# nLab angular velocity

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

The analog of velocity for rotational movement.

For rotation in a plane inside a Cartesian space $\mathbb{R}^n$ the angular velocity is a bivector in $\wedge^2 \mathbb{R}^n$ of the form

$\dot \omega \,\, e_1 \wedge e_2 \,,$

where $e_1$ and $e_2$ are unit vector spanning the plane of rotation, and where $\dot \omega$ is the magnitude of the angular velocity.

Of $n = 3$ (and only then) can we identify bivectors with plain vectors (by the dual operation induced by the Hodge star operator). Often in the literature only this “angular velocity vector” in 3 dimensions is considered.

Standard discussion of angular velocity in $d \leq 3$ is for instance in

The more general discussion in terms of bivectors is found for instance in Geometric Algebra-style documents, such as

• Chris Doran, Anthony Lasenby, Geometric Algebra for Physicists Cambridge University Press

Physical applications of geometric algebra (pdf)