Z] = X, case 19
I_{2} = I_{6} = I_{8} = I_{9} = I_{10} = 0.  For I_{5} ≠ 0:

[W, X] = Y, [W, Y] = X, [W, Z] = [X, Z] = [Y, Z] = 0 [X, Y] = W, case 22

I_{3} = 2,  [X, Y] = W, [X, Z] = 4X,
I_{5} = \frac{1}{40} = \frac{−1}{2}I_{1} = \frac{1−I_{4}}{32} = \frac{−I_{6}}{8} = −I_{7} = I_{10}  [W, X] = [W, Y] = 0,
I_{2} = 0,  [W, Z] = 2W, [Y, Z] = −2Y,
I_{8} = \frac{1}{50},  case 16 with lambda=3

I_{9} = 144 I_{5}^{2} − 6 I_{5} + 3 I_{8}.

I_{3} = \frac{1}{2},  [X, Y] = W, [X, Z] = 4X,
I_{5} = \frac{−1}{2} = −2 I_{1} = −2 I_{10} = \frac{1−I_{4}}{5} = \frac{−I_{6}}{2} = −I_{7} [W, X] = [W, Y] = 0,
I_{2} = I_{9} = 0,  [W, Z] = 2W
I_{8} = \frac{−5}{2}.  [Y, Z] = −2Y,

case 16 with lambda=3

I_{1} = −I_{3} I_{5}, I_{2} = 0, I_{4} = −4 I_{3}^{2} I_{5} − 8 I_{3} I_{5} + 1, I_{6} = −4 I_{3} I_{5}, I_{7} = −I_{5}, I_{8} = −8 I_{3}^{2} I_{5}^{2} − 8 I_{3} I_{5}^{2} + 2 I_{5}, I_{9} = 0,

I_{10} = I_{3} I_{5} − I_{5},  No real Lie algebra.

I_{3} ∈ |C I_{5} = \frac{I_{3}−1}{2 I_{3}^{2} (2 I_{3}+1)}.