Differential invariants and equivalence of ODEs $y''=a^3(x,y)y'^3+a^2(x,y)y'^2+a^1(x,y)y'+a^0(x,y)$

TL;DR

This paper develops the algebra of differential invariants for a class of third-order polynomial ODEs and solves their equivalence problem under point transformations, advancing classification methods in differential geometry.

Contribution

It constructs the algebra of differential invariants and solves the equivalence problem for a specific class of third-order polynomial ODEs.

Findings

Algebra of differential invariants is explicitly constructed.

Equivalence problem for the class of equations is solved.

Provides a classification framework for these ODEs.

Abstract

This paper is devoted to ordinary differential equations of the form $$y''=a^3(x,y)y'^3+a^2(x,y)y'^2+a^1(x,y)y'+a^0(x,y)$$ The algebra of all differential invariants of point transformations is constructed for these equations in general position and the equivalence problem is solved for them.

Equations

$$y^{\prime \prime }=a^3(x,y)(y^{\prime }{)}^3+a^2(x,y)(y^{\prime }{)}^2+a^1(x,y)y^{\prime }+a^0(x,y).$$

$$y^{\prime \prime }=a^3(x,y)(y^{\prime }{)}^3+a^2(x,y)(y^{\prime }{)}^2+a^1(x,y)y^{\prime }+a^0(x,y)$$

$$\pi :E⟶M\pi :(p,u^1,\dots ,u^4)\mapsto p,p=(x,y),$$

$$S_{\mathcal{E}}\colon M\longrightarrow E,\quad S_{\mathcal{E}}\colon p\mapsto\big{(}\,p,\,a^{0}(p),\,\dots,\,a^{3}(p)\,\big{)}.$$

$${\tilde{y}}^{\prime \prime }={\tilde{a}}^3(\tilde{x},\tilde{y})({\tilde{y}}^{\prime }{)}^3+{\tilde{a}}^2(\tilde{x},\tilde{y})({\tilde{y}}^{\prime }{)}^2+{\tilde{a}}^1(\tilde{x},\tilde{y}){\tilde{y}}^{\prime }+{\tilde{a}}^0(\tilde{x},\tilde{y}).$$

$$\tilde{a}^{i}(\tilde{p})=\Phi^{i}\bigl{(}\,a^{0}(f^{-1}(\tilde{p})),\ldots,a^{3}(f^{-1}(\tilde{p})),\,j^{2}_{\tilde{p}}f^{-1}\,\bigr{)},\quad i=0,1,2,3.$$

$$\tilde{p}=f(p),\quad\tilde{u}^{i}=\Phi^{i-1}\bigl{(}\,u^{1},\ldots,u^{4},\,j^{2}_{f(p)}f^{-1}\bigr{)},\,\;i=1,\ldots,4$$

$$\pi \circ f^{(0)}=f\circ \pi ,{\mathrm{id}}_M^{(0)}={\mathrm{id}}_E,(f\circ h{)}^{(0)}=f^{(0)}\circ h^{(0)},f,h\in \mathcal{G}(M).$$

$$S_{\tilde{\mathcal{E}}}=f^{(0)}\circ S_{\mathcal{E}}\circ f^{-1},f\in \mathcal{G}(M).$$

$$f^{(k)}(\,j^{k}_{p}S\,)=j^{k}_{f(p)}\bigl{(}f^{(0)}\circ S\circ f^{-1}\bigr{)},\quad k=0,1,2,\ldots.$$

$$(j_{k+1}S)^{*}(\widehat{d}f)=d\bigl{(}(j_{k}S)^{*}(f)\bigr{)}$$

$$\hat{d}f=\frac{df}{dx}dx+\frac{df}{dy}dy,$$

$$\hat{d}f=\frac{df}{d{I}_i}\hat{d}{I}_i.$$

$${\xi }_1=\frac{1}{(L_3{)}^{2/5}}(L_2\frac{\partial }{\partial x}-L_1\frac{\partial }{\partial y}),{\xi }_2=\frac{1}{(L_3{)}^{4/5}}({\Psi }_2\frac{\partial }{\partial x}-{\Psi }_1\frac{\partial }{\partial y}),$$

$$L_1=3u_{yy}^1-2u_{xy}^2+u_{xx}^3$$

$$+3u^4{u}_x^1-3u^3{u}_y^1+2u^2{u}_y^2-u^2{u}_x^3-3u^1{u}_y^3+6u^1{u}_x^4,$$

$$L_2=3u_{xx}^4-2u_{xy}^3+u_{yy}^2$$

$$-3u^1{u}_y^4+3u^2{u}_x^4-2u^3{u}_x^3+u^3{u}_y^2+3u^4{u}_x^2-6u^4{u}_y^1,$$

$$\begin{gathered} L_3=L_2(L_1\frac{d{L}_2}{dx}-L_2\frac{d{L}_1}{dx})-L_1(L_1\frac{d{L}_2}{dy}-L_2\frac{d{L}_1}{dy}) \\ +L_1^3{u}^4-L_1^2{L}_2{u}^3+L_1{L}_2^2{u}^2-L_2^3{u}^1, \end{gathered}$$

$${\Psi }_1=-L_1^2{u}^3+2L_1{L}_2{u}^2-3L_2^2{u}^1-L_1\frac{d{L}_1}{dy}+4L_1\frac{d{L}_2}{dx}-3L_2\frac{d{L}_1}{dx},$$

$${\Psi }_2=-L_2^2{u}^2+2L_1{L}_2{u}^3-3L_1^2{u}^4+L_2\frac{d{L}_2}{dx}-4L_2\frac{d{L}_1}{dy}+3L_1\frac{d{L}_2}{dy}.$$

$$\nu =L_3^{1/5}(d{x}_1\wedge d{x}_2),$$

$$I_1=\frac{{\xi }_1(\nu )}{\nu },I_2=\frac{{\xi }_2(\nu )}{\nu }$$

$$j_4{S}_{\mathcal{E}}(O^{\prime })\subset O.$$

$$(j_4{S}_{\mathcal{E}}{)}^{\ast }{I}_i=I_i\circ j_4{S}_{\mathcal{E}},i=1,2,$$

$$(j_{5}S_{\mathcal{E}})^{*}(\widehat{d}I_{1}\wedge\widehat{d}I_{2})=(j_{5}S_{\mathcal{E}})^{*}\widehat{d}I_{1}\wedge(j_{5}S_{\mathcal{E}})^{*}\widehat{d}I_{2}=\\ d\big{(}(j_{4}S_{\mathcal{E}})^{*}I_{1}\big{)}\wedge d\big{(}(j_{4}S_{\mathcal{E}})^{*}I_{2}\big{)})\neq 0.$$

$$g=(j_{4}S_{\mathcal{E}})^{*}(I_{1},I_{2})\colon O^{\prime}\longrightarrow O^{\prime},\;\;g\colon(x,y)\mapsto\big{(}\tilde{x}(x,y),\tilde{y}(x,y)\big{)}$$

$${\tilde{y}}^{\prime \prime }={\tilde{J}}_3(\tilde{x},\tilde{y})({\tilde{y}}^{\prime }{)}^3+{\tilde{J}}_2(\tilde{x},\tilde{y})({\tilde{y}}^{\prime }{)}^2+{\tilde{J}}_1(\tilde{x},\tilde{y}){\tilde{y}}^{\prime }+{\tilde{J}}_0(\tilde{x},\tilde{y}).$$

$$S_{\tilde{\mathcal{E}}}=\big{(}\,\tilde{J}_{3}(\tilde{x},\tilde{y}),\,\tilde{J}_{2}(\tilde{x},\tilde{y}),\,\tilde{J}_{1}(\tilde{x},\tilde{y}),\,\tilde{J}_{0}(\tilde{x},\tilde{y})\,\big{)}.$$

$$S_{\tilde{\mathcal{E}}}=g^{(0)}\circ S_{\mathcal{E}}\circ g^{-1}.$$

Full text

[TABLE]

The algebra of all differential invariants of point transformations is constructed for these equations in general position and the equivalence problem is solved for them.

1. Introduction

This paper is devoted to differential invariants of point transformations of equations

[TABLE]

in general position.

There are different approaches to construct differential invariants of these equations: E. Cartan [3], R.B. Gardner [4], S. Lie [5, 6], R. Liouville [7], G. Thomsen [11], A. Tresse [12, 13], R. Sharipov [9, 10], and V. Yumaguzhin [14].

In work [14] a complete family of generators of the algebra of all differential invariants under consideration was constructed for the first time. In addition, a complete family of all differential syzygies of these generators are obtained.

In this paper, it is represented: a direct description of this algebra and a solution of the equivalence problem of equations (1) in general position under point transformations.

1.1. Preliminaries

1.1.1. Notations

Let $M$ be a connected smooth $n$-dimensional manifold, $x_{1},\ldots,x_{n}$ be local coordinates in $M$, $x=(x_{1},\ldots,x_{n})$, and ${\mathcal{G}(M)}$ be a pseudogroup of all diffeomorphisms of $M$.

Let $\pi\colon E\rightarrow M$ be a $m$-dimensional vector bundle over $M$, $x_{1},\ldots,x_{n}$, $u^{1}$, $\ldots$, $u^{m}$ be local coordinates in $\pi$, and $C^{\infty}(\pi)$ be the $C^{\infty}(M)$-module of smooth sections of $\pi$.

Let $\pi_{k}\colon J^{k}\pi\rightarrow M$ be the bundle of $k$-jets of sections of $\pi$, $\pi_{k,l}\colon J^{k}\pi\rightarrow J^{l}\pi$,  $k>l$, be the projection defined by reductions of $k$-jets to their $l$-jets, and $x_{1},\dots,x_{n}$, $u^{i}_{\sigma}$, $i=1,\ldots,m$, $0\leq|\sigma|\leq k$, where $\sigma$ is the multi-index $\{j_{1}\ldots j_{r}\}$, $|\sigma|=r$,  $j_{1},\ldots,j_{r}=1,\ldots,n$, be canonical coordinates in $J^{k}\pi$.

Any sections $S\in C^{\infty}(\pi)$ defines its $k$-jet prolongation $j_{k}S\in C^{\infty}(\pi_{k})$ by the formula $j_{k}S(a)=j_{a}^{k}S$.

1.1.2. Natural bundles

A bundle $\pi$ is called natural if for each $f\in{\mathcal{G}(M)}$ there is defined a diffeomorphism $f^{(0)}\colon E\rightarrow E$ such that the following conditions are satisfied:

1. (1)

$\pi\circ f^{(0)}=f\circ\pi$, 2. (2)

$\mathop{\rm id}\nolimits_{M}^{(0)}=\mathop{\rm id}\nolimits_{E}$, where $\mathop{\rm id}\nolimits_{M}$ and $\mathop{\rm id}\nolimits_{E}$ are identity maps of $M$ and $E$ respectively, 3. (3)

$(f\circ g)^{(0)}=f^{(0)}\circ g^{(0)}$  for all $f,g\in{\mathcal{G}(M)}$,

Corollary 1.1**.**

If $\pi$ is a natural bundle, then for each $f\in{\mathcal{G}(M)}$ its diffeomorphism $f^{(0)}\colon E\rightarrow E$ is uniquely determined.

For this reason we will call this unique diffeomorphism $f^{(0)}$ as the natural lift of $f$ to the bundle $\pi$.

A section of a natural bundle is called a differential structure (on the base of $\pi$), see [1].

Differential order of a natural bundle is the smallest natural number $r$ such that for any $f\in\mathcal{G}(M)$ the value $f^{(0))}(\theta_{0})$ at a point $\theta_{0}\in E$, where $p=\pi(\theta_{0})$ is a point from domain of $f$, is determined by the $r$-jet $j_{p}^{r}f$.

1.1.3. Natural bundles of ODEs (1)

In the rest of this article we will denote by $\pi$ the following vector bundle

[TABLE]

where $M=\mathbb{R}^{2}$, $E=M\times\mathbb{R}^{4}$, $x,y$ are standard coordinates on $M$, $u^{1},\dots,u^{4}$ are standard coordinates on the fiber $\mathbb{R}^{4}$ of $\pi$.

Let $\mathcal{E}$ be an arbitrary ODE (1). Following [14], we will consider every equation $\mathcal{E}$ as a geometric structure. Namely, identify every equation $\mathcal{E}$ with the section $S_{\mathcal{E}}$ of $\pi$

[TABLE]

As a result, the set of all equations $\mathcal{E}$ is identified with the set $C^{\infty}(\pi)$ of all sections $S_{\mathcal{E}}$ of $\pi$.

It is well known, [2], that each diffeomorphism f\in\mathcal{G}(M),\;f\colon(x,y)\mapsto\big{(}\tilde{x}(x,y),\tilde{y}(x,y)\big{)} generates the transformation of every equation $\mathcal{E}$ to the equation $\tilde{\mathcal{E}}$ of the same form:

[TABLE]

The coefficients of $\tilde{\mathcal{E}}$ are expressed in terms of the coefficients of $\mathcal{E}$ and the $2$-jets of the inverse transformation $f^{-1}$ by formulas:

[TABLE]

It follows that the equations

[TABLE]

define the diffeomorphism $f^{(0)}$ of the total space $E$ of $\pi$.

These diffeomorphisms $f^{(0)}$ satisfy the conditions of naturalness:

[TABLE]

Thus, the bundle $\pi$ is a natural bundle (of equations $\mathcal{E}$), and every equation $\mathcal{E}$, considering as a section of $\pi$, is a geometric structure on $M$.

It follows from (4) that differential order of the natural bundle $\pi$ is 2.

1.1.4. Natural bundles of jets of ODEs (1)

Equations (4), represented in the terms of the corresponding sections, has the form:

[TABLE]

Every $f\in\mathcal{G}(M)$ is lifted to the diffeomorphism $f^{(k)}$ of the bundle $J^{k}\pi$ by the formula

[TABLE]

Lemma 1.2**.**

The bundle $\pi_{k}\colon J^{k}\pi\rightarrow M$ with lifting (5) of diffeomorphisms $\mathcal{G}(M)$ in the bundle $\pi_{k}$ is natural.

1.1.5. Orbits, equations, and differential invariants in general position

As a result of the action $\mathcal{G}(M)$ on $J^{k}\pi$, the latter is divided into orbits $\mathop{\rm Orb}\nolimits^{k}_{r}$, where $r$ is the dimension of an isotropy algebra of point of $\mathop{\rm Orb}\nolimits^{k}_{r}$.

We will call an orbit of codimension zero as an orbit of general position, other ones we will call as degenerate orbits.

We get from [14]:

• •

The bundles $J^{0}\pi$ and $J^{1}\pi$ are orbits of the action of $\mathcal{G}(\mathbb{R}^{2})$.

• •

The bundle $J^{2}\pi$ is the union of two orbits: $\mathop{\rm Orb}\nolimits^{2}_{4}$ is an orbit of general position and $\mathop{\rm Orb}\nolimits^{2}_{6}$ is a degenerate orbit. The first one is defined by the inequality $(L_{1},L_{2})\neq 0$, where $L_{1},L_{2}$ are defined by equations (9).

• •

The bundle $J^{3}\pi$ is the union of four orbits: $\mathop{\rm Orb}\nolimits^{3}_{0}$, $\mathop{\rm Orb}\nolimits^{3}_{1}$, $\mathop{\rm Orb}\nolimits^{3}_{2}$, and $\mathop{\rm Orb}\nolimits^{3}_{6}$. The first one is an orbit of general position, it is defined by inequalities $(L_{1},L_{2})\neq 0$ and $L_{3}\neq 0$, where $L_{3}$ is defined by equation (10), the remaining orbits are degenerate.

• •

The bundle $J^{k}\pi$, $k\geq 4$, is union of degenerate orbits.

By a differential invariant of order $k$, $k=0,1,2,\ldots$, we mean a smooth function on $J^{k}\pi$ invariant with respect to all lifted diffeomorphisms $f^{(k)}$, $f\in\mathcal{G}(M)$.

By a differential invariant of order $k$ of the equation $\mathcal{E}$ we mean the restrictions of some differential invariant of order $k$ on the graph of the section $j_{k}S_{\mathcal{E}}$.

Let $I_{1}$ and $I_{2}$ be differential invariants of order $k$. We will say that they are in general position in a domain $O\subset J^{k}\pi$, if $\widehat{d}\,I_{1}\wedge\widehat{d}\,I_{2}\neq 0$ in this domain.

By a tensor differential invariant of order $k$ we mean a smooth tensor field on $J^{k}\pi$ invariant with respect to all lifted diffeomorphisms $f^{(k)}$, $f\in\mathcal{G}(M)$.

1.1.6. Total differentials

Let $f$ be a smooth function on the manifold $J^{k}\pi$. Then its total differential $\widehat{d}f$ is the horizontal differential 1-form on the manifold $J^{k+1}\pi$, which satisfies the following condition

[TABLE]

for all $S\in C^{\infty}(\pi)$.

In canonical coordinates the total differential has the form

[TABLE]

where $\displaystyle\frac{d}{dx}=\frac{\partial}{\partial x}+u_{i}^{j}\frac{\partial}{\partial u^{j}}+\ldots+u^{j}_{x\,\sigma}\frac{\partial}{\partial u^{j}_{\sigma}}+\ldots$ and $\displaystyle\frac{d}{dy}=\frac{\partial}{\partial y}+u_{i}^{j}\frac{\partial}{\partial u^{j}}+\ldots+u^{j}_{\sigma\,y}\frac{\partial}{\partial u^{j}_{\sigma}}+\ldots$. are total derivatives.

1.1.7. Tresse derivatives

Let differential invariants $I_{1}$ and $I_{2}$ be in general position. Then Tresse derivatives $\displaystyle\frac{df}{dI_{i}}$, $i=1,2$, where $f\in C^{\infty}(J^{\infty}(\pi))$, are defined by the relation

[TABLE]

1.2. Tensor differential invariants

In these section, we introduce necessary for us tensor differential invariants. Their detailed computations can be found in [14].

1.2.1. Invariant vector fields

The following vector fields are tensor differential invariants defined on $\mathop{\rm Orb}\nolimits^{3}_{0}$, (see [14], p. 302, ref. (49))

[TABLE]

where $L_{1}$, $L_{2}$, $L_{3}$, and $\Psi_{1}$, $\Psi_{2}$ are defined by the following equations (see [14], p. 291, ref. (26); p. 292, ref. (28); p. 301 )

[TABLE]

[TABLE]

[TABLE]

The fields $\xi_{1}$ and $\xi_{2}$ are linear independent in every point $\theta_{3}\in\mathop{\rm Orb}\nolimits^{3}_{0}$ (see [14], p. 302, Proposition 15).

These fields are known tensor differential invariants, see [9, 10].

1.2.2. The invariant volume form

The following tensor differential invariant on $\mathop{\rm Orb}\nolimits^{3}_{0}$ (see [14], p. 301, ref. (47) )

[TABLE]

was first obtained by R. Liouville in [7].

2. The algebra of all differential invariants of ODEs (1)

2.1. Coordinates generated by invariants

Taking into account that $\xi_{1}$, $\xi_{2}$, and $\nu$ are tensor differential invariants of order 3 defined on $\mathop{\rm Orb}\nolimits_{0}^{3}$, we get that Lie derivatives $\xi_{1}(\nu)$, $\xi_{2}(\nu)$ are tensor differential invariants of order 4 defined on $\pi_{4,3}^{-1}(\mathop{\rm Orb}\nolimits_{0}^{3})$. These invariants define differential invariants $I_{1}$ and $I_{2}$ of order 4 by the formulas:

[TABLE]

which are also defined on $\pi_{4,3}^{-1}(\mathop{\rm Orb}\nolimits_{0}^{3})$.

Let $O$ be a domain in $\pi_{4,3}^{-1}(\mathop{\rm Orb}\nolimits^{3}_{0})$, where the invariants $I_{1}$ and $I_{2}$ are in general position, i.e. $\widehat{d}\,I_{1}\wedge\widehat{d}\,I_{2}\neq 0$ in this domain.

Let $S_{\mathcal{E}}\in C^{\infty}(\pi)$ and $O^{\prime}$ be a domain in $M$ such that

[TABLE]

Then we will say that the equation $\mathcal{E}$ (the section $S_{\mathcal{E}}$) is in general position in $O^{\prime}$.

The function

[TABLE]

is the restriction of invariant $I_{i}$ on the graph of section $j_{4}S_{\mathcal{E}}$. Thus it is a differential invariants of order 4 of the equation $\mathcal{E}$.

Proposition 2.1**.**

The differential invariants of $\mathcal{E}$: $(j_{4}S_{\mathcal{E}})^{*}I_{1}$, $(j_{4}S_{\mathcal{E}})^{*}I_{2}$, are independent coordinates on $O^{\prime}$.

Proof.

General position of the invariants $I_{1}$, $I_{2}$ means that $\widehat{d}I_{1}\wedge\widehat{d}I_{2}\neq 0$. It follows that

[TABLE]

The last inequality means that the differential invariants of $\mathcal{E}$: $(j_{4}S_{\mathcal{E}})^{*}I_{1}$ and $(j_{4}S_{\mathcal{E}})^{*}I_{2}$ are independent coordinates on the domain $O^{\prime}\subset M$. ∎

Let us denote these coordinates as $\tilde{x}=(j_{4}S_{\mathcal{E}})^{*}I_{1}$ and $\tilde{y}=(j_{4}S_{\mathcal{E}})^{*}I_{2}$.

Corollary 2.2**.**

Let (j_{4}S_{\mathcal{E}})^{*}(I_{1},I_{2})_{=}\big{(}(4S_{\mathcal{E}})^{*}I_{1},(4S_{\mathcal{E}})^{*}I_{2}\big{)}. Then the mapping

[TABLE]

is a diffeomorphism.

In the coordinates $\tilde{x},\tilde{y}$, the ODE $\mathcal{E}$ has the form $\tilde{\mathcal{E}}$:

[TABLE]

where the coefficients $\tilde{J}_{3}(\tilde{x},\tilde{y}),\ldots,\tilde{J}_{0}(\tilde{x},\tilde{y})$ are differential invariants. The section $S_{\tilde{\mathcal{E}}}$, see (2), has the form

[TABLE]

Let $g$ be the diffeomorphism defined by (13). Then

[TABLE]

Proposition 2.3**.**

The section $S_{\tilde{\mathcal{E}}}$ is invariant w.r.t. the action of $\mathcal{G}(M)$ on $C^{\infty}(\pi)$, i.e. $f^{0}\circ S_{\tilde{\mathcal{E}}}\circ f^{-1}=S_{\tilde{\mathcal{E}}}$ for all $f\in\mathcal{G}(M)$.

Proof.

f^{0}\circ S_{\tilde{\mathcal{E}}}\circ f^{-1}=f^{0}\circ\big{(}g^{(0)}\circ S_{\mathcal{E}}\circ g^{-1}\big{)}\circ f^{-1}=(f\circ g)^{(0)}\circ S_{\mathcal{E}}\circ(f\circ g)^{-1}.

Taking into account that $\tilde{x}$ and $\tilde{y}$ are differential invariants we get that $f\circ g=g$. This leads to the equality $f^{0}\circ S_{\tilde{\mathcal{E}}}\circ f^{-1}=S_{\tilde{\mathcal{E}}}$. ∎

Corollary 2.4**.**

Differential equation (14) is invariant w.r.t. the action of $\mathcal{G}(M)$ on differential equations (1).

2.2. The algebra of all differential invariants

The $n$-invariants principle, see [8], in our case of equations (1) leads to the following result.

Theorem 2.5**.**

Let an equation $\mathcal{E}$ of form (1) be in general position in the domain $O^{\prime}$ and $\tilde{\mathcal{E}}$ be its expression (14) in coordinates $\tilde{x},\tilde{y}$. Then the algebra of all differential invariants of the equation $\mathcal{E}$ consist of all smooth functions of differential invariants $\tilde{x},\tilde{y},\tilde{J}_{3},\tilde{J}_{2},\tilde{J}_{1},\tilde{J}_{0}$, and their Tresse derivatives.

2.3. The equivalence problem

Let $\mathcal{E}_{1}$, $\mathcal{E}_{2}$ be equations of form (1) in general position and $\tilde{\mathcal{E}}_{1}$, $\tilde{\mathcal{E}}_{2}$ be their forms in coordinates $(I_{1}\circ j_{4}S_{\mathcal{E}_{1}}$, $I_{2}\circ j_{4}S_{\mathcal{E}_{1}})$ and $(I_{1}\circ j_{4}S_{\mathcal{E}_{2}}$, $I_{2}\circ j_{4}S_{\mathcal{E}_{2}})$ respectively.

Theorem 2.6**.**

Let $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be equations of form (1) in general positions in domains $O^{\prime}_{1}\subset M$ and $O^{\prime}_{2}\subset M$ respectively. Then there exists a diffeomorphism $f\colon O^{\prime}_{1}\rightarrow O^{\prime}_{2}$ transforming $\mathcal{E}_{1}$ to $\mathcal{E}_{2}$ if and only if $\tilde{\mathcal{E}}_{1}=\tilde{\mathcal{E}}_{2}$.

Proof.

Suppose $\tilde{\mathcal{E}}_{1}=\tilde{\mathcal{E}}_{2}$, i.e. $S_{\tilde{\mathcal{E}}_{2}}=S_{\tilde{\mathcal{E}}_{1}}$. Then $g_{2}^{(0)}\circ S_{\mathcal{E}_{2}}\circ g_{2}^{-1}=g_{1}^{(0)}\circ S_{\mathcal{E}_{1}}\circ g_{1}^{-1}$ or $S_{\mathcal{E}_{2}}=(g_{2}^{-1}\circ g_{1})^{(0)}\circ S_{\mathcal{E}_{1}}\circ(g_{2}^{-1}\circ g_{1})^{-1}$. Therefore, the diffeomorphism $f$ transforming $\mathcal{E}_{1}$ to $\mathcal{E}_{2}$ is defined by the formula $f=g_{2}^{-1}\circ g_{1}$.

Inversely, suppose that there is $f\in\mathcal{G}(M)$ transforming $\mathcal{E}_{1}$ to $\mathcal{E}_{2}$, i.e. $S_{\mathcal{E}_{2}}=f^{(0)}\circ S_{\mathcal{E}_{1}}\circ f^{-1}$. Then S_{\tilde{\mathcal{E}}_{2}}=g_{2}^{(0)}\circ S_{\mathcal{E}_{2}}\circ g_{2}^{-1}=g_{2}^{(0)}\circ(f^{(0)}\circ S_{\mathcal{E}_{1}}\circ f^{-1})\circ g_{2}^{-1}=(g_{2}\circ f)^{(0)}\circ S_{\mathcal{E}_{1}}\circ(g_{2}\circ f)^{-1}=(g_{2}\circ f)^{(0)}\circ\big{(}(g_{1}^{-1})^{(0)}\circ S_{\tilde{\mathcal{E}}_{1}}\circ g_{1}\big{)}\circ(g_{2}\circ f)^{-1}=(g_{2}\circ f\circ g_{1}^{-1})^{(0)}\circ S_{\tilde{\mathcal{E}}_{1}}\circ(g_{2}\circ f\circ g_{1}^{-1})^{-1}=S_{\tilde{\mathcal{E}}_{1}}. The last equality follows from Proposition 2.3. ∎