Areal velocity

Areal velocity is the rate at which area is swept out by a particle as it moves along a curve. In many applications, the curve lies in a plane, but in others, it is a space curve.

The adjoining figure shows a continuously differentiable curve in blue. At time "t", a moving particle is located at point "B", and at time "t + Δt" the particle has moved to point "C".

The area swept out during time period Δt by the particle is approximately equal to the area of triangle "ABC". As "Δt" approaches zero this near equality becomes exact as a limit.

The vectors "AB" and "AC" add up by the parallelogram rule to vector "AD", so that point "D" is the fourth corner of parallelogram "ABDC" shown in the figure.

The area of triangle "ABC" (in yellow) 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". One may form a vector whose magnitude is the area "ABCD", and which is perpendicular to the parallelogram "ABCD". Then,

:$hbox\{vector-area\}\; (ABDC)\; =\; vec\{r\}(t)\; imes\; vec\{r\}(t+Delta\; t).$

Hence,

:$hbox\{vector-area\}\; (ABC)\; =\; \{vec\{r\}(t)\; imes\; vec\{r\}(t\; +\; Delta\; t)\; over\; 2\}\; =\; Delta\; vec\{A\}.$

The areal velocity vector is

:$frac\{d\; vec\{A\{d\; t\}\; =\; lim\_\{Delta\; t\; ightarrow\; 0\}\; \{Delta\; vec\{A\}\; over\; Delta\; t\}\; =\; lim\_\{Delta\; t\; ightarrow\; 0\}\; \{vec\{r\}(t)\; imes\; vec\{r\}(t\; +\; Delta\; t)\; over\; 2\; Delta\; t\}$

::$=\; lim\_\{Delta\; t\; ightarrow\; 0\}\; \{vec\{r\}(t)\; imes\; [\; vec\{r\}(t)\; +\; vec\{r\},\text{'}(t)\; Delta\; t\; ]\; over\; 2\; Delta\; t\}$

::$=\; lim\_\{Delta\; t\; ightarrow\; 0\}\; \{vec\{r\}(t)\; imes\; vec\{r\},\text{'}(t)\; over\; 2\}\; left(\; \{Delta\; t\; over\; Delta\; t\}\; ight)$

::$=\; \{vec\{r\}(t)\; imes\; vec\{r\},\text{'}(t)\; over\; 2\}.$

But, $vec\{r\},\text{'}(t)$ is the velocity vector $vec\{v\}(t)$of the moving particle, so that

:$frac\{d\; vec\{A\{d\; t\}\; =\; \{vec\{r\}\; imes\; vec\{v\}\; over\; 2\}.$

The areal velocity vector can be placed at the moving point B. As the particle moves along its path in space, it sweeps out a cone-shaped surface. The areal velocity vector is perpendicular to this surface, and, in general, varies in both magnitude and direction. In planar problems, such as the orbit of a planet about the sun, the direction of the areal velocity vector is perpendicular to the orbital plane. Kepler's second law is a statement of the constancy of the rate at which the position vector of a planet sweeps out area, with the sun taken as origin. The path of the planet is an ellipse, with the sun at one focus (Kepler's first law).

The angular momentum of the particle is

:$vec\{L\}\; =\; vec\{r\}\; imes\; m\; vec\{v\},$

and hence

:$vec\{L\}\; =\; 2\; m\; frac\{d\; vec\{A\{d\; t\}$.

The direction of the angular momentum vector L is always the same as that of the areal velocity vector. Angular momentum is conserved if and only if the areal velocity is a constant vector. Thus, in the Kepler problem, the angular momentum of a planet about the sun is conserved and the areal velocity is a vector of constant magnitude perpendicular to the orbital plane. In some problems, a component of angular momentum is conserved, and in these cases, the corresponding component of areal velocity is constant. For a single particle, areal velocity provides a geometrical interpretation of angular momentum.

The concept of areal velocity is closely linked historically with the concept of angular momentum. 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 areas." Since the concept of angular momentum includes more than just geometry, the designation "principle of areas" has been dropped in modern works.

References

* F.R. Moulton, An Introduction to Celestial Mechanics, Macmillan (1914), Dover Reprint (1970).
* H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley (1980).
* J. Casey, Areal Velocity and Angular Momentum for Non-Planar Problems in Particle Mechanics, American Journal of Physics, Vol. 75, 2007, pp. 677-685.
* J.B. Brackenridge, The Key to Newton's Dynamics: The Kepler Problem and the Principia, University of California Press, Berkeley (1995).

ee also

* angular momentum