# Combining 3-Momentum and Kinetic Energy on Galilei/Newton Spacetime

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Relativistic Classical Mechanics on Minkowski Spacetime

#### 2.1. Minkowski Spacetime $\mathbb{M}$

**∇**. We have

#### 2.2. Particle Mechanics on $\mathbb{M}$

#### 2.3. Continuum Mechanics on $\mathbb{M}$

**∇**is the spacetime covariant derivative operator. There is of course more than one way to arrive at these relativistic conservation laws. One approach is to derive the mechanics of one kind of material continuum—a gas of classical particles—from relativistic kinetic theory (e.g., [13]). This is a very direct and “hands-on” way of developing intuition for the physical meaning of the several scalars and tensors into which $\mathit{N}$ and $\mathit{T}$ can be decomposed. However, not all material continua are gases, and the kinetic theory of classical particles is not a fundamental physical theory. In fact Equations (18) and (19) do not depend on any particular microphysical model, and can be motivated on more general grounds (e.g., [2]). Here I present a modified version of this latter approach, streamlined with physical reasoning.

## 3. Galilei/Newton Spacetime, Baryon Conservation, and Mass Conservation

#### 3.1. Galilei/Newton Spacetime $\mathbb{G}$ and Its Geometric Consequences

#### 3.2. Mass Conservation on $\mathbb{G}$

## 4. A More Unified View of Classical Mechanics on Minkowski and Galilei/Newton Spacetimes

## 5. Conclusions

**P**and (55) for

**S**). Defining $\mathit{P}$ as a linear form instead of as a vector geometrizes the deep principle that momentum is conjugate to displacement (a vector). Additionally, as noted by Weyl [4], it is natural that force be regarded as a linear form, so that direct contraction—without a scalar product—with displacement (or velocity) yields work (or power).

## Funding

## Conflicts of Interest

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Cardall, C.Y.
Combining 3-Momentum and Kinetic Energy on Galilei/Newton Spacetime. *Symmetry* **2020**, *12*, 1775.
https://doi.org/10.3390/sym12111775

**AMA Style**

Cardall CY.
Combining 3-Momentum and Kinetic Energy on Galilei/Newton Spacetime. *Symmetry*. 2020; 12(11):1775.
https://doi.org/10.3390/sym12111775

**Chicago/Turabian Style**

Cardall, Christian Y.
2020. "Combining 3-Momentum and Kinetic Energy on Galilei/Newton Spacetime" *Symmetry* 12, no. 11: 1775.
https://doi.org/10.3390/sym12111775