Areal velocity

In classical mechanics, areal velocity (also called sector velocity or sectorial velocity) is a pseudovector whose length equals the rate of change at which area is swept out by a particle as it moves along a curve. In the adjoining figure, suppose that a particle moves along the blue curve. At a certain time ''t'', the particle is located at point ''B'', and a short while later, at time ''t'' + Δ''t'', the particle has moved to point ''C''. The region swept out by the particle is shaded in green in the figure, bounded by the line segments ''AB'' and ''AC'' and the curve along which the particle moves. The areal velocity magnitude (i.e., the ''areal speed'') is this region's area divided by the time interval Δ''t'' in the limit that Δ''t'' becomes vanishingly small. The vector direction is postulated normal to the plane containing the position and velocity vectors of the particle, following a convention known as the right hand rule.

The concept of areal velocity is closely linked historically with the concept of angular momentum. Kepler's second law states that the areal velocity of a planet, with the sun taken as origin, is constant. Isaac Newton was the first scientist to recognize the dynamical significance of Kepler's second law. With the aid of his laws of motion, he proved in 1684 that any planet that is attracted to a fixed center sweeps out equal areas in equal intervals of time. By the middle of the 18th century, the principle of angular momentum was discovered gradually by Daniel Bernoulli and Leonhard Euler and Patrick d'Arcy; d'Arcy's version of the principle was phrased in terms of swept area. For this reason, the principle of angular momentum was often referred to in the older literature in mechanics as "the principle of equal areas." Since the concept of angular momentum includes more than just geometry, the designation "principle of equal areas" has been dropped in modern works.

Connection with angular momentum

In the situation of the first figure, the area swept out during time period Δ''t'' by the particle is approximately equal to the area of triangle ''ABC''. As Δ''t'' approaches zero this near-equality becomes exact as a limit.

Let the point ''D'' be the fourth corner of parallelogram ''ABDC'' shown in the figure, so that the vectors ''AB'' and ''AC'' add up by the parallelogram rule to vector ''AD''. Then the area of triangle ''ABC'' is half the area of parallelogram ''ABDC'', and the area of ''ABDC'' is equal to the magnitude of the cross product of vectors ''AB'' and ''AC''. This area can also be viewed as a (pseudo)vector with this magnitude, and pointing in a direction perpendicular to the parallelogram (following the right hand rule); this vector is the cross product itself:
$\textABCD = \vec(t) \times \vec(t + \Delta t).$

Hence
$\textABC = \frac.$

The areal velocity is this vector area divided by Δ''t'' in the limit that Δ''t'' becomes vanishingly small:
$\begin \text &= \lim_ \frac \\ &= \lim_ \frac \\ &= \lim_ \frac \left( \right) \\ &= \frac. \end$

But, $\vec\,'(t)$ is the velocity vector $\vec(t)$ of the moving particle, so that
$\frac = \frac.$

On the other hand, the angular momentum of the particle is

$\vec = \vec \times m \vec,$

and hence the angular momentum equals 2''m'' times the areal velocity.

Conservation of areal velocity is a general property of central force motion, and, within the context of classical mechanics, is equivalent to the conservation of angular momentum.

See also

* Angular momentum
* Specific angular momentum
* Elliptic coordinate system

References

Further reading

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* {{cite book, first=J. B., last= Brackenridge, title= The Key to Newton's Dynamics: The Kepler Problem and the Principia, url=https://archive.org/details/keytonewtonsdyna0000brac, url-access=registration, publisher= University of California Press, location= Berkeley , isbn=978-0-520-20217-7, year=1995, doi= , jstor= 10.1525/j.ctt1ppn2m

Curves
Kinematic properties