# Quotients of Euler Equations on Space Curves

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## Abstract

**:**

## 1. Introduction

## 2. PDE Quotients

#### 2.1. Algebraic Structures in PDE Geometry

- If $dim\pi =1$, it is defined by the pseudogroup $\mathrm{Cont}\left(\pi \right)$ of the local contact transformations of the manifold ${\mathbf{J}}^{1}$.
- For $dim\pi \ge 2$, the jet geometry is defined by the pseudogroup $\mathrm{Point}\left(\pi \right)$ of the local point transformations, i.e., local diffeomorphisms of the manifold ${\mathbf{J}}^{0}$.

#### 2.2. The Rosenlicht Theorem

**Example**

**1.**

**Example**

**2.**

#### 2.3. Quotients of Algebraic Differential Equations

#### 2.4. Tresse Derivatives

**Proposition**

**1.**

#### 2.5. The Lie–Tresse Theorem

**Theorem**

**1.**

#### 2.6. Relations between Differential Equations and Their Quotients

- Let $u=f\left(x\right)$ be a solution of differential equation ${\mathcal{E}}_{}$ and let ${a}_{i}\left(f\right)$ and ${b}^{j}\left(f\right)$ be values of the invariants ${a}_{i}$ and ${b}^{j}$ on the section f. Then, locally, ${b}^{j}\left(f\right)={B}^{j}\left(a\left(f\right)\right)$, and therefore, ${b}^{j}={B}^{j}\left(a\right)$ is the solution of the quotient equation.
- The above construction is local. In general, the correspondence between solutions is valid on the level of generalized solutions, i.e., on the level of integral manifolds of the Cartan distributions. In addition, the correspondence will lead us to integral manifolds with singularities.
- Now let ${b}^{j}={B}^{j}\left(a\right)$ be a solution of the quotient equation. Then, considering equations ${b}^{j}-{B}^{j}\left(a\right)=0$ as a differential constraint for the equation ${\mathcal{E}}_{}$, we get a finite-type equation ${\mathcal{E}}_{}\cap \left\{{b}^{j}-{B}^{j}\left(a\right)=0\right\}$ with a solution that is a $\mathfrak{g}$-orbit of a solution of ${\mathcal{E}}_{}$.
- Symmetries of the quotient equation are Bäcklund-type transformations of the original equation ${\mathcal{E}}_{}$.

**Example**

**3.**

**Example**

**4.**

## 3. Euler Equations on a Curve

## 4. Quotient Equation

**Proposition**

**2.**

**Proposition**

**3.**

- $L=0$, $M=f\left(x\right)$.
- $L=\frac{{c}_{1}x}{y}$, $f\left(M\right){x}^{\frac{M}{{c}_{1}}}=y$.
- $L=\frac{{c}_{3}x}{{\left(lnx-{c}_{2}\right)}^{{c}_{1}}}{\left(\frac{-{c}_{5}lny+{c}_{4}}{{c}_{1}}{y}^{-{{c}_{1}}^{-1}}\right)}^{{c}_{1}},M=\frac{{c}_{3}y\left(-{c}_{5}lny+{c}_{4}\right)}{{\left(lnx-{c}_{2}\right)}^{{c}_{1}}\left(-lnx+{c}_{2}\right){c}_{5}}{\left(\frac{-{c}_{5}lny+{c}_{4}}{{c}_{1}}{y}^{-1/{c}_{1}}\right)}^{{c}_{1}}$.

#### Virial Expansion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Duyunova, A.; Lychagin, V.; Tychkov, S.
Quotients of Euler Equations on Space Curves. *Symmetry* **2021**, *13*, 186.
https://doi.org/10.3390/sym13020186

**AMA Style**

Duyunova A, Lychagin V, Tychkov S.
Quotients of Euler Equations on Space Curves. *Symmetry*. 2021; 13(2):186.
https://doi.org/10.3390/sym13020186

**Chicago/Turabian Style**

Duyunova, Anna, Valentin Lychagin, and Sergey Tychkov.
2021. "Quotients of Euler Equations on Space Curves" *Symmetry* 13, no. 2: 186.
https://doi.org/10.3390/sym13020186