nLab areal velocity

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 $n$-dimensional Euclidean space $\mathbb{R}^n$, one could select an orthonormal basis on $\mathbb{R}^n$ by postulating an origin $0$ and a function $\hat{i}:[1, n] \to \mathbb{R}^n$ such that for all $m, p \in [1, n]$ the pair of vectors $\hat{i}_m$ and $\hat{i}_p$ is mutually orthonormal. There is an geometric algebra $\mathbb{G}^n$ on $\mathbb{R}^n$ defined by the equations $\hat{i}_m^2 = 1$ for all $m \in [1, n]$, and $\hat{i}_m \hat{i}_p = -\hat{i}_p \hat{i}_m$ for all $m, p \in [1, n]$.

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

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

where $a \wedge b$ is the wedge product of two vectors $a$ and $b$, and $\overrightarrow{v}$ is the velocity.

In 3 dimensions

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

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

with the areal velocity:

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

Conservation of areal velocity

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

See also

• area, velocity

• area of a circle

• Euclidean space

• geometric algebra

• classical mechanics

• angular momentum

• torque

References

See also:

• Wikipedia, Areal velocity