The Diffiety Institute Preprint Series

Preprint DIPS–3/2001 April 2, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Contact classification of linear ordinary differential equations. I.

by

V.A. YUMAGUZHIN

Contact classification of linear ordinary differential equations. I.

V.A. YUMAGUZHIN

Abstract. It is known that a linear ordinary differential equation of order n ≥ 3 can be transformed to the Laguerre–Forsyth form y^{(n)} = ∑_{i=3}^{n} a_{n−i}(x)y^{(n−i)} by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.

1. Introduction

This paper is devoted to the problem of local classification of nth order linear ordinary differential equations (ODE) up to a contact transformation. For n ≤ 2, it is well known (for example, see [2]) that any n-th order linear ODE can be transformed locally to the form y^{(n)} = 0 by a point transformation. For n ≥ 3, this statement is incorrect: there is infinite number of different equivalence classes of linear ODEs.

First this problem was posed by classics of the XIX century E. Laguerre, G.-H. Halphen and others. They obtained results concerning classification of third and forth orders linear ODEs, see [7, 3]. Essentially, this problem was forgotten after that.

Here, we solve the problem for n ≥ 3 in a neighborhood of a regular point. We considered the case n = 3 in [12, 13]. Our approach to the problem is as follows.

In their paper [8], F.M. Mahomed and P.G.L. Leach proved that dimension of the algebra of point symmetries of an n−th order linear ODE equals either n + 4 or n + 2, or n + 1. We prove (Theorem 3.2) that dimension of the algebra of point symmetries of a linear ODE is an invariant of contact transformations that take the set of linear ODEs to itself.

It is well known (see [11, 4, 9]) that any linear ODE can be transformed by a point transformation to the Laguerre–Forsyth form

(1)  y^{(n)} = a_{n−3}(x) y^{(n−3)} + a_{n−4}(x)y^{(n−4)} + ⋯ + a_{0}(x)y.

2000 Mathematics Subject Classification. 34A30, 34A26, 34C20,53A55, 58F35.

Key words and phrases. Linear ordinary differential equation, contact transformation, point transformation, projective transformation, equivalence problem, scalar differential invariants, classical symmetry, point symmetry, Laguerre–Forsyth form.

This work was supported by RFBR Grant 99-01-00353.

We prove (Theorem 3.3) that the Laguerre–Forsyth form of a linear ODE with n + 4-dimensional algebra of point symmetries is y^{(n)} = 0.

For linear ODEs with n + 2 and n + 1-dimensional algebras of point symmetries, we prove (Theorem 3.6) that a contact transformation that takes one of these equation to the other one is a point transformation. Further, for any two equations ℰ_{1} and ℰ_{2} of the form (1), we prove (Theorem 3.7) that if there exists a point transformation that takes ℰ_{1} to ℰ_{2}, then there exists a point transformation of the form

(2) f(x) = \frac{αx + β}{γx + δ} , f̂(x, y) = |f′|^{(n−1)/2} · y, α, β, γ, δ ∈ ℝ

that takes ℰ_{1} to ℰ_{2}. Transformations (2) take the set of all ODEs of the form (1) to itself. Thus, the problem of local classification of linear ODEs with respect to contact transformation is reduced to classification of ODEs (1) with respect to transformations (2).

A transformation (f, f̂) of the form (2) is generated by a projective transformation f of ℝ^{1}. The correspondence f ↦ (f, f̂) is an isomorphism from the group G of projective transformations of ℝ^{1} to the group of point transformations of form (2).

Further, we identify ODE (1) with the section

S_{ℰ} : x ↦ (a_{n−3}(x), a_{n−4}(x), …, a_{0}(x))

of the trivial bundle π: E = ℝ^{1} × ℝ^{n−2} → ℝ^{1}. The transformation law of coefficients of ODEs (1) under transformations (2) defines the lifting of every transformation f ∈ G of the base of π to a diffeomorphism f^{(0)} of the total space E of π. So, it is possible to consider the transformation law of coefficients of ODEs (1) under (2) as the transformation law S ↦ f(S) =^{def} f^{(0)} ∘ S ∘ f^{−1} for sections of π under projective transformations of the base ℝ^{1}. Obviously, the transformation (f, f̂) takes ℰ_{1} of the form (1) to ODE ℰ_{2} iff f(S_{ℰ_{1}}) = S_{ℰ_{2}}. Thus the problem of local classification of linear ODEs reduces to the one of classification of germs of sections of π w.r.t. the group G. Since this group is transitive, the last problem is reduces to classification of germs at 0 ∈ ℝ^{1} of sections of π w.r.t. the isotropy group G_{0} ⊂ G of 0 ∈ ℝ^{1}. We obtain this classification for regular germs in Theorem 4.8 and Corollary 4.9.

Finally, we calculate (Theorem 5.2) scalar differential invariants of the action of the group G_{+} = { f ∈ G | f′ > 0 } on π. This gives a solution to the equivalence problem for regular linear ODEs (Theorem 5.4) resulting in canonical forms for all nonequivalent regular linear ODEs (Theorem 5.8 and its corollary).

All manifolds below and maps are assumed to be smooth; ℝ^{n} denotes the n-dimensional arithmetical space.

2. Preliminaries

Let us recall necessary notation and results of the geometry of differential equations [5, 6] and some facts concerning linear ODEs [8, 11].

2.1. Jet bundles.

2.1.1. Cartan distribution. Let E and M be smooth manifolds of dimensions n+m and n respectively and π: E → M be a smooth bundle. Denote by [S]_{x}^{k} the k-jet of a section S at x ∈ M. Let π_{k}: J^{k}π → M, π_{k}: [S]_{x}^{k} ↦ x, k = 0, 1, 2, …, ∞, be the bundle of k-jets of all sections of π. Moreover, the projection π_{k,r}: J^{k}π → J^{r}π, k > r, is defined by π_{k,r}([S]_{x}^{k}) = [S]_{x}^{r}. Every section S of π generates the section j_{k}S of π_{k} by the formula j_{k}S: x ↦ [S]_{x}^{k}. Denote by L_{S}^{(k)} the image of the section j_{k}S.

Let T_{x_{k}}(J^{k}π) denote the tangent space to J^{k}π at x_{k} ∈ J^{k}π, T_{x_{k}}(L_{S}^{(k)}) denote the tangent space to L_{S}^{(k)} at x_{k} ∈ L_{S}^{(k)}. Consider all L_{S}^{(k)} containing x_{k}. The subspace 𝒞_{x_{k}} ⊂ T_{x_{k}}(J^{k}π) spanning the union of T_{x_{k}}(L_{S}^{(k)}) is called the Cartan plane at x_{k}. The distribution 𝒞: x_{k} ↦ 𝒞_{x_{k}} is called the Cartan distribution on J^{k}π.

2.1.2. Lie transformations. A (local) diffeomorphism of J^{k}π that takes the Cartan distribution to itself is called a Lie transformation. A Lie transformation of J^{0}π (that is an arbitrary diffeomorphism of J^{0}π) is called a point transformation. A Lie transformation of J^{1}π is called a contact transformation if m = 1.

Every Lie transformation f: U → U′ of J^{k}π can be lifted canonically to the Lie transformation f^{(r)} : π_{k+r,k}^{−1}(U) → π_{k+r,k}^{−1}(U′) of J^{k+r}π, r = 1, 2, …, such that π_{k+r,k+l} ∘ f^{(r)} = f^{(l)} ∘ π_{k+r,k+l} for r ≥ l. Indeed, f^{(r)} is defined in the following way. A point x_{k+1} = [S]_{x}^{k+1} ∈ J^{k+1}π is identified by K_{x_{k+1}} = T_{x_{k}}(L_{S}^{(k)}), where x_{k} = π_{k+1,k}(x_{k+1}). The differential f_{∗} maps K_{x_{k+1}} to the subspace f_{∗}(K_{x_{k+1}}). If f_{∗}(K_{x_{k+1}}) is projected on M nondegenerately, then there is x′_{k+1} ∈ J^{k+1}π such that K_{x′_{k+1}} = f_{∗}(K_{x_{k+1}}) and we set f^{(1)}(x_{k+1}) = x′_{k+1}. Obviously, f^{(1)} is a Lie transformation of J^{k+1}π defined almost everywhere in π_{k+1,k}^{−1}(U). Setting f^{(r+1)} = (f^{(r)})^{(1)}, we define the Lie transformation f^{(r)} for all r = 1, 2, … Clearly, f^{(r)} is defined almost everywhere in π_{k+r,k}^{−1}(U). We shall say for brevity that f^{(r)} is defined in π_{k+r,k}^{−1}(U).

It is well known [5, 6]) that any Lie transformation is the lifting of some point transformation if m > 1 and, if m = 1, any Lie transformation is the lifting of a contact transformation.

2.1.3. Lie fields. A vector field ξ on J^{k}π is called a Lie field if its flow is generated by Lie transformations. A vector field in J^{0}π is said to be a point vector field. A Lie field on J^{1}π is called a contact vector field if m = 1. Let ξ be a Lie field in J^{k}π and let f_{t} be its flow. Then the flow f_{t}^{(r)} of Lie transformations defines the Lie field ξ^{(r)} on J^{k+r}π, r = 1, 2, … such that (π_{k+r,k+l})_{∗}ξ^{(r)} = ξ^{(l)}, r ≥ l.

2.2. Ordinary differential equations.

2.2.1. Contact classification. Let π: ℝ^{1} × ℝ^{1} → ℝ^{1} and x, y, p_{1}, …, p_{k} be standard coordinates on J^{k}π. Any k-th order ODE

F (x, y(x), \frac{dy}{dx}, …, \frac{d^{k}y}{dx^{k}}) = 0

is identified with the submanifold

ℰ = { F (x, y, p_{1}, …, p_{k}) = 0 } ⊂ J^{k}π.

A “usual” solution S(x) is identified with the submanifold L_{S}^{(k)} ⊂ ℰ corresponding to a section S: x ↦ S(x) of π. Obviously, L_{S}^{(k)} is a 1-dimensional integral manifold of the Cartan distribution on J^{k}π. A “multivalued” solution of ℰ is a 1-dimensional integral manifold L of the Cartan distribution on J^{k}π such that L ⊂ ℰ. Locally, almost everywhere, a “multivalued” solution has the form L_{S}^{(k)}.

It is natural to classify k-th order ODEs up to a diffeomorphism of J^{k}π that takes the set of all solutions of ODEs to itself. Such diffeomorphisms are Lie transformations. Hence, they are liftings of contact transformation. So, we come to the problem of ODE classification up to a contact transformation.

Let ℰ_{1}, ℰ_{2} ⊂ J^{k}π be k-th order ODEs and f be a point (contact) transformation. We say that f (locally) takes ℰ_{1} to ℰ_{2} if f^{(k)} (f^{(k−1)}) takes (locally) the submanifold ℰ_{1} ⊂ J^{k}π to the submanifold ℰ_{2} ⊂ J^{k}π. We say that ODEs ℰ_{1} and ℰ_{2} are equivalent if there exists a point (contact) transformation that takes (locally) ℰ_{1} to ℰ_{2}.

2.2.2. Point and contact transformations. Any point transformation f is defined in coordinates by the formulas

(3)  X = X(x, y), Y = Y (x, y).

Obviously, the lifting f^{(k)} is defined in standard coordinates by

(4) X = X(x, y), Y = Y (x, y), P_{1} = \frac{DY}{DX} , …, P_{k} = \frac{DP_{k−1}}{DX} ,

where

(5)  D = \frac{∂}{∂x} + p_{1} \frac{∂}{∂y} + p_{2} \frac{∂}{∂p_{1}} + ⋯ + p_{k+1} \frac{∂}{∂p_{k}} + ⋯

is the operator of the total derivative over x.

It is easy to show that a contact transformation is defined in standard coordinates by the formulas

(6)  X = X(x, y, p_{1}), Y = Y (x, y, p_{1}), P_{1} = \frac{Y_{x} + p_{1}Y_{y}}{X_{x} + p_{1}X_{y}} ,

where the functions X(x, y, p_{1}), Y (x, y, p_{1}) are connected by the relation

Y_{p_{1}}(X_{x} + p_{1}X_{y}) − X_{p_{1}}(Y_{x} + p_{1}Y_{y}) = 0.

2.2.3. Point and contact vector fields. Let ξ be a contact vector field. Then ξ can be represented in standard coordinates as

(7)  ξ = ξ_{φ} = −φ_{p_{1}} \frac{∂}{∂x} + (φ − p_{1}φ_{p_{1}})\frac{∂}{∂y} + (φ_{x} + p_{1}φ_{y})\frac{∂}{∂p_{1}} ,

where the function φ = φ(x, y, p_{1}) is the generating function of ξ. Let ζ be an arbitrary point vector field. It has the form

(8)  ζ = a(x, y)\frac{∂}{∂x} + b(x, y)\frac{∂}{∂y}

in standard coordinates. The lifting ζ^{(1)} is a contact vector field. The generating function of ζ^{(1)} is

(9)  b(x, y) − a(x, y) · p_{1}.

Conversely, if the generating function of a contact vector field has the form (9), then this vector field is the lifting of some point vector field.

Let us transform a contact vector field by contact transformation (6). It is easy to verify that the generating functions Φ of the obtained vector field and φ are connected by the formula

(10)  \frac{X_{x} + p_{1}X_{y}}{X_{x}Y_{y} − X_{y}Y_{x}} Φ(X, Y, P_{1}) = φ(x, y, p_{1}).

2.2.4. Classical symmetries. A point vector field ζ in J^{0}π is called a point symmetry of a differential equation ℰ ⊂ J^{n}π if ζ^{(n)} is tangent to ℰ. By Pnt ℰ we denote the set of all point symmetries of ℰ. A contact vector field ξ in J^{1}π is called a contact symmetry of ℰ ⊂ J^{n}π if ξ^{(n−1)} is tangent to the submanifold ℰ. Point and contact symmetries are called classical symmetries. By Sym ℰ we denote the set of all classical symmetries of ℰ.

The space of generating functions of classical symmetries for an ODE p_{n} − F (x, y, p_{1}, …, p_{n−1}) = 0 coincide with the space of solutions φ = φ(x, y, p_{1}) of the linear PDE [5, 6]

(11)  (D̄^{n} − \frac{∂F}{∂p_{n−1}} D̄^{n−1} − ⋯ − \frac{∂F}{∂p_{1}} D̄^{1} − \frac{∂F}{∂y})(φ) = 0,

where

D̄ = \frac{∂}{∂x} + p_{1} \frac{∂}{∂y} + p_{2} \frac{∂}{∂p_{1}} + ⋯ + p_{n−1} \frac{∂}{∂p_{n−2}} + F \frac{∂}{∂p_{n−1}} .

2.3. Linear ordinary differential equations.

2.3.1. Point transformations of linear ODEs. Any linear ODE can reduced to the form

(12)  P_{n} = A_{n−2}(X)P_{n−2} + A_{n−3}(X)P_{n−3} + ⋯ + A_{0}(X)Y

by a point transformation. It is known [11] that an arbitrary equation (12) of order n ≥ 3 is reduced to the Laguerre–Forsyth form (1)

p_{n} = a_{n−3}(x)p_{n−3} + a_{n−4}(x)p_{n−4} + ⋯ + a_{0}(x)y

by the point transformation

X = f(x), Y = |f′|^{(n−1)/2}y,

where f is a solution of the ODE

2f′f′′′ − 3(f′′)^{2} − 24\frac{(n − 2)!}{(n + 1)!} (f′)^{4}A_{n−2}(f) = 0.

This transformation is called the Laguerre–Forsyth transformation of (12). It follows from this result that the problem of local classification of linear ODEs up to a contact transformation reduces to classification of ODEs of the form (1). The following proposition holds [11]:

Proposition 2.1. Let ℰ be an ODE of the form (1). Then a point transformation X takes ℰ to an ODE of the same form iff

(13) X = \frac{α · x + β}{γ · x + δ} , Y = C · |X′|^{(n−1)/2} · y, α, β, γ, δ, C ∈ ℝ .

2.3.2. Point symmetries of linear ODEs. Let ℰ be an arbitrary ODE (1)). In [8] it was proved that a point symmetry of ℰ has the form

(φ(x)\frac{∂}{∂x} + \frac{n − 1}{2}φ′y \frac{∂}{∂y}) + Cy \frac{∂}{∂y} + γ(x)\frac{∂}{∂y} ,

where γ(x) is a solution of ℰ, C ∈ ℝ, and φ(x) is a solution of

(14) \begin{cases}φ′′′ = 0 \\ 3a_{n−3}φ′ + a′_{n−3}φ = 0 \\ \frac{(k − 1)(n − (k − 1))}{2}a_{n−k+1}φ′′ + ka_{n−k}φ′ + a′_{n−k}φ = 0, \\ k = 4, 5, …, n. \end{cases}

Dimension of the solution space of system (14) can be equal to either 3, 1, or 0. It follows that dim Pnt ℰ can be equal to either n + 4, n + 2, or n+1. Obviously, dimension of the algebra of point symmetries is an invariant of point transformations. Below, we prove (Theorem 3.2) that this dimension is an invariant of contact transformations that take the set of ODEs of the form (1) to itself. Thus the set of all linear ODEs ℰ of the form (1) is divided into three nonintersecting families according to dim Pnt ℰ. These families are invariant w.r.t. contact transformations.

3. Symmetries and transformations of linear ODEs

3.1. Classical symmetries. Let ℰ be an n-th order linear ODE. In [13], we proved for n = 3 the following results: dim Sym ℰ can be 10, 5, or 4; if dim Sym ℰ = 10, then Sym ℰ is generated by three contact and seven point symmetries; if dim Sym ℰ = 5 or 4, then Sym ℰ = Pnt ℰ.

Proposition 3.1. If n > 3, then Sym ℰ = Pnt ℰ.

Proof. We can assume without loss of generality that ℰ has the form (1).

Let φ(x, y, y^{(1)}) be the generating function of a classical symmetry. The generating function of a point symmetry is α(x, y)y^{(1)} + β(x, y). Hence, we must check that φ_{y^{(1)}y^{(1)}} ≡ 0. To this end, let us consider equation (11) for ℰ:

(15)  (D̄^{n} − a_{n−3}D̄^{n−3} − a_{n−4}D̄^{n−4} − ⋯ − a_{1}D̄^{1} − a_{0})φ = 0.

Obviously,

D̄^{n}(φ) = D̄^{n−2}(φ_{y^{(1)}}y^{(3)} + φ_{y^{(1)}y^{(1)}}(y^{(2)})^{2} + low degree terms) =

= \begin{cases}D̄(3φ_{y^{(1)}y^{(1)}}y^{(2)}y^{(3)} + l.d.t.), & n = 4 \\ D̄^{n−3}(φ_{y^{(1)}}y^{(4)} + 3φ_{y^{(1)}y^{(1)}}y^{(2)}y^{(3)} + l.d.t.), & n > 4 \end{cases}

= \begin{cases}3φ_{y^{(1)}y^{(1)}}(y^{(3)})^{2} + l.d.t., & n = 4 \\ (\binom{n − 3}{n − 5} + 3\binom{n − 3}{n − 4} + 3) φ_{y^{(1)}y^{(1)}}y^{(3)}y^{(n−1)} + l. d. t., & n > 4 \end{cases}

It now follows from (15) that φ_{y^{(1)}y^{(1)}} ≡ 0. □

Obviously, dimension of the algebra of classical symmetries is invariant under contact transformations. From the above mentioned results of [13] and this proposition, we get

Theorem 3.2. Dimension of the algebra of point symmetries of a linear ODE is invariant under contact transformations that preserve the set of linear ODEs.

3.2. Linear ODEs with n+4-dimensional point symmetry algebra. The following theorem gives local classification of all linear ODEs with n + 4-dimensional algebra of point symmetries.

Theorem 3.3. The Laguerre–Forsyth form of a linear ODE with n+4dimensional algebra of point symmetries is p_{n} = 0.

Proof. Let ℰ = { p_{n} = a_{n−3}(x)p_{n−3} + a_{n−4}(x)p_{n−4} + ⋯ + a_{0}(x)y } be anODE of the form (1) with dim Pnt ℰ = n + 4. From the result of [8] cited in Subsection 2.3.2, we get that Pnt ℰ contains symmetries

φ_{i}(x)\frac{∂}{∂x} + \frac{n − 1}{2}φ′_{i}y \frac{∂}{∂y} ,

where φ_{i}, i = 1, 2, 3, are linear independent solutions of system (14). From the second equation of the system, we have a_{n−3} ≡ 0; from the third equation, we have a_{n−4} ≡ 0, etc. □ Corollary 3.4. Let ℰ_{1} and ℰ_{2} be n-the order linear ODEs with n + 4dimensional algebras of point symmetries. Then there exists a point transformation that takes (locally) one of them to the other. Corollary 3.5. 1. The equation ℰ = { p_{n} = 0 } is the only one in

the set of all ODEs of the form (1) that has n + 4-dimensional algebra of point symmetries.

2. The equation ℰ = { p_{n} = 0 } is invariant w.r.t. all contact transformations that preserve the equations (1).

3.3. Contact transformations of linear ODEs. Theorem 3.6. Let ℰ_{1}, ℰ_{2} be linear ODEs with n + 2 or n + 1-dimen-sional algebras of point symmetries and f be a contact transformation that takes ℰ_{1} to ℰ_{2}. Then f is the lifting of a point transformation. Proof. We can assume without loss of generality that ℰ_{1} and ℰ_{2} have the form (1).

Assume dim Pnt ℰ_{1} = n + 2. The transformation f is defined in standard coordinates by (6). Let Γ_{1}(X), Γ_{2}(X), Γ_{3}(X) be linear independent solutions of ℰ_{1}. We can consider these solutions as generating functions of point symmetries of ℰ_{1} (see Subsection 2.3.2). Eaech of these functions is connected with the corresponding generating function of a point symmetry of ℰ_{2} by (10). Taking into account the form of generating functions of the n + 2-dimensional algebra Pnt ℰ_{2} (see Subsection 2.3.2), we obtain:

ΔΓ_{1}(X(x, y, y^{(1)})) = K_{1}(φ(x)y^{(1)} − \frac{n − 1}{2}φ′y) + C_{1}y + γ_{1}(x)
ΔΓ_{2}(X(x, y, y^{(1)})) = K_{2}(φ(x)y^{(1)} − \frac{n − 1}{2}φ′y) + C_{2}y + γ_{2}(x)
ΔΓ_{3}(X(x, y, y^{(1)})) = K_{3}(φ(x)y^{(1)} − \frac{n − 1}{2}φ′y) + C_{3}y + γ_{3}(x)

where Δ = \frac{X_{x} + y^{(1)}X_{y}}{X_{x}Y_{y} − X_{y}Y_{x}}; K_{j}, C_{j} ∈ ℝ, j = 1, 2, 3. If one of the

numbers K_{1}, K_{2}, K_{3} does not vanish, say K_{1} ≠ 0, then

Δ(Γ_{2} − \frac{K_{2}}{K_{1}} Γ_{1}) = (C_{2} − \frac{K_{2}}{K_{1}} C_{1})y + γ_{2} − \frac{K_{2}}{K_{1}} γ_{1}
Δ(Γ_{3} − \frac{K_{3}}{K_{1}} Γ_{1}) = (C_{3} − \frac{K_{3}}{K_{1}} C_{1})y + γ_{3} − \frac{K_{2}}{K_{1}} γ_{1}

Then \frac{K_{1}Γ_{2} − K_{2}Γ_{1}}{K_{1}Γ_{3} − K_{3}Γ_{1}} is independent of y^{(1)}. Therefore

\frac{∂}{∂y^{(1)}} (\frac{K_{1}Γ_{2} − K_{2}Γ_{1}}{K_{1}Γ_{3} − K_{3}Γ_{1}} ) = \frac{d}{dX} (\frac{K_{1}Γ_{2} − K_{2}Γ_{1}}{K_{1}Γ_{3} − K_{3}Γ_{1}} ) X_{y^{(1)}} = 0.

Suppose that X_{y^{(1)}} ≠ 0. Then

K_{1}Γ_{2} − K_{2}Γ_{1} = K(K_{1}Γ_{3} − K_{3}Γ_{1}),

where K ∈ ℝ. This means that the solutions Γ_{1}, Γ_{2}, Γ_{3} are linear dependent. From this contradiction, we have X_{y^{(1)}} ≡ 0. Hence f is the lifting of some point transformation.

Obviously, the proofs for K_{1} = K_{2} = K_{3} = 0 and dim Pnt ℰ = n + 1 are similar. □

From this result, Theorem 3.3, and Corollary 3.5, we have that classification problem for linear ODEs w.r.t. contact transformations reduces to that for equations of the form (1) w.r.t. point transformations. Proposition 2.1 shows that the last problem reduced to classification of equation of the form (1) w.r.t. transformations (13).

3.4. Reduction to the projective group. From (4), we have that the lifting of transformation (13) to Lie transformation of J^{n}π is defined by

X = \frac{α · x + β}{γ · x + δ} , Y = C · |X′|^{(n−1)/2} · y,

P_{k} = C · ∇^{k}(|X′|^{(n−1)/2} · y), k = 1, 2, …, n,

where ∇ = \frac{1}{DX} · D and D is operator (5). Consider two ODEs of the form (1):

ℰ_{1} = { P_{n} = A_{n−3}(X)P_{n−3} + A_{n−4}(X)P_{n−4} + ⋯ + A_{0}(X)Y }

and

ℰ_{2} = { p_{n} = a_{n−3}(x)p_{n−3} + a_{n−4}(x)p_{n−4} + ⋯ + a_{0}(x)y }.

Suppose, transformation (13) takes ℰ_{1} to ℰ_{2}. This means that

C · ∇^{n}(|X′|^{(n−1)/2} · y) = C · ∑_{i=3}^{n} A_{n−i}(X(x)) · ∇^{n−i}(|X′|^{(n−1)/2} · y).

Theorem 3.7. Let ℰ_{1} and ℰ_{2} be ODEs of the form (1). Then if there exists a point transformation that takes ℰ_{1} to ℰ_{2}, then there exist a point transformation of the form (2)

f(x) = \frac{αx + β}{γx + δ} , f̂(x, y) = |f′|^{(n−1)/2} · y, α, β, γ, δ ∈ ℝ,

that takes ℰ_{1} to ℰ_{2}.

Thus the problem of local contact classification of linear ODEs reduces to that of local classification of ODEs of the form (1) w.r.t. point transformations (2). From (2), we have that the map f ↦ (f, f̂) is an isomorphism from the group of projective transformations of ℝ^{1} to the group of point transformations of the form (2).

4. Classification of linear ODEs

Here we reduce the classification problem for linear ODEs to that for germs of sections of the bundle, related to an ODE, up to projective transformations.

4.1. Bundles of linear ODEs.

4.1.1. The projective group. Denote by G the Lie group of all projective transformations of ℝ^{1}, i.e.,

G = \{ f(x) = \frac{α x + β}{γ x + δ} | α, β, γ, δ ∈ ℝ and det (\begin{matrix}α & β \\ δ & γ \end{matrix}) ≠ 0 \} .

It is easy to check that the set of nonconstant solutions of the equation

(16)  2f′′′f′ − 3(f′′)^{2} = 0

coincides with G. Let

G_{+} = { f ∈ G | f′ > 0 }, G_{−} = { f ∈ G | f′ < 0 }.

Obviously, G_{+} is the connected component of the unit in G, G = G_{+} ∪ G_{−}. Let μ ∈ G_{−} be defined by μ(x) = −x, x ∈ ℝ. Then G_{−} = μ ∘ G_{+}.

Denote by 𝔤 the Lie algebra of G. It is easy to check that 𝔤 as a vector space over ℝ is generated by the vector fields

(17)  ξ_{0} = \frac{∂}{∂x}, ξ_{1} = x\frac{∂}{∂x}, ξ_{2} = x^{2} \frac{∂}{∂x}.

4.1.2. Bundles of linear ODEs of the Laguerre–Forsyth form. Let π: E = ℝ^{1} × ℝ^{n−2} → ℝ^{1} be the product bundle. Denote by x coordinates on the base ℝ^{1} and by a_{n−3}, a_{n−4}, …, a_{0} coordinates on the fiber ℝ^{n−2}.

We identify any linear ODE of the form (1)

ℰ = { p_{n} = a_{n−3}(x)p_{n−3} + a_{n−4}(x)p_{n−4} + ⋯ + a_{0}(x)y }

with the section S_{ℰ} of π defined by the formula

S_{ℰ} : x ↦ (x, a_{n−3}(x), a_{n−4}(x), …, a_{0}(x)).

This identification ℰ ↦ S_{ℰ} is a bijection. We denote by ℰ_{S} the equation corresponding to the section S under this identification.

Let

ℰ_{2} = { P_{n} = A_{n−3}(X)P_{n−3} + A_{n−4}(X)P_{n−4} + ⋯ + A_{0}(X)Y }

be an ODE of form the (1). Subjecting ℰ_{2} to an arbitrary transformation (f, f̂) of the form (2), we obtain linear ODE

ℰ_{1} = { p_{n} = a_{n−3}(x)p_{n−3} + a_{n−4}(x)p_{n−4} + ⋯ + a_{0}(x)y }.

The coefficients of ℰ_{1} are expressed in terms of the coefficients of ℰ_{2} and the projective transformation f by equations of the following form

a_{n−j} = F_{n−j}(A_{n−3}, …, A_{n−j}; \frac{df}{dx}, …, \frac{d^{j+1}f}{dx^{j+1}} ), j = 3, 4, …, n.

Obviously, the coefficients of ℰ_{2} are expressed in terms of the coefficients of ℰ_{1} and the projective transformation f^{−1} by the same equations

(18)  A_{n−j} = F_{n−j}(a_{n−3}, …, a_{n−j}; \frac{df^{−1}}{dX} , …, \frac{d^{j+1}f^{−1}}{dX^{j+1}} ),

j = 3, 4, …, n. Equations (18) define the lifting of any projective transformation f to a diffeomorphism f^{(0)} of the bundle π such that π ∘ f^{(0)} = f ∘ π (in the domain of f^{(0)}).

For any f ∈ G, we define the transformation of sections of π by the formula

(19)  S ↦ f(S) = f^{(0)} ∘ S ∘ f^{−1}.

Now equations (18) can be represented as S_{ℰ_{2}} = f(S_{ℰ_{1}}). Obviously, the following statement holds.

Proposition 4.1. Let ℰ_{1}, ℰ_{2} be equations of the form (1). Then a transformation (f, f̂) of the form (2) takes ℰ_{1} to ℰ_{2} iff f(S_{ℰ_{1}}) = S_{ℰ_{2}}.

Thus, the problem of local classification of linear ODEs of the form (1) up to transformations of the form (2) reduces to classification of germs of sections of π up to a projective transformation of ℝ^{1}.

Since the action of G in ℝ^{1} is transitive, the last problem reduces to classification of germs at 0 ∈ ℝ^{1} of sections of π w.r.t. the isotropy group G_{0} = { f ∈ G | f(0) = 0 } ⊂ G of 0 ∈ ℝ^{1}.

4.1.3. Jet bundles. Let x; a_{n−3}, …, a_{0}; a′_{n−3}, …, a′_{0}; …; a_{n−3}^{(k)}, …, a_{0}^{(k)} be standard coordinates on the jet bundle π_{k}: J^{k}π → ℝ^{1}, k = 0, 1, 2, …, ∞. Any diffeomorphism f^{(0)} f ∈ G can be lifted to the Lie transformation f^{(k)} of J^{k}π, k = 1, 2, …, ∞ by the formula

(20)  f^{(k)}([S]_{p}^{k}) = [f^{(0)} ∘ S ∘ f^{−1}]_{f(p)}^{k}.

Obviously, for any l > m, one has π_{l,m} ∘ f^{(l)} = f^{(m)} ∘ π_{l,m} (in the domains of f^{(l)}). In particular, μ^{(k)} is defined in standard coordinates by the formula

(21)  μ^{(k)}((x, a_{n−j}^{(r)})) = (−x, (−1)^{j+r}a_{n−j}^{(r)}),

j = 3, 4, …, n, r = 0, 1, …, k.

Let G^{(k)} = {f^{(k)} | f ∈ G }, k = 0, 1, 2, …, ∞, and

G_{+}^{(k)} = { f^{(k)} | f ∈ G_{+} }, G_{−}^{(k)} = { f^{(k)} | f ∈ G_{−} }.

Obviously, G_{+}^{(k)} is the connected component of the unit of G^{(k)} and

G^{(k)} = G_{+}^{(k)} ∪ G_{−}^{(k)}, G_{−}^{(k)} = μ^{(k)} ∘ G_{+}^{(k)}.

The lifting of projective transformations of the base ℝ^{1} to diffeomorphisms of J^{k}π generates the lifting of any vector field ξ ∈ 𝔤 to the vector field ξ^{(k)} on J^{k}π. By definition, ξ^{(k)} is the vector field defined by the flow f_{t}^{(k)}, where f_{t} is the flow of ξ. Obviously (π_{l,m})_{∗}(ξ^{(l)}) = ξ^{(m)} for l > m.

Let ξ = φ(x)\frac{∂}{∂x} be an arbitrary element of 𝔤. The vector field ξ^{(∞)} is defined by the formula (see [5])

(22)  ξ^{(∞)} = φD_{x} + �_{ψ},

where

D_{x} = \frac{∂}{∂x} + ∑_{k=0}^{∞} ∑_{j=3}^{n} a_{n−j}^{(k+1)} \frac{∂}{∂a_{n−j}^{(k)}}

is the operator of total derivative over x in J^{∞}π,

�_{ψ} = ∑_{k=0}^{∞} ∑_{j=3}^{n} D_{x}^{k}(ψ_{n−j})\frac{∂}{∂a_{n−j}^{(k)}}

is the evolutionary derivation with ψ = (ψ_{n−3}, …, ψ_{0})^{t} being its generating function. This function is defined in the following way. Let x_{1} = [S]_{x}^{1} ∈ J^{1}π, x = π_{1}(x_{1}); then

(23)  ψ(x_{1}) = (\begin{matrix}ψ_{n−3}(x_{1}) \\ . & . & . & . & . & . & . & . \\ ψ_{0}(x_{1}) \end{matrix}) = \frac{d}{dt}(f_{t}^{(0)} ∘ S ∘ f_{t}^{−1})|_{t=0}(x)

Let S(x) = (x, a_{n−3}(x), …, a_{0}(x)). Then, taking into account that

\frac{df_{t}}{dt} |_{t=0} = φ and φ′′′ = 0, we obtain

(24)  ψ = (\begin{matrix}−3a_{n−3}φ′ − a_{n−3}^{(1)}φ \\ −\frac{3(n−3)}{2}a_{n−3}φ′′ − 4a_{n−4}φ′ − a_{n−4}^{(1)}φ \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ −\frac{(k−1)(n−(k−1))}{2}a_{n−k+1}φ′′ − ka_{n−k}φ′ − a_{n−k}^{(1)}φ \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ −\frac{n−1}{2} · 1 · a_{1}φ′′ − na_{0}φ′ − a_{0}^{(1)}φ \end{matrix})

Now it follows from (22) and (24) that for any k = 0, 1, 2, …, ∞,

(25)  ξ_{0}^{(k)} = \frac{∂}{∂x},
(26)  ξ_{1}^{(k)} = x\frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=3}^{n}(j + r)a_{n−j}^{(r)} \frac{∂}{∂a_{n−j}^{(r)}} ,

ξ_{2}^{(k)} = x^{2} \frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=3}^{n}[2x(j + r)a_{n−j}^{(r)} \frac{∂}{∂a_{n−j}^{(r)}}

+ (j − 1)(n − (j − 1))a_{n−(j−1)}^{(r)} \frac{∂}{∂a_{n−j}^{(r)}}

+ (2j + r − 1)ra_{n−j}^{(r−1)} \frac{∂}{∂a_{n−j}^{(r)}} ], (27)

where a_{n−2}^{(r)} = 0.

4.2. Projective symmetries. Let S be a section of π and ξ be a vector field from 𝔤. By f_{t} we denote the flow of ξ. We say that ξ is a projective symmetry of S if one of the following equivalent conditions is fulfilled:

(1) the vector field ξ^{(0)} is tangent to the image L_{S}^{(0)} of S;

(2) f_{t}(S) =^{def} f_{t}^{(0)} ∘ S ∘ f_{t}^{−1} = S;

(3) \frac{d}{dt}(f_{t}(S)|_{t=0} = 0.

Denote by Prj S the Lie algebra of all projective symmetries of S.

Proposition 4.2. Consider the section S(x) = (x, a_{n−3}(x), …, a_{0}(x))

and let ξ = φ(x)\frac{∂}{∂x}. Then:

1. ξ is a projective symmetry of S iff φ(x) is a solution of system (14);

2. dim Prj S is equal to either 3 or 1, or 0;

3. φ(x)\frac{∂}{∂x} is a projective symmetry of S iff φ(x)\frac{∂}{∂x} + \frac{n−1}{2}φ′y \frac{∂}{∂y}

is a point symmetry of the equation ℰ_{S};

4. dim Pnt ℰ_{S} = dim Prj S_{ℰ} + n + 1.

Proof. The first statement follows from (23), (24), and (14). The second one follows from (17) and (14). From Proposition 4.1, we obtain the third statement. The last statement follows from the results of [8]. □

4.3. Invariant subbundles. Let E^{i}, i = n − 3, n − 4, …, 0, −1, be the subspaces of the total space E of π defined by

E^{i} = { (x, a_{n−3}, a_{n−4}, …, a_{0}) ∈ E | a_{j} = 0 if j > i }.

Consider the subbundle π|_{E^{i}}: E^{i} → ℝ of the bundle π.

Proposition 4.3. Every subbundle E^{i} is G^{(0)}-invariant.

Proof. From (25)–(27), we have that the restrictions of the vector fields ξ_{0}^{(0)}, ξ_{1}^{(0)}, ξ_{2}^{(0)} to E^{i} are defined by

(28)  ξ_{0}^{(0)}|_{E^{i}} = \frac{∂}{∂x},
(29)  ξ_{1}^{(0)}|_{E^{i}} = x\frac{∂}{∂x} − ((n − i)a_{i} \frac{∂}{∂a_{i}} + ⋯ + na_{0} \frac{∂}{∂a_{0}} ),
ξ_{2}^{(0)}|_{E^{i}} = x^{2} \frac{∂}{∂x} − 2x((n − i)a_{i} \frac{∂}{∂a_{i}} + ⋯ + na_{0} \frac{∂}{∂a_{0}} )

− (i(n − i)a_{i} \frac{∂}{∂a_{i−1}} + ⋯ + (n − 1)a_{1} \frac{∂}{∂a_{0}} ). (30)

Clearly, ξ_{0}^{(0)}|_{E^{i}}, ξ_{1}^{(0)}|_{E^{i}}, ξ_{2}^{(0)}|_{E^{i}} are tangent to E^{i}. Therefore every sub-

bundle E^{i} is G_{+}^{(0)}-invariant. From (21), we have μ^{(0)}(E^{i}) = E^{i}. □

Thus, we have the following sequence of the G^{(0)}-invariant subbundles: E = E^{n−3} ⊃ E^{n−4} ⊃ ⋯ ⊃ E^{0} ⊃ E^{−1}. Let E_{i}, i = n − 3, n − 4, …, 0, −1, be the subsets of the total space E of π defined by

E_{i} = E^{i} \ E^{i−1} if i ≥ 0 and E_{−1} = E^{−1}.

Consider the subbundle π^{i} = π|_{E_{i}}: E_{i} → ℝ of the bundle π.

Corollary 4.4. Every subbundle E_{i} is G^{(0)}-invariant.

Thus, E is the union

(31)  E = E_{n−3} ∪ E_{n−4} ∪ ⋯ ∪ E_{0} ∪ E_{−1}

of nonintersecting G^{(0)}-invariant subbundles.

The following proposition is needed for the sequel.

Proposition 4.5. The symmetric differential n − i-form ω_{i} = a_{i} dx^{n−i} on E_{i} is G^{(0)}-invariant.

Proof. Let us calculate the Lie derivatives of ω_{i} w.r.t. vector fields ξ_{0}^{(0)}|_{E_{i}}, ξ_{1}^{(0)}|_{E_{i}}, ξ_{2}^{(0)}|_{E_{i}}. From (28)–(30), we have

ξ_{0}^{(0)}|_{E_{i}}(ω_{i}) = 0,
ξ_{1}^{(0)}|_{E_{i}}(ω_{i}) = a_{i}(n − i) dx^{n−i} − (n − i)a_{i} dx^{n−i} = 0,
ξ_{2}^{(0)}|_{E_{i}}(ω_{i}) = a_{i}(n − i)2x dx^{n−i} − 2x(n − i)a_{i} dx^{n−i} = 0.

Hence ω_{i} is G_{+}^{(0)}-invariant. It follows from (21) that (μ^{(0)})^{∗}(ω_{i}) = ω_{i}. Thus ω_{i} is G^{(0)}-invariant. □

This result gives us the transformation law for the first nonzero component:

Corollary 4.6. Let θ_{0} = (x, 0, …, 0, a_{i}, …, a_{0}) ∈ E_{i}, let f ∈ G, and f^{(0)}(θ_{0}) = (f(x), 0, …, 0, A_{i}, …, A_{0}) ∈ E_{i}. Then a_{i} = (f′(x))^{n−i}A_{i}.

4.4. Classification of regular germs.

4.4.1. Regular germs. Let S be a section of π, p be a point in a domain of S. Denote by {S}_{p} the germ of S at p. Let {S_{1}}_{p_{1}} and {S_{2}}_{p_{2}} be germs of sections S_{1} and S_{2} respectively. We say that {S_{1}}_{p_{1}} and {S_{2}}_{p_{2}} are G_{+}(G)-equivalent if there exists f ∈ G_{+}(G) with { f(S_{1}) }_{f(p_{1})} = {S_{2}}_{p_{2}}. A germ {S}_{p} is regular of class i if there exist a neighborhood V of p and subbundle E_{i} with Im S|_{V} ⊂ E_{i}.

If {S}_{p} is a regular germ of class i ≥ 0, then one has S(x) = (x, 0, …, 0, a_{i}(x), …, a_{0}(x)) in a neighborhood of p. For this reason, we will often denote {S}_{p} by { a_{i}, …, a_{0} }_{p}. If {S}_{p} is a regular germ, then p is a regular point of S (regular point of ℰ_{S}).

Let ℱ be the set of all regular germs at 0 ∈ ℝ^{1} of sections of π. Obviously, dimension of the algebra of projective symmetries of a section of π^{i} is an invariant of transformations (19). Then ℱ = ℱ_{3} ∪ ℱ_{1} ∪ ℱ_{0}, where the subsets

ℱ_{r} = { {S}_{0} | dim Prj S = r }, r = 3, 1, 0,

are nonintersecting G_{0}-invariant subsets. Obviously, ℱ_{3} consists of the germ of the zero section only.

4.4.2. Classification. Here we classify regular germs from ℱ_{r}, r = 0, 1, w.r.t. G_{0}. Recall that

G_{0} = \{ f ∈ G | f(0) = 0 \} = \{ \frac{βx}{γx + 1}, β, γ ∈ ℝ, β ≠ 0 \} .

By definition, put G_{0+} = G_{0} ∩ G_{+}, G_{0−} = G_{0} ∩ G_{−}. Then G_{0} = G_{0+} ∪ G_{0−} and G_{0−} = μ ∘ G_{0+}.

Let ℱ_{r,i} ⊂ ℱ_{r} be the subset of all regular germs of class i. It follows from Corollary 4.4 that ℱ_{r,i} is G_{0}-invariant. Thus ℱ_{r} is the union ℱ_{r} = ∪_{i=0}^{n−3}ℱ_{r,i} of nonintersecting invariant subsets.

Let ℝ_{+} = { a ∈ ℝ | a > 0 } and let ℝ_{−} = { a ∈ ℝ | a < 0 }. Define the map ℓ_{r,i}: ℱ_{r,i} → (ℝ \{0}) × ℝ by the formula

{ a_{i}, …, a_{0} }_{0} ↦ (a_{i}(0), a′_{i}(0)).

Consider the action G_{0+} × ℱ_{r,i} → ℱ_{r,i}, (f, { S}_{0}) ↦ {f(S)}_{0}, of the group G_{0+} on ℱ_{r,i}. This action divides ℱ_{r,i} into nonintersecting orbits. Let Θ be one of these orbits.

Proposition 4.7. The map ℓ_{r,i}|_{Θ} is a bijection from the orbit Θ either to (ℝ_{+}) × ℝ or to (ℝ_{−}) × ℝ.

Proof. Let \{S\}_{0} = \{ a_{i}, …, a_{0} \}_{0} ∈ Θ, let f = \frac{βx}{γx+1} be an element of G_{0+}, and let f({S}_{0}) = { A_{i}, …, A_{0} }_{0}. Then from Corollary 4.6 we have a_{i}(x) = (f′(x))^{n−i}A_{i}(f(x)). Since β > 0, the points A_{i}(0) and a_{i}(0) belong either to ℝ_{+} or to ℝ_{−}. This means that ℓ_{r,i}(Θ) belongs either to (ℝ_{+}) × ℝ or to (ℝ_{−}) × ℝ.

Assume ℓ_{r,i}(Θ) ⊂ (ℝ_{+}) × ℝ and prove that the map ℓ_{r,i}|_{Θ} : Θ → (ℝ_{+}) × ℝ is an injection. Let {S_{1}}_{0} = { a_{i}, …, a_{0} }_{0} ∈ Θ and {S_{2}}_{0} = \{ A_{i}, …, A_{0} \}_{0} ∈ Θ. Then there exist a transformation f(x) = d\frac{βx}{γx+1} ∈ G_{0+} that takes {S_{1}}_{0} to {S_{2}}_{0}. This means that in some neighborhood of 0, we have

a_{i}(x) = (f′(x))^{n−i}A_{i}(f(x)).

Differentiating both sides of this equation w.r.t. x, we obtain a′_{i}(x) = −2(n − i)(f′(x))^{n−i−1}f′′(x)A_{i}(f(x))

+ (f′(x))^{n−i}A′_{i}(f(x))f′(x).

Suppose a_{i}(0) = A_{i}(0) and a′_{i}(0) = A′_{i}(0). Then from the last two equations we have

\begin{cases}A_{i}(0) = & β^{n−i}A_{i}(0), \\ A′_{i}(0) = & −2(n − i)β^{n−i}γA_{i}(0) + β^{n−i+1}A′_{i}(0). \end{cases}

From this system, we obtain β = 1 and γ = 0. This means that f is the identical transformation. It follows that {S_{1}}_{0} = {S_{2}}_{0}. Therefore ℓ_{r,i}|_{Θ} is an injection.

Let us prove that ℓ_{r,i}|_{Θ} is a surjection. Let (a, a′) ∈ (ℝ_{+}) × ℝ and let {S_{1}}_{0} = {A_{i}, …, A_{0}}_{0} ∈ Θ. Obviously the equations

\begin{cases}a = & β^{n−i}A_{i}(0), \\ a′ = & −2(n − i)β^{n−i}γA_{i}(0) + β^{n−i+1}A′_{i}(0). \end{cases}

define the transformation f(x) = \frac{βx}{γx + 1} ∈ G_{0+} uniquely. Clearly,

{S_{2}}_{0} = {(f^{−1})^{(0)} ∘ S_{1} ∘ f}_{0} ∈ Θ and ℓ_{r,i}({S_{2}}_{0}) = (a, a′).

Obviously, the proof for the case ℓ_{r,i}(Θ) ⊂ (ℝ_{−}) × ℝ is the same. □

Let ℒ_{r,i}^{+} = ℓ_{r,i}^{−1}((1, 0)) and let ℒ_{r,i}^{−} = ℓ_{r,i}^{−1}((−1, 0)). Denote by ℳ_{r,i} the subset of ℒ_{r,i}^{+} ∪ ℒ_{r,i}^{−} defined in the following way:

(1) if i = 0 , then ℳ_{r,0} = ℒ_{r,0}^{+} ∪ ℒ_{r,0}^{−},

(2) if i > 0, then ℳ_{r,i} consists of all germs { a_{i}, a_{i−1}, …, a_{0} }_{0} from ℒ_{r,i}^{+} ∪ ℒ_{r,i}^{−} satisfying one of the following conditions:

(a) a_{i−j}(0) = 0 for all odd numbers j with 1 ≤ j ≤ i,

(b) there exists an odd number r with 1 ≤ r ≤ i such that a_{i−r}(0) > 0 and if r > 1, then a_{i−j}(0) = 0 for all odd numbers j with 1 ≤ j < r.

Classification of regular germs from the family ℱ_{r,i} is given by

Theorem 4.8. 1. The set ℒ_{r,i}^{+} ∪ ℒ_{r,i}^{−} is a family of all germs from

ℱ_{r,i} nonequivalent w.r.t. G_{0+}.

2. If n − i is odd, then ℒ_{r,i}^{+} is a family of all germs from ℱ_{r,i} nonequivalent w.r.t. G_{0}.

3. If n − i is even, then ℳ_{r,i} is a family of all germs from ℱ_{r,i} nonequivalent w.r.t. G_{0}.

Proof. From Proposition 4.7, we have the first statement. From (21), we have μ(ℒ_{r,i}^{−}) = ℒ_{r,i}^{+} from where the second statement follows. The third statement also follows immediately from (21). □

Corollary 4.9. Classification of regular germs of sections is as follows:

1. The family of germs of the form

{ ±1 + b(x)x^{2}, a_{i−1}(x), …, a_{0}(x) }_{0}

is a family of all regular germs of class i nonequivalent w.r.t. G_{0+}.

2. If n − i is odd, then the family of germs of the form

{ 1 + b(x)x^{2}, a_{i−1}(x), …, a_{0}(x) }_{0}

is a family of all regular germs of class i nonequivalent w.r.t. G_{0}.

3. If n − i is even, then the family of germs of the form

{ ±1 + b(x)x^{2}, a_{i−1}(x), …, a_{0}(x) }_{0},

satisfying to one of the following conditions:

a) a_{i−j}(0) = 0 for all odd numbers j with 1 ≤ j ≤ i,

b) there exist an odd number r with 1 ≤ r ≤ i such that a_{i−r}(0) > 0 and if r > 1, then a_{i−j}(0) = 0 for all odd numbers j with 1 ≤ j < r

is the family of all regular germs of class i nonequivalent w.r.t. G_{0}.

5. The equivalence problem

5.1. Scalar differential invariants of linear ODEs. Here we calculate scalar differential invariants of linear ODEs. For a general theory of scalar differential invariants refer to [1, 10].

It was proved in Subsection 4.3 that the bundle π^{i} = π|_{E_{i}}: E_{i} → ℝ is G^{(0)}-invariant. It follows that the jet bundles J^{k}π^{i} are G^{(k)}-invariant, k = 1, 2 …, ∞. Hence J^{k}π^{i} are invariant w.r.t. the subgroup G_{+}^{(k)} ⊂ G^{(k)}, k = 0, 1, 2 …, ∞.

A function I ∈ C^{∞}(J^{k}π^{i}) is called a scalar differential invariant of G (G_{+}) if

(f^{(k)})^{∗}I = I ∀f ∈ G (G_{+}).

Let I be a scalar differential invariant of G (G_{+}) and S be a section of π^{i}. By definition, put I(S) = (j_{k}S)^{∗}I. For any f ∈ G, we have

(32)  I(f(S)) ∘ f = I(S).

Indeed,

I(f(S)) = (j_{k}f(S))^{∗}I = (j_{k}(f^{(0)} ∘ S ∘ f^{−1}))^{∗}I

= (f^{(k)} ∘ j_{k}S ∘ f^{−1})^{∗}I = (f^{−1})^{∗} ∘ (j_{k}S)^{∗} ∘ (f^{(k)})^{∗}I

= (f^{−1})^{∗} ∘ (j_{k}S)^{∗}I = (f^{−1})^{∗}I(S) = I(S) ∘ f^{−1}.

Let S be a section of π admitting a 1-dimensional algebra of projective symmetries. Then I(S) is a constant for any scalar differential invariant I. Indeed, let ξ be a projective symmetry of S and let f_{t} be its flow. Then

I(S) = I(f_{t}(S)) = I(f_{t}(S)) ∘ f_{t} = I(S) ∘ f_{t}.

It is not hard to prove that I ∈ C^{∞}(J^{k}π^{i}) is a scalar differential invariant of G_{+} iff I is a solution of the system of linear PDEs

(33)  \begin{cases}ξ̄_{0}^{(k)}(I) & = 0 \\ ξ̄_{1}^{(k)}(I) & = 0 \\ ξ̄_{2}^{(k)}(I) & = 0, \end{cases}

where ξ̄_{0}^{(k)}, ξ̄_{1}^{(k)} ξ̄_{2}^{(k)} are the restrictions of ξ_{0}^{(k)}, ξ_{1}^{(k)}, ξ_{2}^{(k)} to J^{k}π^{i}. From (25)–(27), we have

ξ̄_{0}^{(k)} = ξ_{0}^{(k)}|_{J^{k}π^{i}} = \frac{∂}{∂x}, (34)

ξ̄_{1}^{(k)} = ξ_{1}^{(k)}|_{J^{k}π^{i}} = x\frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=i}^{0}(n − j + r)a_{j}^{(r)} \frac{∂}{∂a_{j}^{(r)}} , (35)

ξ̄_{2}^{(k)} = ξ_{2}^{(k)}|_{J^{k}π^{i}} = x^{2} \frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=0}^{i}[2x(n − j + r)a_{j}^{(r)} \frac{∂}{∂a_{j}^{(r)}}

(36) + (n − j − 1)(j + 1)a_{j+1}^{(r)} \frac{∂}{∂a_{j}^{(r)}} + (2(n − j) + r − 1)ra_{j}^{(r−1)} \frac{∂}{∂a_{j}^{(r)}} ].

Denote by 𝒜_{i}^{k} the algebra of scalar differential invariants of G_{+} on J^{k}π^{i}. We identify A_{i}^{k} with its image (π_{l,k}^{i})^{∗}(𝒜_{i}^{k}), l > k. As a result, we have the following filtration

A_{i} = A_{i}^{∞} ⊃ ⋯ ⊃ A_{i}^{k} ⊃ ⋯ ⊃ A_{i}^{1} ⊃ A_{i}^{0}.

Let 𝒟_{i}^{k} be the distribution on J^{k}π^{i} generated by vector fields ξ̄_{0}^{(k)}, ξ̄_{1}^{(k)}, ξ̄_{2}^{(k)}. From (34)–(36), we have that dim 𝒟_{i}^{k} = 2 if i = 0 and k = 0 otherwise dim 𝒟_{i}^{k} = 3.

Denote by N_{i}^{k} the number of functionally independent scalar differential invariant in A_{i}^{k}. Clearly,

N_{i}^{k} = dim J^{k}π^{i} − dim 𝒟_{i}^{k}.

It is easy to prove that

(37)  N_{0}^{0} = 0, N_{0}^{1} = 0, N_{0}^{k} = k − 1,  if k ≥ 2,
(38)  N_{1}^{0} = 0, N_{1}^{k} = 2k,  if k ≥ 1,
(39)  N_{i}^{0} = i − 1, N_{i}^{k} = (k + 1)(i − 1) + 2k,  if k ≥ 1.

Consider the vector field on J^{∞}π^{i}

(40)  ζ_{i} = |a_{i}|^{−1/(n−i)}D̄_{x},

where D̄_{x} = D_{x}|_{J^{∞}π^{i}} = ∂/∂x+∑_{r=0}^{∞} ∑_{j=i}^{0} a_{j}^{(r+1)}∂/∂a_{j}^{(r)} is the operator of total derivative over x restricted to J^{∞}π^{i}.

Proposition 5.1. The vector field ζ_{i} is invariant w.r.t. G_{+}^{(∞)}.

Proof. By ξ̄_{r}^{(∞)}, r = 0, 1, 2, we denote the restriction of ξ_{r}^{(∞)} to J^{∞}π^{i}. Let us check that [ζ_{i}, ξ̄_{r}^{(∞)}] = 0 for all r. By (25), we have

[ζ_{i}, ξ̄_{0}^{(∞)}] = [|a_{i}|^{−1/(n−i)}D̄_{x}, \frac{∂}{∂x}] = 0.

Using (24), consider the vector fields ξ̄_{1}^{(∞)} and ξ̄_{2}^{(∞)} in the form (22):

ξ̄_{1}^{(∞)} = xD̄_{x} + ¯�_{((n−i)a_{i}+xa_{i}^{(1)})} ,
ξ̄_{2}^{(∞)} = x^{2}D̄_{x} + ¯�_{((i+1)(n−i+1)a_{i+1}+2(n−i)xa_{i}+x^{2}a_{i}^{(1)})} ,

where ¯�_{ψ} is the restriction of �_{ψ} on J^{∞}π^{i}. Now taking into account that [D̄_{x}, ¯�_{ψ} ] = 0 for any ψ, we easily obtain that [ζ_{i}, ξ̄_{1}^{(∞)}] = 0 and [ζ_{i}, ξ̄_{2}^{(∞)}] = 0. □

Obviously, for any I ∈ 𝒜_{i}, its Lie derivative ζ_{i}(I) ∈ 𝒜_{i}. Thus, ζ_{i} and I generate the sequence I, ζ_{i}(I), …, ζ_{i}^{k}(I), … of scalar differential invariants from 𝒜_{i}.

Theorem 5.2. The algebra 𝒜_{i} is generated by the following free generators

ζ_{i}^{k}(I_{i−m}), m = 0, 1, …, i, k = 0, 1, 2, …,

where

I_{i} = [2a_{i}a_{i}^{(2)} − \frac{2(n − i) + 1}{n − i}(a_{i}^{(1)})^{2}] · (a_{i})^{−2(n−i+1)/(n−i)}; (41)

I_{i−1} = [a_{i−1} − \frac{i}{2} a_{i}^{(1)}] · |a_{i}|^{−(n−i+1)/(n−i)}; (42)

for 2 ≤ m ≤ i,

I_{i−m} = [a_{i−m} + \frac{(−1)^{m}}{m!} ∏_{r=n−i+1}^{n−i+m−1} \frac{(n − r)r}{(n − i)i}(a_{i})^{1−m}(a_{i−1})^{m}

+ ∑_{l=n−i+1}^{n−i+m−1} \frac{(−1)^{n−i+m−l}}{(n − i + m − l)!} ∏_{r=l}^{n−i+m−1} \frac{(n − r)r}{(n − i)i}(a_{i})^{i−n+l−m} ·

· (a_{i−1})^{n−i+m−l}a_{n−l}] · |a_{i}|^{−(n−i+m)/(n−i)}. (43)

Proof. It is not hard to check that I_{i}, …, I_{0} are solutions of (33).

Let i = 0, then I_{0} ∈ 𝒜_{0}^{2}. For any k = 0, 1, 2, …, the invariants I_{0}, ζ_{0}(I_{0}), ζ_{0}^{2}(I_{0}), …, ζ_{0}^{k}(I_{0}) belong to 𝒜_{0}^{k+2} and are functionally independent. The number of them equals (k +2)−1. Now from (37) we obtain that (k + 2) − 1 = N_{0}^{k+2}. This concludes the proof for i = 0.

Suppose i ≥ 1. We have

ζ_{i}(I_{i−1}) = [−\frac{i}{2}|a_{i}|a_{i}^{(2)} + …] · (a_{i})^{−2(n−i+1)/(n−i)}

The manifold J^{∞}π^{i} has two connected components defined by the inequalities a_{i} > 0 and a_{i} < 0. Comparing ζ_{i}(I_{i−1}) with I_{i}, we can define the scalar differential invariant J ∈ 𝒜_{i}^{1} by the formula

J = \begin{cases}I_{i} + \frac{4}{i}ζ_{i}(I_{i−1}), & if a_{i} > 0 \\ I_{i} − \frac{4}{i}ζ_{i}(I_{i−1}), & if a_{i} < 0. \end{cases}

It is easy to see that

J = [\frac{4}{i}a_{i}a_{i−1}^{(1)} − \frac{4(n − i + 1)}{i(n − i)}a_{i}^{(1)}a_{i−1}

+ \frac{1}{n − i}(a_{i}^{(1)})^{2}] (a_{i})^{−2(n−i+1)/(n−i)}.

Let i = 1. Then I_{i−1}, J ∈ 𝒜_{i}^{1} and they are functionally independent. The invariants

I_{i−1}, J, ζ_{i}(I_{i−1}), ζ_{i}(J), …, ζ_{i}^{k}(I_{i−1}), ζ_{i}^{k}(J)

belong to 𝒜_{i}^{k+1}, k = 0, 1, 2, …, they are functionally independent, and the number of them equals 2(k + 1). Now from (38), we obtain 2(k + 1) = N_{1}^{k+1}. This concludes the proof for i = 1.

Let i > 1. Then the invariants I_{i−2}, …, I_{0} are functionally independent and they belong to 𝒜_{i}^{0}. The invariants I_{i−2}, …, I_{0}, I_{i−1}, J are functionally independent and lie in 𝒜_{i}^{1}. Finally, the invariants

I_{i−2}, …, I_{0}, I_{i−1}, J, …, ζ_{i}^{k}(I_{i−2}), …, ζ_{i}^{k}(I_{0}), ζ_{i}^{k}(I_{i−1}), ζ_{i}^{k}(J)

are functionally independent, they belong to 𝒜_{i}^{k}, k = 1, 2, …, and the number of them is equal to (k + 1)(i − 1) + 2k. Now from (39), we obtain (k + 1)(i − 1) + 2k = N_{i}^{k}. □

Remark 5.3. From (21), we obtain that

I_{i} = [2a_{i}a_{i}^{(2)} − \frac{2(n − i) + 1}{n − i}(a_{i}^{(1)})^{2}] · (a_{i})^{−2(n−i+1)/(n−i)}

is an invariant of the group G.

5.2. The equivalence problem of linear ODEs. Let

S_{1} : x ↦ (x, 0, …, 0, a_{i}(x), …, a_{0}(x))

and

S_{2} : X ↦ (X, 0, …, 0, A_{i}(X), …, A_{0}(X))

be sections of π^{i} in neighborhoods of points p ∈ ℝ and P ∈ ℝ respectively. The sections S_{1} and S_{2} are locally G_{+}-equivalent at (p, P) if there exist f ∈ G_{+} and neighborhoods V of p and U of P such that

f(p) = P and f(S_{1}|_{V}) =^{def} f^{(0)} ∘ S_{1}|_{V} ∘ f^{−1} = S_{2}|_{U}. G-equivalence is

defined in the same way.

Theorem 5.4. Sections S_{1} and S_{2} of π^{i} are locally G_{+}-equivalent at (p, P) iff the following conditions hold:

1. a_{i}(p) · A_{i}(P) > 0,

2. the solution f of the Cauchy problem

(44)  \{ \begin{matrix}f′ = |a_{i}(x)|^{1/(n−i)} · |A_{i}(f(x))|^{−1/(n−i)}, \\ f(p) = P \end{matrix}

satisfies to the equations

(45)  I_{m}(S_{2}) ∘ f = I_{m}(S_{1}), m = i, i − 1, …, 0

in some neighborhood of p.

Proof. Suppose S_{1} and S_{2} are locally G_{+}-equivalent at (p, P). Then there exist f ∈ G_{+} and neighborhoods V of p and U of P such that f(p) = P and f(S_{1}|_{V}) = S_{2}|_{U}. Consider the symmetric differential n − i-form ω_{i} on E_{i} (Proposition 4.5). We have

(46)  f^{∗}(S_{2}^{∗}(ω_{i})) = S_{1}^{∗}(ω_{i}).

Indeed,

f^{∗}(f(S_{1})^{∗}(ω_{i})) = f^{∗}((f^{(0)} ∘ S_{1} ∘ f^{−1})^{∗}(ω_{i}))

= S_{1}^{∗}((f^{(0)})^{∗}(ω_{i})) = S_{1}^{∗}(ω_{i}).

Equality (46) means that a_{i}(x) = (f′)^{n−i}A_{i}(f(x)). It also follows that either a_{i}(p), A_{i}(P) > 0 or a_{i}(p), A_{i}(P) < 0 and that f is a solution of Cauchy problem (44). Further, from (32), we have that equations (45) hold.

Conversely, let a_{i}(p) · A_{i}(P) > 0, f be a solution of Cauchy problem (44), and f be a solution of equations (45). Let us show that f ∈ G_{+}. From (44), we can obtain f′′ and f′′′ in terms of a_{i}, A_{i} and their 1-st and 2-nd derivatives:

(47) f′′ = \frac{1}{r}[ |a_{i}|^{\frac{1−r}{r}} |A_{i}|^{\frac{−1}{r}} sgn(a_{i})a′_{i} − |a_{i}|^{\frac{2}{r}} |A_{i}|^{\frac{−r−2}{r}} sgn(A_{i})A′_{i}] ,

f′′′ = \frac{1}{r}[ \frac{1 − r}{r}|a_{i}|^{\frac{1−2r}{r}} |A_{i}|^{\frac{−1}{r}} (a′_{i})^{2} + |a_{i}|^{\frac{1−r}{r}} |A_{i}|^{\frac{−1}{r}} sgn(a_{i})a′′_{i}

− \frac{3}{r}|a_{i}|^{\frac{2−r}{r}} |A_{i}|^{\frac{−r−2}{r}} a′_{i}A′_{i} + \frac{2 + r}{r}|a_{i}|^{\frac{3}{r}} |A_{i}|^{\frac{−2r−3}{r}} (A′_{i})^{2}

−|a_{i}|^{\frac{3}{r}} |A_{i}|^{\frac{−r−3}{r}} sgn(A_{i})A′′_{i}] , (48)

where r = n − i. Substituting expressions (44), (47), and (48) for f′, f′′, and f′′′ in the left-hand side of equation (16), we obtain

2f′′′f′ − 3(f′′)^{2} = \frac{1}{n − i}|a_{i}|^{4/(n−i)}|A_{i}|^{−2/(n−i)}(I_{i}(S_{1}) − I_{i}(S_{2}) ∘ f) = 0.

Thus, f ∈ G_{+}.

Let S_{3} = f(S_{1}). Then

I_{m}(S_{3}) ∘ f = I_{m}(f(S_{1})) ∘ f = I_{m}(S_{1}) = I_{m}(S_{2}) ∘ f,

m = i, i − 1, …, 0. Hence

I_{m}(S_{3}) = I_{m}(S_{2}) m = i, i − 1, …, 0.

Let S_{3} : X ↦ (X, 0, …, 0, B_{i}, …, B_{0}). Then one obviously has B_{i} = (f′)^{−(n−i)}a_{i} = A_{i} in some neighborhood of P. It now follows from I_{i−1}(S_{3}) = I_{i−1}(S_{2}) that B_{i−1} = A_{i−1} in this neighborhood. From I_{i−2}(S_{3}) = I_{i−2}(S_{2}), we have B_{i−2} = A_{i−2} in this neighborhood and so on. Thus S_{3} = S_{2} in some neighborhood of P. □

Corollary 5.5. Sections S_{1}, S_{2} of π^{i} are locally G-equivalent at (p, P) iff S_{1} locally G_{+}-equivalent either to S_{2} at (p, P) or to μ(S_{2}) at (p, −P).

Corollary 5.6. Let the invariants I_{m}(S_{1}), I_{m}(S_{2}), m = i, i − 1, …, 0 be constants. Then S_{1}, S_{2} are G_{+}-locally equivalent at (p, P) iff the following conditions hold:

1. either a_{i}(p) · A_{i}(P) > 0,

2. I_{m}(S_{1}) = I_{m}(S_{2}), m = i, i − 1, …, 0.

Proposition 5.7. Let S be a section of π^{i}. Then dim Prj S = 1 iff I_{i}(S), I_{i−1}(S), …, I_{0}(S) are constants.

Proof. The necessity was proved in the beginning of this subsection. Prove the sufficiency. Let S(x) = (x, 0, …, 0, a_{i}(x), …, a_{0}(x)) and invariants I_{i}(S), I_{i−1}(S), …, I_{0}(S) be constants. From Proposition 4.2, we have that a vector field φ(x)∂/∂x ∈ 𝔤 is a projective symmetry of section S iff φ(x) is a solution of system (14):

(49)  \begin{cases}φ′′′ = 0 \\ (n − i)a_{i}φ′ + a′_{i}φ = 0 \\ \frac{(n − j − 1)(j + 1)}{2}a_{j+1}φ′′ + (n − j)a_{j}φ′ + a′_{j}φ = 0, \end{cases}

j = i − 1, i − 2, …, 0. From the 2-nd equation of the system, we have that φ = C|a_{i}|^{−1/(n−i)}, C ∈ ℝ.

From the identities dI_{m}(S)/dx ≡ 0, m = i, i−1, …, 0, we can obtain by direct calculations that |a_{i}|^{−1/(n−i)} is a solution of system (49). Thus, the vector field |a_{i}|^{−1/(n−i)}∂/∂x is a symmetry of S. □

5.3. The canonical forms of linear ODEs. In this subsection, we use the invariants I_{i}, I_{i−1}, …, I_{0} to obtain canonical form of all nonequivalent regular germs at 0 of sections of π.

Denote by 𝒴_{i}^{±} the system of equations on unknown section S: x ↦ (x, 0, …, 0, a_{i}(x), …, a_{0}(x)) of π^{i} in a neighborhood of 0 ∈ ℝ^{1}:

\begin{cases}I_{i}(S)(x) = K_{i}(x), a_{i}(0) = ±1 , a′_{i}(0) = 0 \\ I_{i−1}(S)(x) = K_{i−1}(x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ I_{0}(S)(x) = K_{0}(x), \end{cases}

where K_{i}, K_{i−1}, …, K_{0} are smooth functions defined in neighborhoods of 0 ∈ ℝ.

There exists a unique solution of the system 𝒴_{i}^{±}. Indeed, it follows from (41) that the first equation of the system is a Cauchy problem for 2-nd order ODE on unknown function a_{i}(x). There exists a unique solution, say a_{i}(x, K_{i}), to this problem. From (42), (43), we have that all other unknown functions a_{i−1}, …, a_{0} are defined recursively, uniquely, and in an explicit form from the next equations of the system. Denote them by a_{i−1}(x, K_{i}, K_{i−1}), …, a_{0}(x, K_{i}, K_{i−1}, …, K_{0}). Thus, the section

S(K_{i}, …, K_{0}): x ↦ (x, 0, …, 0, a_{i}(x, K_{i}), …, a_{0}(x, K_{i}, …, K_{0}))

is a unique solution of system 𝒴_{i}^{+} (𝒴_{i}^{−}).

Below we denote by S^{+}(K_{i}, …, K_{0}) and S^{−}(K_{i}, …, K_{0}) the solutions of 𝒴_{i}^{+} and 𝒴_{i}^{−} respectively. Now from Theorem 4.8, we obtain canonical forms of nonequivalent linear ODEs:

Theorem 5.8. 1. The set { S^{±}(K_{i}, …, K_{0}) }_{0} is the family of all

regular germs of class i nonequivalent w.r.t. G_{0+}.

2. If n − i is odd, then { S^{+}(K_{i}, …, K_{0}) }_{0} is the family of all regular germs of class i nonequivalent w.r.t. G_{0}.

3. If n − i is even, then { S^{±}(K_{i}, …, K_{0}) }_{0} satisfying one of the conditions:

a) a_{i−j}(0, K_{i}(0), …, K_{i−j}(0)) = 0 for all odd j, 1 ≤ j ≤ i,

b) there exist an odd number r with 1 ≤ r ≤ i such that

a_{i−r}(0, K_{i}(0), …, K_{i−r}(0)) > 0

and if r > 1, then for all odd numbers j with 1 ≤ j < r, a_{i−j}(0, K_{i}(0), …, K_{i−j}(0)) = 0

is a family of all regular germs of class i nonequivalent w.r.t. G_{0}.

Let {K_{i}, …, K_{0}}_{0} be germ of the vector function (K_{i}, …, K_{0}) at 0 ∈ ℝ^{1}, ℳ be the set of all these germs, and ℳ_{1} be the subset of ℳ consisting of germs of constant vector functions. Then canonical forms of nonequivalent linear ODEs with 1 and 0-dimensional algebras of projective symmetries are given by

Corollary 5.9. 1. Suppose all vector functions (K_{i}, …, K_{0}) are constant in Theorem 5.8; then the theorem gives the family of all regular germs of class i from ℱ_{1} nonequivalent w.r.t. G_{0}^{+} and G_{0}.

2. Suppose all vector functions (K_{i}, …, K_{0}) in Theorem 5.8 satisfy the condition (K_{i}, …, K_{0})_{0} ∈ ℳ \ ℳ_{1}. Then the theorem gives the family of all regular germs of class i from ℱ_{0} nonequivalent w.r.t. G_{0+} and G_{0}.

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Contact classification of linear odes 25

E-mail address: yuma@diffiety.botik.ru

Program System Institute, m. Botik, Pereslavl-Zalessky, 152020, Russia.