# Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems

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## Abstract

**:**

## 1. Introduction

## 2. Notation, Basic Definitions, and Geometric Preliminaries

#### 2.1. Riemannian Manifolds and Levi-Civita Connection

#### 2.2. Geodesics: Definition and Second-Order Geodesic Vector Field

**Definition**

**1.**

#### 2.3. Lagrangian Dynamical Systems

- 1.
- The Cartan Lagrangian one-form ${\theta}_{L}={d}^{v}L=dL\circ S$, with S being the vertical endomorphism on the tangent bundle $\mathrm{T}M$.
- 2.
- The Cartan Lagrangian two-form ${\omega}_{L}=-d{\theta}_{L}$.
- 3.
- The energy function ${E}_{L}=\Delta L-L$, with $\Delta $ the Liouville vector field on $\mathrm{T}M$.
- 4.
- Assuming that ${\omega}_{L}$ is a symplectic form (see below), a unique vector field $\mathsf{\Gamma}\in \mathfrak{X}\left(\mathrm{T}M\right)$ solution to the Hamilton dynamical equation is defined:$$i\left(\mathsf{\Gamma}\right){\omega}_{L}=d{E}_{L}\phantom{\rule{0.166667em}{0ex}}.$$

#### 2.4. The Geodesic Flow as a Lagrangian Dynamical System

#### 2.5. Hamiltonian Dynamical Systems

#### 2.6. The Geodesic Flow as a Hamiltonian Dynamical System

## 3. Distinguished Vector Fields in a Riemann Manifold

#### 3.1. Definitions and Initial Relations

**Definition 2.**

**Definition**

**3.**

**Definition**

**4.**

#### 3.2. Some Properties of Geodesic Vector Fields

**Proof.**

#### 3.3. Another Interesting Property of Killing Vector Fields

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

#### 3.4. Killing Vector Fields and the Geodesic Flow

**Proposition**

**2.**

- (i)
- If ${\mathsf{\Gamma}}_{g}\in \mathfrak{X}\left(\mathrm{T}M\right)$ is the second-order geodesic vector field, then ${\mathcal{L}}_{{X}^{c}}{\mathsf{\Gamma}}_{g}=0$.
- (ii)
- If $\gamma :\mathbb{R}\to M$ is a geodesic curve, then ${\varphi}_{t}^{X}\circ \gamma $ is also a geodesic curve.
- (iii)
- If $Y\in \mathfrak{X}\left(M\right)$ is a geodesic vector field, i.e., ${\nabla}_{Y}Y=0$, then ${\left({\varphi}_{t}^{X}\right)}_{*}Y$ is a geodesic vector field, that is$${\nabla}_{{\left({\varphi}_{t}^{X}\right)}_{*}Y}{\left({\varphi}_{t}^{X}\right)}_{*}Y=0\phantom{\rule{0.166667em}{0ex}}.$$

**Proof.**

## 4. Hamilton–Jacobi Vector Fields

#### 4.1. Geometric Hamilton–Jacobi Theory

#### 4.2. Lagrangian Theory

#### 4.3. Hamiltonian Theory

**Proposition**

**3.**

**Proof.**

## 5. Hamilton–Jacobi Vector Fields for the Second-Order Geodesic Vector Field

**Definition**

**5.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 6. Hamilton–Jacobi Vector Fields and Complete Lift of Vector Fields

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

## 7. General Newtonian Dynamical Systems and Newtonian Vector Fields

#### 7.1. Newtonian Systems and Their Dynamical Equations

**Definition**

**6.**

**Definition**

**7.**

#### 7.2. Conserved Quantities and Dynamical Symmetries

**Lemma**

**2.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Comments**:

- 1.
- Observe that $i\left({X}^{c}\right)F=i\left(X\right)F=-g({Z}_{F},X)$; hence, the condition in the above proposition is that ${Z}_{F}$ and X are orthogonal.
- 2.
- If F is a basic one-form, that is $(M,g,F)$ is a Newtonian dynamical system, the result is the same.
- 3.
- In the case of the geodesic systems, $F=0$, the condition $g(X,{Z}_{F})=0$ is trivially satisfied, and we obtain the result of Proposition 1 as a particular case.
- 4.
- If the system is a conservative simple mechanical one and its potential function is $V:M\to \phantom{\rule{0.166667em}{0ex}}\mathbb{R}$, then ${Z}_{F}=\mathrm{grad}\phantom{\rule{0.166667em}{0ex}}V$ and $F=-dV$. In this situation, the condition $i\left({X}^{c}\right)F=0$ is ${\mathcal{L}}_{X}V=0$, that is the potential function V is invariant by the vector field X.

**Definition**

**8.**

**Proposition 10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

**Proposition**

**12.**

**Proof.**

**Comments**:

- 1.
- We remark about the difference between Proposition 9 and Proposition 12 for the characterisation of the vector field X as a generator of a conserved quantity or as a dynamical symmetry.
- 2.
- In the case of a conservative simple mechanical system, that is $F=-d\phantom{\rule{0.166667em}{0ex}}V$, if X is a Killing vector field, we have that:(a) If ${\mathcal{L}}_{X}V=0$, then the quantity $\widehat{i\left(X\right)g}$ is conserved by the dynamics;(b) If ${\mathcal{L}}_{X}dV=0$, then the vector field ${X}^{c}$ is a dynamical symmetry of $\mathsf{\Gamma}$. This condition on V and X is equivalent to $d{\mathcal{L}}_{X}V=0$, which is a condition weaker than Condition (a).In fact, we have that Condition (a) implies Condition (b), then we can have a dynamical symmetry X such that $\widehat{i\left(X\right)g}$ is not a conserved quantity. In these circumstances, the vector fields X satisfying Condition (a) are usually called Noether symmetries. Obviously, a Noether symmetry is a dynamical symmetry, but not the contrary.

#### 7.3. Newtonian Vector Fields

**Definition 9.**

**Proposition**

**13.**

**Proof.**

**Proposition 14.**

**Proof.**

#### 7.4. Hamilton–Jacobi Vector Fields for Generalised Newtonian Systems

**Proposition**

**15.**

**Proof.**

**Proposition**

**16.**

- (i)
- X is a generalised Newtonian vector field, ${\nabla}_{X}X={Z}_{F}\circ X$.
- (ii)
- The function ${T}_{g}\circ X$ on M satisfies the condition $d({T}_{g}\circ X)=-F\circ X$, where ${T}_{g}:\mathrm{T}M\to \mathbb{R}$ is the kinetic energy function defined by (14).

**Proof.**

#### 7.5. The Case of Nonholonomic Generalised Newtonian Systems

**Comment**: For every $(q,u)\in C$, we used the notation ${({V}_{(q,u)}\mathrm{T}M\cap {\mathrm{T}}_{(q,u)}C)}_{q}$ to denote the subspace of ${\mathrm{T}}_{q}M$ such that its vertical lifting gives $({V}_{(q,u)}\mathrm{T}M\cap {\mathrm{T}}_{(q,u)}C)$. As ${({V}_{(q,u)}\mathrm{T}M\cap {\mathrm{T}}_{(q,u)}C)}_{q}$ is a linear subspace of ${\mathrm{T}}_{q}M$, the meaning of its orthogonal complement with respect to the metric g is clear.

**Proposition**

**17.**

**Proposition**

**18.**

**Proposition**

**19.**

## 8. Application to Conformal Metrics

#### 8.1. Conformal Metrics: Definitions and Geodesics

**Definition**

**10.**

#### 8.2. Newtonian Vector Fields and Hamilton–Jacobi Equation for Geodesics of the Metric $\overline{g}=exp\left(2\phi \right)\phantom{\rule{0.166667em}{0ex}}g$

## 9. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Sections along Maps and External Forces

**Local expressions:**

**Definition**

**A1.**

#### Appendix A.1. The Case of a Riemannian Manifold

#### Appendix A.2. External Forces Depending on Velocities

#### Appendix A.3. External Forces as Semibasic Forms on TM

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Cariñena, J.F.; Muñoz-Lecanda, M.-C.
Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems. *Symmetry* **2023**, *15*, 181.
https://doi.org/10.3390/sym15010181

**AMA Style**

Cariñena JF, Muñoz-Lecanda M-C.
Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems. *Symmetry*. 2023; 15(1):181.
https://doi.org/10.3390/sym15010181

**Chicago/Turabian Style**

Cariñena, José F., and Miguel-C. Muñoz-Lecanda.
2023. "Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems" *Symmetry* 15, no. 1: 181.
https://doi.org/10.3390/sym15010181