# Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Affine Lagrangians on ${\mathbb{R}}^{\mathbf{2}}$

**Remark 1.**

## 3. Hamiltonian Formulation

**Remark 2.**

## 4. The Inverse Problem for First-Order Systems

#### 4.1. Theory of the Jacobi Multipliers

**Definition 1.**

#### 4.2. The Inverse Problem on ${\mathbb{R}}^{2}$

**Theorem 1.**

**Proof.**

## 5. Applications in Mechanical and Biological Systems

#### 5.1. Mechanical Systems

#### 5.2. Biological Systems

## 6. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Cariñena, J.F.; Fernández-Núñez, J.
Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems. *Symmetry* **2022**, *14*, 2520.
https://doi.org/10.3390/sym14122520

**AMA Style**

Cariñena JF, Fernández-Núñez J.
Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems. *Symmetry*. 2022; 14(12):2520.
https://doi.org/10.3390/sym14122520

**Chicago/Turabian Style**

Cariñena, José F., and José Fernández-Núñez.
2022. "Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems" *Symmetry* 14, no. 12: 2520.
https://doi.org/10.3390/sym14122520