International Conference on

Secondary calculus and cohomological Physics Moscow, August 1997

CONTACT TRANSFORMATIONS AND LOCAL REDUCIBILITY OF ODE TO THE FORM Y′′′ = 0

V.N.Gusyatnikova and V.A.Yumaguzhin

Program System Institute

m.Botik, Pereslavl-Zalessky, 152140, Russia E-mail: yuma@diffiety.botik.ru

Abstract. Contact group orbits of the ODE y′′′ = 0 in corresponding jet bundles are investigated. Explicit relations for the right-hand side of an arbitrary 3 -order ODE necessary and sufficient for the existence of a contact transformation reducing this equation locally to the form y′′′ = 0 are obtained.

1. Introduction

In [1], Chern obtained conditions for an ODE to be locally reducible by a contact transformation to the form y′′′ = 0. His approach is based on the Cartan’s exterior form method.

In this paper we use another approach to this problem. It is as follows.

An arbitrary contact transformation transforms the equation y′′′ = 0 to the form

y′′′ = u^{3}(x, y, y′) · y^{′′3} + u^{2}(x, y, y′) · y^{′′2} + u^{1}(x, y, y′) · y′′ + u^{0}(x, y, y′)  (1)

We identify every equation Y of form (1) with the section S_{Y}(p) = ( u^{3}(p), u^{2}(p), u^{1}(p), u^{0}(p) ) of the product bundle π : R^{4} × R^{3} ⟶ R^{3}. An arbitrary contact transformation transforms an arbitrary equation (1) to an equation of that form. Therefore, the action of the pseudogroup Γ of all contact transformation on the bundle π is defined. Differential prolongations of this action define the action of Γ on the jet bundles J^{k}π of π, k = 0, 1, 2, …, ∞. As a result, every J^{k}π may be represented as an union of nonintersecting orbits. Denote by Im j_{k}S_{Y} the image of k-jet prolongation of S_{Y}. We call the orbit of the subset Im j_{k}S_{Y} ⊂ J^{k}π the k-orbit of Y.

Here we prove that Y is locally reducible by a contact transformation to the form y′′′ = 0 iff Im j_{2}S_{Y} is a subset of the 2-orbit of the equation y′′′ = 0. We also obtain explicit equations defining this orbit as a submanifold in J^{2}π.

In [2], we used this approach to obtain the well known conditions (see [3], [4]) for a second-order ODE to be locally reducible by a point transformation to a linear form.

Key words and phrases: Ordinary differential equation, jet bundle, contact transformation, equivalence problem, linearizability.

In this paper, all complicated calculations are esecuted with the help of the computer algebraic system REDUCE 3.6. Corresponding programs are collected in the appendix to this paper.

Below, everything is supposed to be smooth.

2. Preliminaries

Let R^{n} denotes the n – dimensional arithmetical space, [f]_{x}^{k} denotes the k–jet of a map f

at the point x. By definition, put A_{x_{1}...x_{k}} = \frac{∂^{k}A}{∂x_{1} … ∂x_{k}}. Below, for any multi-index σ =

( i_{1}, i_{2}, …, i_{r}) we put |σ| = i_{1} + i_{2} + … + i_{r} and i_{s} ≥ 0, s = 1, 2, …, r.

2.1.

Let λ : R^{m} × R^{n} ⟶ R^{n} be a product bundle. Denote by [S]_{p}^{k} the k-jet of the section S of λ at p ∈ R^{n}, k = 0, 1, 2, …, ∞. The bundle of all these k-jets is denoted. by λ_{k} : J^{k}λ → R^{n}.

Let x^{1}, …, x^{n} be standard coordinates on the base R^{n} of λ and let u^{1}, …, u^{m} be standard coordinates on the fiber R^{m} of λ. Then x^{i}; u_{σ}^{j} denote standard coordinates of J^{k}λ, where i = 1, 2, …, n; j = 1, 2, …, m, σ = ( i_{1}, i_{2}, …, i_{n}) and 0 ≤ |σ| ≤ k.

The natural projection λ_{k_{1},k_{2}} : J^{k_{1}}λ ⟶ J^{k_{2}}λ, k_{2} ≤ k_{1} ≤ ∞ projects each k_{1}-jet to its k_{2}-jet.

Every section S of λ defines the section j_{k}S of λ_{k} by the formula j_{k}S(p) = [S]_{p}^{k}, p ∈ R^{n}.

Let L_{S}^{(k)} = Im j_{k}S. Any k-jet θ_{k} = [S]_{p}^{k} can be identified with the tangent space to the submanifold L_{S}^{(k−1)} at the point θ_{k−1} = λ_{k,k−1}(θ_{k}). This space is denoted by K_{θ}.

We treat any submanifold E ⊂ J^{k}λ as a k-th order PDE system with n independent and m dependent variables. Then a solution of E is a section S of λ with L_{S}^{(k)} ⊂ E.

Let E ⊂ J^{k}λ be a PDE system. The first prolongation of E is the submanifold E^{(1)} ⊂ J^{k+1}λ defined by

E^{(1)} = { θ_{k+1} ∈ J^{k+1}λ | K_{θ_{k+1}} ⊂ T_{θ_{k}}(E), θ_{k−1} = λ_{k+1,k}(θ_{k+1}) }.

The r-th prolongation E^{(r)}, r ≥ 2 is defined by induction on r E^{(r)} = ( E^{(r−1)} )^{(1)}. If E is defined by the equations F^{α}(x^{i}, u_{σ}^{j}) = 0, α = 1, 2, …, r, then E^{(r)} is defined by the equations D_{σ}F^{α} = 0, α = 1, 2, …, r, 0 ≤ |σ| ≤ r, where D_{σ} = D_{(i_{1},i_{2},...,i_{n})} = D_{x^{1}}^{i_{1}} ∘ ⋯ ∘ D_{x^{n}}^{i_{n}},

D_{x^{j}} = \frac{∂}{∂x^{j}} + ∑_{|σ|=0}^{∞} u_{σ+1_{j}}^{i} \frac{∂}{∂u_{σ}^{i}} , j = 1, 2, …, n

is the total derivative operator with respect to x^{j}, σ + 1_{j} = ( i_{1}, …, i_{j−1}, i_{j}, i_{j+1}, …, i_{n}) + 1_{j} = ( i_{1}, …, i_{j−1}, i_{j} + 1, i_{j+1}, …, i_{n}).

We use the following fact from the theory of completely integrable PDE systems. Let θ_{k} ∈ E and θ_{k−1} = λ_{k,k−1}(θ_{k}). The tangent space to E ∩ λ_{k,k−1}^{−1}(θ_{k−1}) is called the symbol of E at θ_{k} and is denoted by Smbl_{θ_{k}}E.

reducibility of ode to the form y′′′ = 0 3

Theorem 2.1 ([5]) Let E ⊂ J^{k}λ be a PDE system. Assume that

1) λ_{k+1,k}|_{E^{(1)}}: E^{(1)} → E is surjective,

2) Smbl_{θ_{k}}E = { 0 } for all θ_{k} ∈ E. Then for any θ_{k} ∈ E there is a solution f of E with [f]_{p}^{k} = θ_{k}, p = λ_{k}(θ_{k}).

2.2.

Let λ : R^{1} × R^{1} → R^{1} be a product bundle, let x be the coordinate on the base of λ and let y be the coordinate on the fiber of λ. We denote the standard coordinates on J^{1}λ by x, y, z.

We recall (see, for example, [6]) that a contact transformation is a most general smooth invertible transformation of variables that preserves the structure of an ODE: it is a diffeomorphism of J^{1}λ defined (in the coordinates x, y, z) by the formula

\begin{aligned}X & = & X(x, y, z) \\ Y & = & Y (x, y, z) \\ Z & = & (Y_{x} + zY_{y})/(X_{x} + zX_{y}), \end{aligned}  (2)

where functions X(x, y, z) and Y (x, y, z) are connected by the relation

X_{z}(Y_{x} + zY_{y}) − Y_{z}(X_{x} + zX_{y}) = 0  (3)

Below, we denote the space J^{1}λ by B. The point (0, 0, 0) ∈ B is denoted by0 .

Lemma 2.2 The Lie pseudogroup of all contact transformations of B is transitive.

Proof. Let p = (x_{0}, y_{0}, z_{0}) be an arbitrary point of B. Then the contact transformation

\{ \begin{matrix}X & = & x − x_{0} \\ Y & = & z_{0} · (x − x_{0}) + y − y_{0} \\ Z & = & z − z_{0} \end{matrix}  (4)

takes p to 0.

A vector field on B is a contact vector field if its flow consists of contact transformations. It is easily shown that a contact vector field ξ can be represented (in coordinates x, y, z) by the formula

ξ = ξ_{φ} = −φ_{z} \frac{∂}{∂x} + (φ − zφ_{z})\frac{∂}{∂y} + (φ_{x} + zφ_{y})\frac{∂}{∂z} ,  (5)

where the function φ = φ(x, y, z) is called the generating function of ξ. Obviously, the generating function defines the contact vector field completely.

2.3.

Let Y_{0} be an arbitrary equation of form (1)

Y′′′ = U^{3}(X, Y, Z) · Y^{′′3} + U^{2}(X, Y, Z) · Y′′2 + U^{1}(X, Y, Z) · Y′′ + U^{0}(X, Y, Z) (6)

Acting on Y_{0} by an arbitrary contact transformation (2), we obtain a new equation Y of form (1)

y′′′ = u^{3}(x, y, z) · y^{′′3} + u^{2}(x, y, z) · y^{′′2} + u^{1}(x, y, z) · y′′ + u^{0}(x, y, z)

Coefficients of the latter equation are expressed in terms of coefficients of the former equation and derivatives of the functions X(x, y, z), Y (x, y, z) defining the contact transformation by the formulas

\begin{aligned}u^{3} & = F · [ & X_{zz}Z_{z} − Z_{zz}X_{z} \\ & & +U^{3}Z_{z}^{3} + U^{2}Z_{z}^{2}X_{z} + U^{1}Z_{z}X_{z}^{2} + U^{2}X_{z}^{3} ], \\ u^{2} & = F · [ & X_{zz}(Z_{x} + zZ_{y}) − Z_{zz}(X_{x} + zX_{y}), \\ & & +2( (X_{xz} + zX_{yz})Z_{z} − (Z_{xz} + zZ_{yz})X_{z}) \\ & & +X_{y}Z_{z} − Z_{y}X_{z} \\ & & +3U^{3}(Z_{x} + zZ_{y})Z_{z}^{2} \\ & & +U^{2}( Z_{x}^{2}(X_{x} + zX_{y}) + 2Z_{z}(Z_{x} + zZ_{y})X_{z}) \\ & & +U^{1}( X_{x}^{2}(Z_{x} + zZ_{y}) + 2X_{z}(X_{x} + zX_{y})Z_{z}) \\ & & +3U^{0}(X_{x} + zX_{y})X_{z}^{2} ], \\ u^{1} & = F · [ & (X_{xx} + 2zX_{xy} + (z)^{2}X_{yy})Z_{z}, \\ & & +(Z_{xx} + 2zZ_{xy} + (z)^{2}Z_{yy})X_{z} \\ & & +2( (X_{xz} + zX_{yz})(Z_{x} + zZ_{y}) \\ & & −(Z_{xz} + zZ_{yz})(X_{x} + zX_{y}) ) \\ & & +(Z_{x} + zZ_{y})X_{y} − Z_{y}(X_{x} + zX_{y}) \\ & & +3U^{3}(Z_{x} + zZ_{y})^{2}Z_{z} \\ & & +U^{2}( X_{z}(Z_{x} + zZ_{y})^{2} + 2Z_{z}(Z_{x} + zZ_{y})(X_{x} + zX_{y}) ) \\ & & +U^{1}( X_{z}(Z_{x} + zZ_{y})(X_{x} + zX_{y}) + 2Z_{z}(X_{x} + zX_{y})^{2} ) \\ & & +3U^{0}(X_{x} + zX_{y})^{2}X_{z} ], \\ u^{0} & = F · [ & (X_{xx} + 2zX_{xy} + (z)^{2}X_{yy})(Z_{x} + zZ_{y}) \\ & & −(Z_{xx} + 2zZ_{xy} + (z)^{2}Z_{yy})(X_{x} + zX_{y}) \\ & & +U^{3}(Z_{x} + zZ_{y})^{3} \\ & & +U^{2}(Z_{x} + zZ_{y})^{2}(X_{x} + zX_{y}) \\ & & +U^{1}(Z_{x} + zZ_{y})(X_{x} + zX_{y})^{2} \\ & & +U^{0}(X_{x} + zX_{y})^{3} ], \end{aligned}  (7)

where F = \frac{X_{x} + zX_{y}}{X_{x}Y_{y} − Y_{x}X_{y}}. These formulas were obtained with the help of REDUCE-program

6.1.

2.4.

Let us consider relations (7), where u^{3}(x, y, z), …, u^{0}(x, y, z) are coefficients of some equation Y of form (1). Then relations (7) may be considered as a PDE system with respect to unknown functions X(x, y, z), Y (x, y, z). Obviously, this system is compatible whenever Y_{0} is locally reducible by a contact transformation to Y. We add equation (3) to this PDE system and denote the resulting system by Ẽ(Y_{0}, Y).

Let τ : R^{2} × B ⟶ B be a product bundle; then Ẽ(Y_{0}, Y) ⊂ J^{3}τ. We say that θ_{3} ∈ J^{3}τ is a contact transformation’s jet iff there is a contact transformation (2) such that θ_{3} is the 3-jet of the section (x, y, z) ↦ ( X(x, y, z), Y (x, y, z) ) of τ.

Let E(Y_{0}, Y) ⊂ Ẽ(Y_{0}, Y) be the subset consisting of all contact transformation’s jets. For Y_{0} = { y′′′ = 0 } we write E(Y) instead of E(Y_{0}, Y). We say that contact transformation (2) is satisfies the PDE system E(Y_{0}, Y) iff the pair

of functions ( X(x, y, z), Y (x, y, z) ) is a solution of this system.

Thus the equations Y_{0} and Y are locally equivalent iff there is a contact transformation satisfying E(Y_{0}, Y).

The main goal of this paper is to obtain the compatibility conditions for the PDE system E(Y).

2.5.

Let π : R^{4} × B ⟶ B be a product bundle. We denote the standard coordinates of the fiber R^{4} of π by u^{3}, u^{2}, u^{1}, u^{0}.

Let Y be an arbitrary equation of form (1) and let its coefficients u^{3}, u^{2}, u^{1}, u^{0} be defined on the domain V ⊂ B. We identify Y with the section S_{Y} of π defined by the formula

S_{Y}(p) = ( u^{3}(p), u^{2}(p), u^{1}(p), u^{0}(p) ), p ∈ V

We denote by S_{0} the trivial section corresponding to the equation y′′′ = 0.

2.6.

Let f be an arbitrary contact transformation on the base B of π. Formulas (2), (3), (7) define the lifting of f to the (local) diffeomorphism f^{(0)} of J^{0}π.

The k-th order differential prolongation of these relations defines the lifting of f^{(0)} to the (local) diffeomorphism f^{(k)} of the jet bundle J^{k}π, k = 1, 2, … ∞ (see, for example, [6]).

Obviously, the lifting of contact transformations generates the lifting of any contact vector fields ξ_{φ} to the vector field ξ_{φ}^{(k)} in J^{k}π k = 0, 1, 2, …, ∞.

It is well known (see, for example, [6]) that ξ_{φ}^{(∞)} is defined by the formula

ξ_{φ}^{(∞)} = −φ_{z} · D_{x} + (φ − zφ_{z}) · D_{y} + (φ_{x} + zφ_{y}) · D_{z} + ∑_{|σ|=0}^{∞} D_{σ}(ψ^{i}(φ))\frac{∂}{∂u_{σ}^{i}} ,

where ψ^{3}(φ), …, ψ^{0}(φ) are components of the ”velocity of deformation”

\frac{d}{dt}( f_{t}^{(0)} ∘ S ∘ f_{t})|_{t=0},  (8)

of an arbitrary section S transformed by the flow f_{t}^{(0)} of ξ_{φ}^{(0)}. The explicit form of the formula (8) is as follows

( \begin{matrix}ψ^{3}(φ) \\ ψ^{2}(φ) \\ ψ^{1}(φ) \\ ψ^{0}(φ) \end{matrix} ) = ( \begin{matrix}u_{x}^{3}φ_{z} − u_{y}^{3}(φ − zφ_{z}) − u_{z}^{3}(φ_{x} + zφ_{y}) \\ −u^{3}(3φ_{xz} + 3zφ_{yz} + 2φ_{y}) + u^{2}φ_{zz} \\ +φ_{zzz} \\ u_{x}^{2}φ_{z} − u_{y}^{2}(φ − zφ_{z}) − u_{z}^{2}(φ_{x} + zφ_{y}) \\ −3u^{3}(φ_{xx} + 2zφ_{xy} + z^{2}φ_{yy}) \\ −u^{2}(φ_{xz} + zφ_{yz} + φ_{y}) + 2u^{1}φ_{zz} \\ +3(φ_{xzz} + zφ_{yzz} + φ_{yz}) \\ u_{x}^{1}φ_{z} − u_{y}^{1}(φ − zφ_{z}) − u_{z}^{1}(φ_{x} + zφ_{y}) \\ −2u^{2}(φ_{xx} + 2zφ_{xy} + z^{2}φ_{yy}) \\ +u^{1}(φ_{xz} + zφ_{yz}) + 3u^{0}φ_{zz} \\ +3(φ_{xy} + zφ_{yy}) + 3(φ_{xxz} + 2zφ_{xyz} + z^{2}φ_{yyz}) \\ u_{x}^{0}φ_{z} − u_{y}^{0}(φ − zφ_{z}) − u_{z}^{0}(φ_{x} + zφ_{y}) \\ −u^{1}(φ_{xx} + 2zφ_{xy} + z^{2}φ_{yy}) \\ +u^{0}(3φ_{xz} + 3zφ_{yz} + φ_{y}) \\ +φ_{xxx} + 3zφ_{xxy} + 3z^{2}φ_{xyy} + z^{3}φ_{yyy} \end{matrix} )  (9)

It can be obtained with the help of REDUCE-program 6.2.

If ξ_{φ}(p) = 0, then the restriction of ξ_{φ}^{(k)} on the fiber π_{k}^{−1}(p) is defined by the formula

ξ_{φ}^{(k)} |_{π_{k}^{−1}(p)} = ∑_{|σ|=0}^{k} D_{σ}(ψ^{i}(φ))(p)\frac{∂}{∂u_{σ}^{i}} .  (10)

2.7.

Let Γ be the Lie pseudogroup of all contact transformation of B. Then Γ^{(k)} = { f^{(k)} | f ∈ Γ } is a Lie pseudogroup acting on J^{k}π, k = 0, 1, 2, … ∞. With respect to this action, J^{k}π splits into a union of nonintersecting orbits. We recall that the subset

Orb (θ_{k}) = { f^{(k)}(θ_{k}) ∈ J^{k}π | f^{(k)} ∈ Γ^{(k)} } ⊂ J^{k}π

is named the orbit of the point θ_{k} ∈ J^{k}π with respect to the action of Γ^{(k)} on J^{k}π.

Let Y be an arbitrary equation of form (1) and let its coefficients be defined on the domain V ⊂ B. The k-orbit of Y is the submanifold Orb_{k}(Y) ⊂ J^{k}π defined by the formula

Orb_{k}(Y) = ⋃_{p∈V} Orb ( [S_{Y}]_{p}^{k} ).

We denote by Orb_{k} the k-orbit of the equation y′′′ = 0.

3. Orbits of the equation y′′′ = 0

In this section we calculate Codim Orb_{k}, k = 0, 1, 2, …

3.1.

Lemma 3.1 Orb_{k} = Orb ( [S_{0}]0^{k} ), k = 0, 1, 2, … ∞.

Proof. Let p = (x_{0}, y_{0}, z_{0}) be an arbitrary point of B. Contact transformation (7) takes p to 0. It follows from relations (28) that this transformation transforms the equation y′′′ = 0 onto itself. Therefore, f^{(k)} takes [S_{0}]0^{k} to [S_{0}]_{p}^{k}.

It follows from the transitivity of Γ that

dim Orb (θ_{k}) = dim( Orb (θ_{k}) ∩ π_{k}^{−1}(p) ) + dim B, ∀p ∈ B, ∀θ_{k} ∈ J^{k}π

Let p be an arbitrary point of B and let θ_{k} be an arbitrary point of π_{k}^{−1}(p). Let us consider the Lie group

G^{k}(p) = { [f]_{p}^{k+2} | f ∈ Γ, f(p) = p }.

This group acts on the fiber π_{k}^{−1}(p) of the bundle π_{k} by the formula

( [f]_{p}^{k+2} × θ_{k} ) ⟼ f^{(k)}(θ_{k}).  (11)

Obviously, Orb (θ_{k}) ∩ π_{k}^{−1}(p) is an orbit of this action.

Let H(θ_{k}) ⊂ G^{k}(p) be the isotropy group of the point θ_{k} ∈ J^{k}π. We denote by H^{k}(p) the isotropy group of the point [S_{0}]_{p}^{k}.

It’s well known that Orb (θ_{k})∩π_{k}^{−1}(p) is diffeomorphic to the quotient group G^{k}(p)/H(θ_{k}). Let G^{k}(p), H(θ_{k}) and H^{k}(p) be the Lie algebras of the groups

G^{k}(p), H(θ_{k}) and H^{k}(p) respectively. Then

dim( Orb (θ_{k}) ∩ π_{k}^{−1}(p) ) = dim G^{k}(p) − dim H(θ_{k}).

3.2.

Then

G^{k}(p) = { [ξ_{φ}]_{p}^{2+k} | ξ_{φ} is a contact vector field and ξ_{φ}(p) = 0 }

It follows from (5) that the condition ξ_{φ}(p) = 0 is equivalent to the relations

φ(p) = 0, φ_{x}(p) = −zφ_{y}(p), φ_{z}(p) = 0.  (12)

Therefore, an element [ξ_{φ}]_{p}^{2+k} ∈ G^{k}(p) is defined by the following collection of derivatives of φ at a point p

φ_{y}, \begin{matrix}φ_{x^{2}}, \\ φ_{xy}, φ_{xz}, \\ φ_{y^{2}}, φ_{yz}, φ_{z^{2}}, \end{matrix} ⋯ \begin{matrix}φ_{x^{3+k}}, \\ φ_{x^{2+k}y}, φ_{x^{2+k}z}, \\ ⋯ \\ φ_{y^{3+k}}, φ_{y^{2+k}z}, ⋯ φ_{yz^{2+k}}, φ_{z^{3+k}} \end{matrix}  (13)

It is easy to prove the following statement.

Lemma 3.2 The space of collections (13) defining 0 ∈ G^{k}(p) is one-dimensional and is generated by the following elements

0, \begin{matrix}0, \\ 0, 0, \\ 0, 0, 0, \end{matrix} ⋯ \begin{matrix}0, \\ 0, 0, \\ ⋯ \\ 0, 0, 0, \end{matrix} \begin{matrix}(−z)^{3+k}, \\ (−z)^{2+k}, 0, \\ ⋯ \\ (−z)^{2}, 0, ⋯ 0, \\ (−z)^{1}, 0, 0, ⋯ 0, \\ (−z)^{0}, 0, 0, 0, ⋯ 0. \end{matrix}

Proof. From (5) it follows that [ξ_{φ}]_{p}^{2+k} = 0 iff

D_{σ}(φ_{z})|_{p}= 0, D_{σ}(φ − zφ_{z})|_{p}= 0, D_{σ}(φ_{x} − zφ_{y})|_{p}= 0.

for every σ = (i_{1}, i_{2}, i_{3}) with 0 ≤ |σ| ≤ 2 + k. Therefore, φ_{x^{i_{1}}y^{i_{2}}z^{i_{3}}} = 0 for i_{3} ≥ 1 and 0 ≤ i_{1} + i_{2} + i_{3} ≤ 3 + k, φ_{x^{i_{1}}y^{i_{2}}z^{i_{3}}} = 0 for 0 ≤ i_{1} + i_{2} + i_{3} ≤ 2 + k, and φ_{x^{i_{1}}y^{i_{2}}} = (−z)^{i_{1}}φ_{y^{i_{2}}} for i_{1} + i_{2} = 3 + k.

Now it follows from (13) and lemma 3.2 that

dim G^{k}(p) = 6 + ∑_{i=0}^{k} \frac{(i + 4)(i + 5)}{2}, k = 0, 1, 2, ….  (14)

3.3.

The action (11) of the group G^{k}(p) on π_{k}^{−1}(p) generates the representation of its Lie algebra G^{k}(p) by vector fields on π_{k}^{−1}(p). It follows from (10) that this representation is defined in coordinates by the formula

[ξ_{φ}]_{p}^{k} ⟼ ∑_{|σ|=0}^{k}( D_{σ}ψ^{i}(φ) )(p) · \frac{∂}{∂u_{σ}^{i}} ,

where ψ^{3}(φ), …, ψ^{0}(φ) are components of the matrix (9).

Therefore, a collection (13) defines an element of H(θ_{k}) iff it is a solution of the following system of homogeneous linear algebraic equations

D_{σ}(ψ^{i}(φ)) |_{θ_{k}} = 0, i = 3, 2, 1, 0; |σ| = 0, 1, 2, …, k  (15)

At the point [S_{0}]0^{k}, this system has the form

D_{σ}( φ_{zzz} )|0= 0,  (16)
D_{σ}( φ_{xzz} + zφ_{yzz} + φ_{yz} )|0= 0,  (17)
D_{σ}( φ_{xxz} + 2zφ_{xyz} + z^{2}φ_{yyz} + φ_{xy} + zφ_{yy} )|0= 0,  (18)
D_{σ}( φ_{xxx} + 3zφ_{xxy} + 3z^{2}φ_{xyy} + z^{3}φ_{yyy} )|0= 0,  (19)
0 ≤ |σ| ≤ k

Now, it is easy to calculate

\begin{aligned}dim H_{p}^{0} & = & 12 \\ dim H_{p}^{1} & = & 15 \end{aligned} ∀p ∈ B  (20)

Lemma 3.3 For 2 ≤ k ≤ ∞, a solution of system (16) – (19) can be represented in the form

\begin{matrix}( φ_{y}, \begin{matrix}φ_{x^{2}}, \\ φ_{xy}, φ_{xz}, \\ φ_{y^{2}}, φ_{yz}, φ_{z^{2}}, \end{matrix} \begin{matrix}∗, \\ ∗, ∗, \\ ∗, ∗, ∗, \\ ∗, ∗, ∗, ∗, \end{matrix} ⋯ \begin{matrix}∗, \\ ∗, ∗, \\ ⋯ \\ ∗, ∗, ⋯, ∗, ∗, \end{matrix} \begin{matrix}∗, \\ ⋯ \\ ∗, ⋯, ∗, \\ φ_{y^{1+k}}, ∗, ⋯, ∗, \end{matrix} \\ \begin{matrix}∗, \\ ⋯ \\ ∗, ⋯, ∗, \\ φ_{xy^{1+k}}, ∗, ⋯, ∗, \\ φ_{y^{2+k}}, φ_{y^{1+k}z}, ∗, ⋯, ∗, \end{matrix} & \begin{matrix}∗, \\ ⋯ \\ ∗, ⋯, ∗, \\ φ_{x^{2}y^{1+k}}, ∗, ⋯, ∗, \\ φ_{xy^{2+k}}, φ_{xy^{1+k}z}, ∗, ⋯, ∗, \\ φ_{y^{3+k}}, φ_{y^{2+k}z}, φ_{y^{1+k}z^{2}}, ∗, ⋯, ∗ ), \end{matrix} \end{matrix}

where unknowns of the form φ_{x^{i_{1}}y^{i_{2}}z^{i_{3}}} are free, while unknowns denoted by ∗ are dependent ones (they are linear functions of free unknowns).

Proof. By S_{k} denote system (16) - (19).

For k = 2 the lemma’s statement is proved by the direct calculation. (It can be done with the help of REDUCE-program (6.4).)

Let r be a natural number and r ≥ 2. Suppose that the lemma’s statement is proved for all 2 ≤ k ≤ r. Let us prove it for k = r + 1.

We assume that an unknown is dependent in S_{r+1} if it is dependent in S_{r} in correspondence to the lemma’s statement.

Step 1. Consider the unknowns φ_{σ} of the highest order (that is, unknowns of order r +4). It follows from the system S_{r+1} that the unknowns

φ_{x^{2}y^{r+2}},
φ_{xy^{r+3}}, φ_{xy^{r+2}z},
φ_{y^{r+4}}, φ_{y^{r+3}z}, φ_{y^{r+2}z^{2}}

are free and all other unknowns of order r + 4 are expressed in terms of unknowns of lower orders.

Step 2. Consider the following unknowns of order r + 3

φ_{x^{2}y^{r+1}},
φ_{xy^{r+2}}, φ_{xy^{r+1}z},
φ_{y^{r+3}}, φ_{y^{r+2}z}, φ_{y^{r+1}z^{2}}.

They are free in S_{r}. We can choose the unknowns φ_{x^{2}y^{r+1}}, φ_{xy^{r+1}z}, φ_{y^{r+1}z^{2}} to be dependent in S_{r+1}. Indeed, the equations (18) with σ = (1, r, 0) and (19) with σ = (0, r, 1) of S_{r+1} are as follows:

φ_{x^{3}y^{r}z} + φ_{x^{2}y^{r+1}} = 0,
φ_{x^{3}y^{r}z} + 3φ_{x^{2}y^{r+1}} = 0.

It follows from these equations that φ_{x^{2}y^{r+1}} = 0.

Equations (17) with σ = (1, r, 0) and (18) with σ = (0, r, 1) of S_{r+1} are as follows

φ_{x^{2}y^{r}z} + φ_{xy^{r+1}z} = 0,
φ_{x^{2}y^{r}z^{2}} + 2φ_{xy^{r+1}z} + φ_{xy^{r+1}z} + φ_{y^{r+2}} = 0.

It follows from these equations that φ_{xy^{r+1}z} = −\frac{1}{2}φ_{y^{r+2}}.

Equation (17) with σ = (0, r, 1) of S_{r+1} is

φ_{y^{r+1}z^{2}} + φ_{y^{r+1}z^{2}} = 0,

Therefore, φ_{y^{r+1}z^{2}} = 0.

We can choose the unknowns

\begin{matrix}φ_{xy^{r+2}}, \\ φ_{y^{r+3}}, φ_{y^{r+2}z}, \end{matrix}

to be free. Indeed, φ_{y^{r+3}} does not enter the equations of S_{r+1}. The unknowns φ_{xy^{r+2}} enters only equation (18) with σ = (0, r + 1, 0)

φ_{x^{2}y^{r+1}z} + φ_{xy^{r+2}} = 0

and φ_{x^{2}y^{r+1}z} enters this equation only.

The unknown φ_{y^{r+2}z} enters only the equation (17) with σ = (0, r + 1, 0)

φ_{xy^{r+1}z^{2}} + φ_{y^{r+2}z} = 0

and φ_{xy^{r+1}z^{2}} enters this equation only.

Step 3. Consider the following unknowns of order r + 2

\begin{matrix}φ_{xy^{r+1}}, \\ φ_{y^{r+2}}, φ_{y^{r+1}z}. \end{matrix}

They are free in S_{r}. The unknowns φ_{xy^{r+1}}, φ_{y^{r+1}z} are dependent in S_{r+1}. Indeed, the equations (18) with σ = (0, r, 0), (19) with σ = (0, r − 1, 2) and (18) with σ = (1, r − 1, 1) from S_{r+1} are as follows

φ_{x^{2}y^{r}z} + φ_{xy^{r+1}} = 0,
φ_{x^{3}y^{r−1}z^{2}} + 6φ_{x^{2}y^{r}z} + 6φ_{xy^{r+1}} = 0,
φ_{x^{3}y^{r−1}z^{2}} + 2φ_{x^{2}y^{r}z} + 4φ_{x^{2}y^{r}z} + φ_{xy^{r+1}} = 0.

It follows from these equations that φ_{xy^{r+1}} = 0.

Equations (17) with σ = (0, r, 0) and (18) with σ = (0, r − 1, 2) of S_{r+1} are as follows

φ_{xy^{r}z^{2}} + φ_{y^{r+1}z} = 0,
4φ_{xy^{r}z^{2}} + 2φ_{y^{r+1}z} + φ_{xy^{r}z^{2}} + 2φy^{r+1}z = 0.

It follows from these equations that φ_{y^{r+1}z} = 0.

The unknown φ_{y^{r+2}} enters only (18) with σ = (0, r, 1)

φ_{x^{2}y^{r}z^{2}} + 3φ_{xy^{r+1}z} + φ_{y^{r+2}} = 0

and φ_{x^{2}y^{r}z} yet enters the equation (17) with σ = (1, r, 0) only. Therefore, we can choose it to be free in S_{r+1}.

Step 4. The unknown φ_{y^{r+1}} is dependent one in S_{r+1}. Indeed, the equations (18) with σ = (0, r − 1, 1), (19) with σ = (0, r − 2, 3) and (17) with σ = (1, r − 1, 0) of S_{r+1} are as follows

φ_{x^{2}y^{r−1}z^{2}} + 3φ_{xy^{r}z} + φ_{y^{r+1}} = 0,
3φ_{x^{2}y^{r−1}z^{2}} + 3φ_{xy^{r}z} + 2φ_{y^{r+1}} = 0,
φ_{x^{2}y^{r−1}z^{2}} + φ_{xy^{r}z} = 0.

It follows from these equations that φ_{y^{r+1}} = 0.

Step 5. At last we can choose the unknowns

φ_{y}, \begin{matrix}φ_{x^{2}}, \\ φ_{xy}, φ_{xz}, \\ φ_{y^{2}}, φ_{yz}, φ_{z^{2}} \end{matrix}

to be free. Indeed, it is obvious for unknowns φ_{y}, φ_{x^{2}}, φ_{xz}, φ_{z^{2}} because they does not enter the equations of S_{r+1}.

The unknown φ_{xy} enters only the equation (18) with σ = (0, 0, 0)

φ_{x^{2}z} + φ_{xy} = 0

and φ_{x^{2}z} enters only this equation too.

The unknown φ_{yz} enters only the equation (17) with σ = (0, 0, 0)

φ_{xz^{2}} + φ_{yz} = 0

and φ_{xz^{2}} enters only this equation too.

The unknown φ_{y^{2}} enters only the equation (18) with σ = (0, 0, 1)

φ_{x^{2}z^{2}} + 3φ_{xyz} + φ_{y^{2}} = 0

and the unknowns φ_{x^{2}z^{2}}, φ_{xyz} yet enters the equation (17) with σ = (1, 0, 0)

φ_{x^{2}z^{2}} + φ_{xyz} = 0

of S_{r+1} only.

Corollary 3.4 For any p ∈ B

dim H^{k}(p) = 16, 2 ≤ k < ∞  (21)
dim H^{∞}(p) = 7.

Let

ρ_{l,m} : G^{l}(p) ⟶ G^{m}(p), ∞ ≥ l ≥ m

be the natural projection taking a l-jet to its m-jet. Let us consider the algebra ρ_{∞,0}( H^{∞}(p) ) and its subalgebra H^{∞,2}(p) defined by the formula

H^{∞,2}(p) = { [ξ_{φ}]_{p}^{2} ∈ ρ_{∞,2}( H^{∞}(p) ) | [ξ_{φ}]_{p}^{1} = 0 }.  (22)

Corollary 3.5 H^{∞,2}(p) = {0}.

Proof. For 0 ∈ B, the statement immediately follows from lemma 3.3. Let p ∈ B and let f be a contact transformation takng p to 0. It is easy to check that H^{∞,2}(p) = f_{∗}^{(∞)}H^{∞,2}(0) = {0}.

3.4.

Now we can calculate the codimension of Orb_{k}, k = 0, 1, 2, … We have Codim Orb_{k} = dim π_{k}^{−1}(p) − ( dim G^{k}(p) − dim H^{k}(p) ). It’s clear that

dim π_{k}^{−1}(p) = 4 ∑_{i=0}^{k} \frac{(i + 1)(i + 2)}{2}, k = 0, 1, 2, ….  (23)

Taking into account formulas (23),(14), (20), and (21), we obtain

Codim Orb_{k} = 0, k = 0, 1,
Codim Orb_{k} = ∑_{i=0}^{k} \frac{4(i + 1)(i + 2) − (i + 4)(i + 5)}{2} + 10, k ≥ 2.  (24)

It follows from (24) that the submanifold Orb_{2} ⊂ J^{2}π is defined by four equations.

4. Explicit relations

In this section, we obtain equations defining the submanifold Orb_{2} ⊂ J^{2}π.

4.1.

Here we find the equations defining the submanifold Orb_{2} ∩ π_{2}^{−1}(0) ⊂ π_{2}^{−1}(0).

Let us consider an isotropy algebra H(θ_{2}) of an arbitrary point θ_{2} ∈ π_{2}^{−1}(0). Each element [ξ_{φ}]0^{2} ∈ H(θ_{2}) is defined by the collection of the derivatives (13, with k = 2) which is a solution of system of homogeneous linear algebraic equations (15, with k = 2)

D_{σ}(ψ^{i}(φ))|_{θ_{2}}= 0, i = 3, 2, 1, 0, |σ| = 0, 1, 2.  (25)

Obviously, the number of equation of this system is equal to 40; it follows from (13) that the number of unknowns in this system is equal to 53. Let us consider the matrix of this system. Its elements are polynomials of u_{σ}^{i}, i = 3, 2, 1, 0, |σ| = 0, 1, 2. System (25) can be reduced by elementary transformations to the equivalent system with coefficient matrix Mat (θ_{2}) of the form

Mat (θ_{2}) = ( \begin{matrix}A \\ B \end{matrix} )  (26)

where A is a 36 × 53-matrix with Rank A = 36 and the matrix B is the 4 × 53-matrix

( \begin{matrix}0 & ⋯ & 0 & F^{3} & 0 & F^{3} & F^{2} \\ 0 & ⋯ & 0 & 2F^{2} & F^{3} & F^{2} & 2F^{1} \\ 0 & ⋯ & 0 & 0 & F^{1} & F^{0} & 0 \\ 0 & ⋯ & 0 & F^{1} & 2F^{2} & F^{1} & F^{0} \end{matrix} )

where

\begin{aligned}F̃^{3} & = & 27u_{xz}^{3} − 9u_{zz}^{2} + 18u_{x}^{3}u^{2} + 81u_{y}^{3} + 18u_{z}^{3}u^{1} \\ & & −18u_{x}^{2}u^{3} − 18u_{z}^{2}u^{2} + 36u_{z}^{1}u^{3} − 54(u^{3})^{2}u^{0} \\ & & +18u^{3}u^{2}u^{1} − 4(u^{2})^{3}, \\ F̃^{2} & = & −9u_{xx}^{3} − 3u_{xz}^{2} + 6u_{zz}^{1} − 18u_{x}^{3}u^{1} \\ & & −18u_{z}^{3}u^{0} + 6u_{x}^{2}u^{2} − 24u_{y}^{2} + 6u_{z}^{2}u^{1} \\ & & −36u_{z}^{0}u^{3} + 18u^{3}u^{2}u^{0} − 12u^{3}(u^{1})^{2} + 2(u^{2})^{2}u^{1}, \\ F̃^{1} & = & 6u_{xx}^{2} − 3u_{xz}^{1} − 9u_{zz}^{0} + 36u_{x}^{3}u^{0} − 6u_{x}^{1}u^{2} \\ & & +21u_{y}^{1} − 6u_{z}^{1}u^{1} + 18u_{x}^{0}u^{3} + 18u_{z}^{0}u^{2} \\ & & +18u^{3}u^{1}u^{0} − 12(u^{2})^{2}u^{0} + 2u^{2}(u^{1})^{2}, \\ F̃^{0} & = & −9u_{xx}^{1} + 27u_{xz}^{0} − 36u_{x}^{2}u^{0} + 18u_{x}^{1}u^{1} \\ & & +18u_{z}^{1}u^{0} − 18u_{x}^{0}u^{2} − 54u_{y}^{0} − 18u_{z}^{0}u^{1} \\ & & −54u^{3}(u^{0})^{2} + 18u^{2}u^{1}u^{0} − 4(u^{1})^{3}. \end{aligned}  (27)

The matrix (26) was obtained with the help of REDUCE-program 6.3.

Thus we have Codim Orb (θ_{2}) = dim π_{k}^{−1}(0) − ( dim G^{k}(p) − dim H^{k}(p) ). It folows from lemma 3.2 that dim H^{k}(p) =(the number of unknowns of the system (25)) −Rank Mat (θ_{2})− 1 = 52 − Rank Mat (θ_{2}). Now, taking into account the formulas (23) and (14), we obtain Codim Orb (θ_{2}) = 40 − Rank Mat (θ_{2})

Hence we have that Codim Orb (θ_{2}) = 4 iff F̃^{i}(θ_{2}) = 0, i = 3, 2, 1, 0.

Theorem 4.1 Orb_{2} ∩ π_{2}^{−1}(0) = { θ_{2} ∈ π_{2}^{−1}(0) | F̃^{i}(θ_{2}) = 0, i = 3, 2, 1, 0 }.

Proof. Let M ⊂ π_{2}^{−1}(0) be the submanifold defined by the equations F̃^{i}(θ_{2}) = 0, i = 3, 2, 1, 0. It can be easily checked that M is a path-connected manifold. Obviously, for every θ_{2} ∈ M the submanifold Orb (θ_{2}) ∩ π_{2}^{−1}(0) is an open subset of M. Therefore Orb_{2} ∩ π_{2}^{−1}(0) = M

4.2.

Let p = (x_{0}, y_{0}, z_{0}) be an arbitrary point of B. Let us consider the contact transformation p̂ defined by the equations

\{ \begin{matrix}X & = & x + x_{0} \\ Y & = & z_{0} · x + y + y_{0} \\ Z & = & z + z_{0} \end{matrix}  (28)

This transformation takes 0 to p. The lifting p̂^{(2)} takes every point θ_{2} ∈ π_{2}^{−1}(p) to the point p̂^{(2)}(θ_{2}) ∈ π_{2}^{−1}(0). Therefore submanifold Orb_{2} ∩ π^{−1}(p) is defined by the equations F̃^{i} ∘ p̂^{(2)} = 0, i = 3, 2, 1, 0. Let functions F^{i}, i = 3, 2, 1, 0, be defined by the formula

F^{i}(θ_{2}) = F̃^{i} ∘ π̂_{2}(θ_{2})^{(2)}(θ_{2}), i = 3, 2, 1, 0, θ_{2} ∈ J^{2}π.

Then we obtain the following theorem.

Theorem 4.2 Orb_{2} = { θ_{2} ∈ J^{2}π | F^{i}(θ_{2}) = 0, i = 3, 2, 1, 0 }.

Now we deduce the explicit form of the functions F^{i}. Suppose the points θ_{2} ∈ π_{2}^{−1}(p) and p̂^{(2)}(θ_{2}) ∈ π_{2}^{−1}(0) in standard coordinates on J^{2}π have the following form.

θ_{2} = ( X, Y, Z; U^{3}, …, U^{0}; …; U_{XX}^{3}, …, U_{ZZ}^{0} )
p̂^{(2)}(θ_{2}) = ( 0, 0, 0; u^{3}, …, u^{0}; …; u_{xx}^{3}, …, u_{zz}^{0} ).

Then it follows from (7) and (28) that

\begin{aligned}u^{i} & = & U^{i}, \\ u_{x}^{i} & = & U_{X}^{i} + ZU_{Y}^{i} , \\ u_{y}^{i} & = & U_{Y}^{i} , \\ u_{z}^{i} & = & U_{Z}^{i}, \\ u_{xx}^{i} & = & U_{XX}^{i} + 2ZU_{XY}^{i} + Z^{2}U_{Y Y}^{i} , \\ u_{xy}^{i} & = & U_{XY}^{i} + ZU_{Y Y}^{i} , \\ u_{xz}^{i} & = & U_{XZ}^{i} + ZU_{Y Z}^{i}, \\ u_{yy}^{i} & = & U_{Y Y}^{i} , \\ u_{yz}^{i} & = & U_{Y Z}^{i}, \\ u_{zz}^{i} & = & U_{ZZ}^{i}, i = 3, 2, 1, 0. \end{aligned}

Substituting these expressions into (27), we get

\begin{aligned}F^{3} & = & 27u_{xz}^{3} + 27zu_{yz}^{3} − 9u_{zz}^{2} + 18u_{x}^{3}u^{2} + 18zu_{y}^{3}u^{2} \\ & & +81u_{y}^{3} + 18u_{z}^{3}u^{1} − 18u_{x}^{2}u^{3} − 18zu_{y}^{2}u^{3} − 18u_{z}^{2}u^{2} \\ & & +36u_{z}^{1}u^{3} − 54(u^{3})^{2}u^{0} + 18u^{3}u^{2}u^{1} − 4(u^{2})^{3}, \\ F^{2} & = & −9u_{xx}^{3} − 18zu_{xy}^{3} − 3u_{xz}^{2} − 3zu_{yz}^{2} + 6u_{zz}^{1} \\ & & −18u_{x}^{3}u^{1} − 18zu_{y}^{3}u^{1} − 18u_{z}^{3}u^{0} + 6u_{x}^{2}u^{2} + 6zu_{y}^{2}u^{2} \\ & & −24u_{y}^{2} + 6u_{z}^{2}u^{1} − 36u_{z}^{0}u^{3} + 18u^{3}u^{2}u^{0} − 12u^{3}(u^{1})^{2} \\ & & +2(u^{2})^{2}u^{1} − 9z^{2}u_{yy}^{3}, \\ F^{1} & = & 6u_{xx}^{2} + 12zu_{xy}^{2} − 3u_{xz}^{1} − 3zu_{yz}^{1} − 9u_{zz}^{0} \\ & & +36u_{x}^{3}u^{0} + 36zu_{y}^{3}u^{0} − 6u_{x}^{1}u^{2} − 6zu_{y}^{1}u^{2} + 21u_{y}^{1} \\ & & −6u_{z}^{1}u^{1} + 18u_{x}^{0}u^{3} + 18zu_{y}^{0}u^{3} + 18u_{z}^{0}u^{2} + 18u^{3}u^{1}u^{0} \\ & & −12(u^{2})^{2}u^{0} + 2u^{2}(u^{1})^{2} + 6z^{2}u_{yy}^{2}, \\ F^{0} & = & −9u_{xx}^{1} − 18zu_{xy}^{1} + 27u_{xz}^{0} + 27zu_{yz}^{0} − 36u_{x}^{2}u^{0} \\ & & −36zu_{y}^{2}u^{0} + 18u_{x}^{1}u^{1} + 18zu_{y}^{1}u^{1} + 18u_{z}^{1}u^{0} − 18u_{x}^{0}u^{2} \\ & & −18zu_{y}^{0}u^{2} − 54u_{y}^{0} − 18u_{z}^{0}u^{1} − 54u^{3}(u^{0})^{2} + 18u^{2}u^{1}u^{0} \\ & & −4(u^{1})^{3} − 9z^{2}u_{yy}^{1}. \end{aligned}  (29)

4.3.

Let Orb _{2}^{(k)} be the k-th order differential prolongation of Orb _{2}^{(k)} (see subsection 2.1).

Orb _{2}^{(k)} = { θ_{2+k} ∈ J^{2+k}π | D_{σ}F^{i}(θ_{2+k}) = 0, |σ| ≤ k, i = 3, 2, 1, 0 }.

Theorem 4.3 Orb _{2}^{(k)} = Orb_{2+k}, k = 0, 1, 2, …, ∞.

Proof. Notice that

Orb_{2+k} ⊂ Orb _{2}^{(k)}, k = 0, 1, 2, …, ∞.

Indeed, if θ_{2+k} ∈ Orb_{2+k}, then there is a contact transformation f such that θ_{2+k} = f^{(2+k)}([S_{0}]0^{2+k}). This transformation transforms Y_{0} to the equation Y with θ_{2+k} = [S_{Y}]_{p}^{2+k}. Therefore, Im (j_{2}S_{Y}) ⊂ Orb_{2}. It follows that Im (j_{2+k}S_{Y}) ⊂ Orb _{2}^{(k)}, k = 0, 1, 2, …, ∞. Hence θ_{2+k} ∈ Orb _{2}^{(k)}.

Suppose that the equality Orb_{2+k} = Orb _{2}^{(k)} is proved for k = 1, 2, …, r. Let us prove it for k = r + 1.

Lemma 4.4 If Orb_{2+r} = Orb _{2}^{(r)} and π_{3+r,2+r}^{−1}([S_{0}]0^{2+r}) ∩ Orb_{3+r} =

π_{3+r,2+r}^{−1}([S_{0}]0^{2+r}) ∩ Orb _{2}^{r+1} then Orb_{3+r} = Orb _{2}^{(r+1)}.

Proof. Let θ_{3+r} ∈ Orb _{2}^{(r+1)}, θ_{2+r} = π_{3+r,2+r}(θ_{3+r}), and let f be a contact transformation such that f^{(2+r)}(θ_{2+r}) = [S_{0}]0^{2+r}. We have K_{θ_{3+r}} ⊂ T_{θ_{2+r}}(Orb _{2}^{(r)}) = T_{θ_{2+r}}(Orb_{2+r}). Therefore,

K_{f^{3+r}(θ_{3+r})} = f_{∗}^{(2+r)}(K_{θ_{3+r}}) ⊂ f_{∗}^{(2+r)}(T_{θ_{2+r}}(Orb_{2+r})) = T_{[S_{0}]0^{2+r}}(Orb_{2}(r))). It follows that

f^{3+r}(θ_{3+r}) ∈ Orb _{2}^{(r+1)}. Obviously f^{3+r}(θ_{3+r}) ∈ π_{3+r,2+r}^{−1}([S_{0}]0^{2+r}) ∩ Orb _{2}^{(r+1)}

= π_{3+r,2+r}^{−1}([S_{0}]0^{2+r}) ∩ Orb_{3+r}. Then it follows θ_{3+r} ∈ Orb_{3+r}. This concludes the proof of the lemma.

Obviously, π_{3+r,2+r}^{−1}([S_{0}]0^{2+r})∩Orb_{3+r} ⊂ π_{3+r,2+r}^{−1}([S_{0}]0^{2+r})∩Orb _{2}^{r+1}. Therefore, to complete the proof of the theorem it is sufficient to show that the dimensions of these manifolds are the same. That is, dim Smbl_{[S_{0}]0^{3+r}}(Orb_{3+r}) = dim Smbl_{[S_{0}]0^{3+r}}(Orb _{2}^{(1+r)}).

The following lemma implies this equality.

Lemma 4.5

Codim Smbl_{[S_{0}]0^{2+k}}(Orb_{2+k}) = Codim Smbl_{[S_{0}]0^{2+k}}(Orb _{2}^{(k)}), k = 0, 1, 2, …

Proof. Notice that Codim Smbl_{[S_{0}]0^{2}}(Orb_{2}) = 4. For k ≥ 1, it follows from (24) that

Codim Smbl_{[S_{0}]0^{2+k}}(Orb_{2+k}) = 4\frac{(k + 3)(k + 4)}{2} − \frac{(k + 6)(k + 7)}{2}
= \frac{3}{2}(k^{2} + 5k + 2).

From theorem 4.2 we obtain that Smbl_{[S_{0}]0^{2}}(Orb_{2}) is defined by the system of linear algbraic equations

\{ \begin{matrix}3u_{x^{2}}^{3} + u_{xz}^{2} − 2u_{z^{2}}^{1}  = 0 \\ 2u_{x^{2}}^{2} − u_{xz}^{1} − 3u_{z^{2}}^{0} = 0 \\ −u_{x^{2}}^{1} + 3u_{xz}^{0} = 0 \\ 3u_{xz}^{3} − u_{z^{2}}^{2}  = 0. \end{matrix}  S_{2}

Hence Smbl_{[S_{0}]0^{2+k}}(Orb _{2}^{(k)}) is defined by the k-th order prolongation of this system, k =

1, 2, . . .. We denote this prolongation by S_{2}^{(k)}. It is obvious that equations of S_{2}^{(k)} generated by the first three equations of S_{2} are linear independent.

Let us consider the equation in S_{2}^{(k)} generated by the last equation of S_{2}.

D_{x}^{a}D_{y}^{b}D_{z}^{c}( 3u_{xz}^{3} − u_{z^{2}}^{2} ) = 0, a + b + c = k.  (30)

Let it be a linear combination of equations of S_{2}^{(k)} generated by the first three equations of S_{2}. Then we have

D_{x}^{a}D_{y}^{b}D_{z}^{c}( 3u_{xz}^{3} − u_{z^{2}}^{2} ) = D_{x}^{a−1}D_{y}^{b}D_{z}^{c+1}( 3u_{x^{2}}^{3} + u_{xz}^{2} − 2u_{z^{2}}^{1} ) + ⋯,

where the dots stand for a linear combination of equations generated by the second and the third equation of S_{2} and a ≥ 1. Transfering D_{x}^{a−1}D_{y}^{b}D_{z}^{c+1}( 3u_{x^{2}}^{3} + u_{xz}^{2} − 2u_{z^{2}}^{1} ) into the left-hand side of this relation, we obtain the following relation

−2u_{x^{a}y^{b}z^{c+2}}^{2} + 2u_{x^{a−1}y^{b}z^{c+3}}^{1} = −D_{x}^{a−2}D_{y}^{b}D_{z}^{c+2}( 2u_{x^{2}}^{2} − u_{xz}^{1} − 3u_{z^{2}}^{0} ) + ⋯,

where the dots stand for a linear combination of equations generated by the third equation of S_{2} and a ≥ 2. Transfering −D_{x}^{a−2}D_{y}^{b}D_{z}^{c+2}( 2u_{x^{2}}^{2} − u_{xz}^{1} − 3u_{z^{2}}^{0} ) into the left-hand side of this relation, we obtain the following

u_{x^{a−1}y^{b}z^{c+3}}^{1} + 3u_{x^{a−2}y^{b}z^{c+4}}^{0} = −D_{x}^{a−3}D_{y}^{b}D_{z}^{c+3}( −u_{x^{2}}^{1} + u_{xz}^{0} ) + ⋯,

where a ≥ 3.

Thus, equations of form (30) are a linear combinations of equations from S_{2}^{(k)} generated by the first three equations of S_{2} if and only if a ≥ 3.

Obviously, the number of these equations in S_{2}^{(k)} is \frac{(k − 2)(k − 1)}{2}. Therefore, the rank

of S_{2}^{(k)} is equal to

4\frac{(k + 1)(k + 2)}{2} − \frac{(k − 2)(k − 1)}{2} = \frac{3}{2}(k^{2} + 5k + 2).

This completes the lemma’s proof.

5. Integrability conditions

Theorem 5.1 Let Y be an ODE of form t (1). Let p be a point in the domain U of S_{Y}. Then Y is reducible by a contact transformation to the form y′′′ = 0 in a neighborhood of p iff there exists a neighborhood V ⊂ U of p such that Im (j_{2}S_{Y} |_{V}) ⊂ Orb_{2}.

Proof. The necessity statement is obvious. Let us prove the sufficiency. We can assume that Im (j_{2}S_{Y}) ⊂ Orb_{2}. Let us consider PDE system E = τ_{∞,3}( E(Y)^{(∞)} ). Obviously, E ⊂ E(Y) and solution sets for these systems are the same. It follows from the theorems 4.2, 4.3 that Im j_{∞}S_{Y} ⊂ Orb_{∞}. Therefore, E(Y)^{(∞)} ≠ ∅. It follows that E ≠ ∅. Now we prove that E has solutions. To this end we prove that E is a completely integrable PDE system (see theorem 2.1).

First, let us prove that τ_{4,3} : E^{(1)} → E is a surjection. Let θ_{3} ∈ E; then there is θ_{∞} ∈ E(Y)^{(∞)} such that τ_{∞,3}(θ_{∞}) = θ_{3}. Let τ_{∞,4}(θ_{∞}) = θ_{4}. Taking into account K_{θ_{∞}} ⊂ T_{θ_{∞}}(E(Y)^{(∞)}) and E = τ_{∞,3}( E(Y)^{(∞)} ), we obtain K_{θ_{4}} = (τ_{∞,3})_{∗}(K_{θ_{∞}}) ⊂ T_{θ_{3}}(E). It follows that θ_{4} ∈ E^{(1)}. Thus the projection τ_{4,3} : E^{(1)} ⟶ E is surjection.

Second, let us prove that Smbl_{θ_{3}}E = {0} for any θ_{3} ∈ E. Let θ_{3} ∈ E. By definition Smbl E = T_{θ_{3}}( τ_{3,2}^{−1}(θ_{2}) ∩ E ), where θ_{2} = τ_{3,2}(θ_{3}). Obviously,

τ_{3,2}^{−1}(θ_{2}) ∩ E = τ_{∞,3}( τ_{∞,2}^{−1}(θ_{2}) ∩ E(Y)^{(∞)} )

and

Smbl_{θ_{3}}E = (τ_{∞,3})_{∗}T_{θ_{∞}}( τ_{∞,2}^{−1}(θ_{2}) ∩ E(Y)^{(∞)} ).

Let p = τ_{3}(θ_{3}). We define the subgroup H^{∞,2}(p) of the isotropy group H^{∞}(p) of the point [S_{0}]_{p}^{∞} ∈ J^{∞}π by the formula

H^{∞,2}(p) = { [f]_{p}^{∞} ∈ H^{∞}(p) | [f]_{p}^{1} = [Id]_{p}^{1} },

where Id is the identity diffeomorphism. Let θ_{∞}, θ̄_{∞} ∈ τ_{∞,2}^{−1}(θ_{2}) ∩ E(Y)^{(∞)}; then θ_{∞}^{−1} · θ̄_{∞} ∈ H^{∞,2}(p). Consequently,

τ_{∞,2}^{−1}(θ_{2}) ∩ E(Y)^{(∞)} = θ_{∞} · H^{∞,2}(p).

Obviously, the Lie algebra of H^{∞,2}(p) is H^{∞,2}(p) defined by (22). Then

T_{θ_{∞}}(τ_{∞,2}^{−1}(θ_{2}) ∩ E(Y)^{(∞)}) = (θ_{∞})_{∗}H^{∞,2}(p).

It follows from corollary 3.5 that H^{∞,2}(p) = { 0 }.

Taking into account theorem 4.2, we deduce the following

Corollary 5.2 Let Y be an ODE of form (1) and let p be a point in the domain of the coefficients u^{3}(x, y, y′), u^{2}(x, y, y′), u^{1}(x, y, y′), u^{0}(x, y, y′) of Y. Then Y is reducible to the form y′′′ = 0 by a contact transformation in a neighborhood of p iff the collection

( u^{3}(x, y, z), u^{2}(x, y, z), u^{1}(x, y, z), u^{0}(x, y, z) )

is a solution of the PDE system

F^{3} = 0, F^{2} = 0, F^{1} = 0, F^{0} = 0

in a neighborhood of p.

6. Appendix

This section contains REDUCE programs that were used in calculations in this paper. In texts of these programs we denote

\frac{∂^{i1+i2+i3}f}{(∂x)^{i1}(∂y)^{i2}(∂z)^{i3}} by f \overbrace{x … x}^{i1} \overbrace{y … y}^{i2} \overbrace{z … z}^{i3} or by f(i1, i2, i3)

and we also denote

\frac{∂^{i1+i2+i3}u^{n}}{(∂x)^{i1}(∂y)^{i2}(∂z)^{i3}} by un \overbrace{x … x}^{i1} \overbrace{y … y}^{i2} \overbrace{z … z}^{i3} or by u(n, i1, i2, i3),

where n = 3, 2, 1, 0.

6.1.

The following program calculates the law of coefficient’s transformation for 3-order ODEs of form (1) with respect to contact transformations. Here, an initial ODE has the form

p3 = W (3) ∗ p2 ∗ ∗3 + W (2) ∗ p2 ∗ ∗2 + W (1) ∗ p2 + W (0);

a contact transformation has the form

( x, y, z) ⟼ ( q(x, y, z), p(x, y, z), r(x, y, z) ),

where r = (df(p, x) + z ∗ df(p, y))/(df(q, x) + z ∗ df(q, y)) and df(p, z)(df(q, x) + z ∗ df(q, y)) − df(q, z) ∗ (df(p, x) + z ∗ df(p, y)) = 0.

The program expresses p3 and p2 in terms of y3, y2 and z and substitutes these expressions in the initial ODE. As a result an ODE of the form

y3 = H ∗ [u(3) ∗ y2 ∗ ∗3 + u(2) ∗ y2 ∗ ∗2 + u(1) ∗ y2 + u0]

is obtained. Next, the program calculates H. At the end it returns the coefficients of the resulting ODE expressed in terms of W (3), W (2), W (1), W (0) and derivatives of the functions q, p, r.

Program 6.1 .

OFF NAT;

DEPEND q,x,y,z; DEPEND p,x,y,z; DEPEND r,x,y,z; OPERATOR DX,W,u,n;

FOR ALL A LET

DX(A)=df(A,x)+df(A,y)*z+df(A,z)*y2+df(A,y2)*y3+df(A,y3)*y4; LET

df(p,z)= df(q,z)*(df(p,x)+z*df(p,y))/(df(q,x)+z*df(q,y)),

p2 = DX(r)/DX(q),

p3 = DX(p2)/DX(q),

n(3)=6, n(2)=2, n(1)=1, n(0)=1; % Here n(i)=i!.

G := p3 - W(3)*p2**3 - W(2)*p2**2 - W(1)*p2 - W(0)$ H:= DF(G,y3)$ yy:= G/H - y3$ % It’s a preliminary form of right-hand side of

% the transformed ODE is

H :=DEN(yy)$ G:=H*yy$ Clear yy$ H:=SUB(r=(z*df(p,y)+df(p,x))/(z*df(q,y)+df(q,x)),H)$ H:=1/H$ Clear r$

OUT result$ % y3:=H*yy; % It’s a final form of the transformed ODE. FACTOR W(3),W(2),W(1),W(0)$ FOR i:=3 STEP -1 UNTIL 0 DO

WRITE u(i),”=H*[ ”,SUB(y2=0,DF(G,y2,i))/n(i),” ],”$ WRITE ”where H=”,H$ CLEAR G,H$

SHUT result$

END;

6.2.

The following program calculates the velocity of deformation of jet-sections of π with respect to contact transformations.

Let ξ_{f} be an arbitrary contact vector field in B; let f_{s}(x, y, z) = ( q(s, x, y, z), p(s, x, y, z), r(s, x, y, z) ) be its flow and let S = ( W (3), W (2), W (1), W (0) ) be an arbitrary section of the bundle π. Then the deformation velocity of S

is defined by formula (8) \frac{d}{ds}( f_{s}^{(0)} ∘ S ∘ f_{s})|_{s=0}. The program substitutes the transformed

section f_{s}^{(0)} ∘ S ∘ f_{s} = ( u(3), u(2), u(1), u(0) ) into the procedure \frac{d}{ds}|_{s=0}. This procedure is denoted by DS in the program. The transformed section ( u(3), u(2), u(1), u(0) ) is obtained with the help of program 6.1.

Program 6.2 .

OFF NAT;

OPERATOR q,p,r,W,u; DEPEND q, s,x,y,z; DEPEND p, s,x,y,z; DEPEND r, s,x,y,z; DEPEND W(3),s; DEPEND W(2),s; DEPEND W(1),s; DEPEND W(0),s;

PROCEDURE DS(AA); BEGIN

SCALAR A; A:= df(AA,s);

A:= sub(df(q,x)=1, sub(df(q,y)=0, sub(df(q,z)=0, sub(df(q,x,2)=0, sub(df(q,x,y)=0, sub(df(q,x,z)=0, sub(df(q,y,2)=0, sub(df(q,y,z)=0, sub(df(q,z,2)=0, sub(df(p,x)=0, sub(df(p,y)=1, sub(df(r,x)=0, sub(df(r,y)=0, sub(df(r,z)=1, sub(df(r,x,2)=0, sub(df(r,x,y)=0, sub(df(r,x,z)=0, sub(df(r,y,2)=0, sub(df(r,y,z)=0, sub(df(r,z,2)=0, A))))))))))))))))))));

A:= sub(df(q,s,x)=-fxz, sub(df(q,s,y)=-fyz, sub(df(q,s,z)=-fzz, sub(df(q,s,x,2)=-fxxz, sub(df(q,s,x,y)=-fxyz, sub(df(q,s,x,z)=-fxzz, sub(df(q,s,y,2)=-fyyz, sub(df(q,s,y,z)=-fyzz, sub(df(q,s,z,2)=-fzzz, sub(df(p,s,x)=fx-z*fxz, sub(df(p,s,y)=fy-z*fyz, sub(df(r,s,x)=fxx + z*fxy, sub(df(r,s,y)=fxy + z*fyy, sub(df(r,s,z)=fxz + z*fyz + fy, sub(df(r,s,x,2)=fxxx + z*fxxy, sub(df(r,s,x,y)=fxxy + z*fxyy, sub(df(r,s,x,z)=fxxz + z*fxyz + fxy, sub(df(r,s,y,2)=fxyy + z*fyyy, sub(df(r,s,y,z)=fxyz + z*fyyz + fyy, sub(df(r,s,z,2)=fxzz + z*fyzz + 2*fyz, A)))))))))))))))))))); A:= sub(W(3)=u3, sub(df(W(3),s)= u3x*fz - u3y*(f-z*fz) - u3z*(fx+z*fy),A));

A:= sub(W(2)=u2,

sub(df(W(2),s)= u2x*fz - u2y*(f-z*fz) - u2z*(fx+z*fy),A));

A:= sub(W(1)=u1,

sub(df(W(1),s)= u1x*fz - u1y*(f-z*fz) - u1z*(fx+z*fy),A));

A:= sub(W(0)=u0,

sub(df(W(0),s)= u0x*fz - u0y*(f-z*fz) - u0z*(fx+z*fy),A));

RETURN A; END;

% Values of functions u^{3}, u^{2}, u^{1}, u^{0} (7) can be obtained with the help of % program 6.1. g(3):=u(3)$ g(2):=u(2)$ g(1):=u(1)$ g(0):=u(0)$

ORDER u3x, u3y, u3z, u2x, u2y, u2z, u1x, u1y, u1z, u0x, u0y, u0z, u3, u2, u1, u0;

OUT result$ DS(g(3)); DS(g(2)); DS(g(1)); DS(g(0));

SHUT result$

END$

6.3.

The following program performs the following tasks:

- it calculates the linear algebraic system (15) with θ_{k} ∈ π_{k}^{−1}(0),

- it reduces this system by elementary transformations to an equivalent system with coefficient matrix of the form

( \begin{matrix}A \\ B \end{matrix} ) ,

where A is a step form matrix of maximal rank such that the first element of every step is a nonzero integer, components of B are zeros or polynomials with of nonzero degree.

Program 6.3 .

OFF NAT;

LET k=2; % k=0,1,... is order of the jet bundle.

OPERATOR f,u,g; DEPEND f(i1,i2,i3), X,Y,Z; DEPEND u(n,i1,i2,i3), n,X,Y,Z;

FOR ALL n,i1,i2,i3 LET DF(f(i1,i2,i3),X)=f(i1+1,i2,i3), DF(f(i1,i2,i3),Y)=f(i1,i2+1,i3), DF(f(i1,i2,i3),Z)=f(i1,i2,i3+1),

DF(u(n,i1,i2,i3),X)=u(n,i1+1,i2,i3),

DF(u(n,i1,i2,i3),Y)=u(n,i1,i2+1,i3),

DF(u(n,i1,i2,i3),Z)=u(n,i1,i2,i3+1);

% Functions ψ^{3}(φ), ψ^{2}(φ), ψ^{1}(φ), ψ^{0}(φ) are the left-hand sides of % equations of system (15) with k = 0. They can be obtained with the help % of program 6.2.

g(0):=ψ^{3}(φ) g(1):=ψ^{2}(φ) g(2):=ψ^{1}(φ) g(3):=ψ^{0}(φ)

% The following procedure creates the array ”arrders” of all Nth % derivatives of the function FN=FN(x,y,z) PROCEDURE DERS(N,FN); begin

scalar j3,der,num,j;

num:=(N+1)*(N+2)/2-1; % num+1 is the number of all N-th deriv-s.

array arrders(num);

j:=-1;

for j1:=N step (-1) until 0 do

≪ j3:=-1; for j2:=N-j1 step (-1) until 0 do ≪ j3:=j3+1; j:=j+1; arrders(j):=df(FN,x,j1,y,j2,z,j3)

≫ ≫;

end;

% The following procedure creates the array ”arrders1” of all N-th % derivatives of the function FN=FN(x,y,z) with % Z=0, f(0,0,0)=0, f(1,0,0)=0, f(0,0,1)=0. PROCEDURE DERS1(N,FN); begin

scalar j3,der,num,j;

num:=(N+1)*(N+2)/2-1; % num+1 is the number of all N-th deriv-s.

array arrders1(num); j:=-1; for j1:=N step (-1) until 0 do ≪ j3:=-1;

for j2:=N-j1 step (-1) until 0 do

≪ j3:=j3+1; j:=j+1; arrders1(j):=SUB(f(0,0,0)=0,SUB(f(1,0,0)=0,SUB(f(0,0,1)=0, SUB(z=0,df(FN,x,j1,y,j2,z,j3) ) )));

≫ ≫;

end;

% The following procedure returns a number of all derivatives of order % ≤ N for the function f(0,0,0) PROCEDURE NUMVARS(N); begin

scalar s; s:=0; for m:=0:N do s:=s+(m+1)*(m+2)/2; return s;

end;

% The following procedure creates the array ”arrvars” of all derivatives % of order ≤ N for the function f(0,0,0) except the derivatives % f(0,0,0), f(1,0,0) and f(0,0,1). PROCEDURE VARS(N); begin

scalar num,j1,j2; num:=NUMVARS(N)-3-1; array arrvars(num); arrvars(0):=f(0,1,0); for m:=2:N do ≪ j1:=NUMVARS(m-1)-3;

j2:=NUMVARS(m)-3-1;

DERS(m,f(0,0,0));

for j:=j1:j2 do arrvars(j):=arrders(j-j1) ≫; clear arrders;

end;

% The following procedure creates an array ”arrpsys”. This array consists % of left-hand sides of equations of system (15) with % θ_{k} ∈ π_{k}^{−1}(0) and k = N. PROCEDURE PSYS(N); begin

scalar num,j1,j2,r,rr; num:=4*NUMVARS(N)-1; array arrpsys(num); for m:=0:N do ≪ r:=(m+1)*(m+2)/2;

rr:=4*NUMVARS(m-1);

for j:=0:3 do

≪ DERS1(m,g(j)); j1:=rr+j*r; j2:=j1+r-1; for s:=j1:j2 do arrpsys(s):=arrders1(s-j1)

≫ ≫; clear arrders1;

end;

VARS(k+3)$ Lngth:=NUMVARS(k+3)-3-1$ % It’s the length of ”arrvars” PSYS(k)$ Len:=4*NUMVARS(k)-1$ % It’s the length of ”arrpsys” q:=0$

OUT result$

for j:=0:Lngth do ≪ Vr:=arrvars(j)$ WRITE Vr;

s:=-1$

REPEAT

≪ s:=s+1$ Eqv:=arrpsys(s)$ Cff:=DF(Eqv,Vr)

≫

UNTIL ( ( NUMBERP(Cff) AND (Cff NEQ 0) ) OR (s=Len) )$ IF ( NUMBERP(Cff) AND (Cff NEQ 0) ) THEN ≪ arrpsys(s):=0$

q:=q+1$

WRITE ” E”,q,”: ”,Eqv,”=0”; FOR r:=0:Len DO ≪ Cff1:=DF(arrpsys(r),Vr)$

IF (Cff1 NEQ 0) THEN arrpsys(r):=Cff*arrpsys(r)-Cff1*Eqv ≫$ CLEAR Eqv,Cff,Cff1$

≫ ≫$ CLEAR arrvars,Lngth$ WRITE ”Conditions”; FOR w:=0:Len DO IF (arrpsys(w) NEQ 0) THEN ≪ q:=q+1$

WRITE ” E”,q,”: ”,arrpsys(w),”=0”; arrpsys(w):=0$

≫$ CLEAR q,Len, arrpsys$

SHUT result$

END$

6.4.

If we add to the program 6.3 the following substitution command

FOR ALL n,i1,i2,i3 LET u(n,i1,i2,i3)=0;

then the resulting program reduces the system of homogeneous linear algebraic equations (16) – (19) to an equivalent system with coefficient matrix of the step form.

References

[1] S.S.Chern. Sur la geometrie d’une equation differentielle du troisieme ordre, Comptes Rendus, 1937, 1227.

[2] V.N.Gusyatnikova, V.A.Yumaguzhin, Point transformations and linearizability of second-order ordinary differential equations, Matematicheskie Zametki, Vol. 49, 1991, No. 1, pp. 146-148 (in Russian, English translation in: Soviet Mat. Zametki).

[3] E.Cartan, Sur les varietes a connexion projective, Bull. Soc. Math. France, Vol. 52, pp. 205-241, 1924.

[4] C.Grissom, G.Tompson, G.Wilkens, Linearization of second order ordinary differential equations via Cartan’s equivalence method, Journal of differential equations, Vol. 77, pp. 1-15, 1989.

[5] M.Kuranishi. Lectures on involutive systems of partial differential equations, São Paulo, 1967.

[6] I.S.Krasil’shchik, V.V.Lychagin, A.M.Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.