nLab
angular velocity

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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)

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The analog of velocity? for rotational movement.

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

ω˙e 1e 2, \dot \omega \,\, e_1 \wedge e_2 \,,

where e 1e_1 and e 2e_2 are unit vector spanning the plane of rotation, and where ω˙\dot \omega is the magnitude of the angular velocity.

Of n=3n = 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.

References

Standard discussion of angular velocity in d3d \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)

Revised on August 29, 2011 21:23:30 by Urs Schreiber (131.211.238.66)