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classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, areal velocity (also called sector velocity or sectorial velocity) is a
pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
whose
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
equals the rate of change at which
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
is swept out by a particle as it moves along a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
. 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 In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...
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 Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
magnitude (i.e., the ''areal
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
'') 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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
to the plane containing the position and velocity vectors of the particle, following a convention known as the
right hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of th ...
. The concept of areal velocity is closely linked historically with the concept of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
.
Kepler's second law In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits ...
states that the areal velocity of a planet, with the sun taken as origin, is constant.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
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 Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...
and
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Patrick d'Arcy Patrick d'Arcy (27 September 1725 – 18 October 1779) was an Irish mathematician born in Kiltullagh, County Galway in the west of Ireland. His family, who were Catholics, suffered under the penal laws. In 1739 d'Arcy was sent abroad b ...
; 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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
. 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 In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
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 In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of th ...
); 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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
*
Specific angular momentum In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative posit ...
* Elliptic coordinate system


References


Further reading

* * * * {{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