nLab areal velocity

Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Analysis

Geometry

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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& \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)

Variational calculus

Contents

Idea

The bivector whose magnitude equals the rate of change at which area is swept out by a point/particle as it moves along a curve.

Definition

Given an nn-dimensional Euclidean space n\mathbb{R}^n, one could select an orthonormal basis on n\mathbb{R}^n by postulating an origin 00 and a function i^:[1,n] n\hat{i}:[1, n] \to \mathbb{R}^n such that for all m,p[1,n]m, p \in [1, n] the pair of vectors i^ m\hat{i}_m and i^ p\hat{i}_p is mutually orthonormal. There is an geometric algebra 𝔾 n\mathbb{G}^n on n\mathbb{R}^n defined by the equations i^ m 2=1\hat{i}_m^2 = 1 for all m[1,n]m \in [1, n], and i^ mi^ p=i^ pi^ m\hat{i}_m \hat{i}_p = -\hat{i}_p \hat{i}_m for all m,p[1,n]m, p \in [1, n].

A smooth curve 𝒞\mathcal{C} in n\mathbb{R}^n could be parameterized by a smooth function r: n\overrightarrow{r}:\mathbb{R} \to \mathbb{R}^n. Then the areal velocity, sector velocity, or sectorial velocity of a point in 𝒞\mathcal{C} in n\mathbb{R}^n is given by the bivector-valued function A:𝔾 n 2A:\mathbb{R} \to \langle \mathbb{G}^n \rangle_2

A(t)=r(t)v(t)2A(t) = \frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}

where aba \wedge b is the wedge product of two vectors aa and bb, and v\overrightarrow{v} is the velocity.

In 3 dimensions

In 3 dimensions, the vector areal velocity a:𝔾 n 1\overrightarrow{a}:\mathbb{R} \to \langle \mathbb{G}^n \rangle_1 is the Hodge dual of the areal velocity, which is the product of the pseudoscalar

I= i:[1,n]i^ iI = \prod_{i:[1, n]} \hat{i}_i

with the areal velocity:

a(t)=IA(t)=I(r(t)v(t)2)=I(r(t)v(t))2=r(t)×v(t)2\overrightarrow{a}(t) = I A(t) = I\left(\frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}\right) = \frac{I(\overrightarrow{r}(t) \wedge \overrightarrow{v}(t))}{2} = \frac{\overrightarrow{r}(t) \times \overrightarrow{v}(t)}{2}

Conservation of areal velocity

Conservation of areal velocity is the same as the conservation of angular momentum.

See also

References

See also:

Last revised on May 17, 2022 at 15:38:25. See the history of this page for a list of all contributions to it.