The concept of areal velocity is intrinsically and historically connected with that of angular momentum. For central force fields the area swept out in the orbital plane by a particle’s radius vector is proportional to the time. For non-planar problems, it is shown that a particle’s position vector sweeps out a general conical surface and that interesting kinematical relations hold. The areal velocity vector is always perpendicular to the conical surface and is proportional to the angular momentum of the particle. In some problems, the magnitude of the areal velocity is constant while its direction changes. In others, only a component of the areal velocity vector is constant. A moving orthonormal basis associated with the areal velocity is defined and a matrix equation for the angular velocities of the basis vectors is found to have an elegant form, involving “sweeping” and “tilting” components. Illustrative examples are provided and some historical background is included.

1.
See, for example,
F. R.
Moulton
,
An Introduction to Celestial Mechanics
(
Macmillan
, New York,
1914
), 2nd revised ed.; reprinted by Dover, 1970;
H.
Goldstein
,
Classical Mechanics
(
Addison-Wesley
, Reading, MA,
1950
) (2nd ed., 1980);
A.
Sommerfeld
,
Mechanics
(
Academic Press
, New York,
1952
), Vol.
1
;
M. R.
Spiegel
,
Schaum’s Outline of Theory and Problems of Theoretical Mechanics, with an Introduction to Lagrange’s Equations and Hamiltonian Theory
(
McGraw-Hill
, New York,
1967
);
T.
Yoshida
, “
A new derivation of the areal velocity
,”
Am. J. Phys.
55
,
752
753
(
1987
).
2.
The concepts and tools of differential geometry and tensor theory shed much light on Lagrangian mechanics. See
J. L.
Synge
,
Tensorial Methods in Dynamics
(
University of Toronto Press
, Toronto,
1936
);
J.
Casey
, “
Geometrical derivation of Lagrange’s equations for a system of particles
,”
Am. J. Phys.
62
,
836
847
(
1994
) which contains an extensive list of references;
J.
Casey
, “
On the advantages of a geometrical viewpoint in the derivation of Lagrange’s equations for a rigid continuum
,”
J. Appl. Math. Phys. (ZAMP)
46
,
S805
S847
(
1995
);
V. I.
Arnold
,
Mathematical Methods of Classical Mechanics
(
Springer-Verlag
, New York,
1989
), 2nd ed.
3.
The tortuous route by which Kepler arrived at his three planetary laws has been well documented. It involved many false starts with incorrect physical and geometrical hypotheses. Kepler continually revised his ideas when he found his predictions to be in disagreement with the unprecedentedly accurate orbital data that he had obtained from Tycho Brahe (1546–1601).
See
E. J.
Dijksterhuis
,
The Mechanization of the World Picture
(
Clarendon Press
, Oxford,
1961
), Part IV, pp.
25
59
;
E. J.
Aiton
, “
Kepler’s second law of planetary motion
,”
Isis
60
,
75
90
(
1969
);
C.
Wilson
, “
How did Kepler discover his first two laws?
Sci. Am.
226
(
3
),
92
106
(
1972
);
A. E. L.
Davis
, “
The mathematics of the area law: Kepler’s successful proof in Epitome Astronomiae Copernicanae (1621)
,”
Arch. Hist. Exact Sci.
57
,
355
393
(
2003
).
The reception to Kepler’s laws during the half century following their announcement was slow and mixed. His second law was generally accepted by astronomers in only an approximate variant form. See
J. L.
Russell
, “
Kepler’s laws of planetary motion
,”
Brit. J. Hist. Sci.
2
,
1
24
(
1964
).
4.
Newton’s De Motu Corporum is transcribed and translated in The Mathematical Papers of Isaac Newton
, edited by
D. T.
Whiteside
(
University Press
, Cambridge,
1974
), Vol.
VI
.
The development of Newton’s ideas on the planetary problem is discussed by
D. T.
Whiteside
, “
Newton’s early thoughts on planetary motion: A fresh look
,”
Brit. J. Hist. Sci.
2
,
117
137
(
1964
);
J. T.
Cushing
, “
Kepler’s laws and universal gravitation in Newton’s Principia
,”
Am. J. Phys.
50
,
617
628
(
1982
);
J. B.
Brackenridge
,
The Key to Newton’s Dynamics: The Kepler Problem and the Principia
(
University of California Press
, Berkeley,
1995
);
B.
Pourciau
, “
Reading the master: Newton and the birth of celestial mechanics
,”
Am. Math. Monthly
104
,
1
19
(
1997
).
The reader is referred also to
S.
Chandrasekhar
,
Newton’s Principia for the Common Reader
(
Clarendon Press
, Oxford,
1995
).
5.
I.
Newton
,
Philosophiae Naturalis Principia Mathematica
(Londini,
1687
).
F.
Cajori’s
revision of A. Motte’s 1729 English translation was published under the title
Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World
(
University of California Press
, Berkeley,
1960
).
A new translation has been prepared by
I. B.
Cohen
and
A.
Whitman
, assisted by
J.
Budenz
,
Isaac Newton: The Principia, Mathematical Principles of Natural Philosophy
(
University of California Press
, Berkeley,
1999
).
6.
The emergence of the principle of angular momentum is traced by C. Truesdell, “Whence the law of moment of momentum?”
L’Aventure de la Science: Mélanges Alexandre Koyré
(
Hermann
, Paris,
1964
), Vol.
I
, pp.
588
612
;
reprinted in
C.
Truesdell
,
Essays in the History of Mechanics
(
Springer-Verlag
, New York,
1968
), pp.
239
271
.
7.
Born in the west of Ireland, Patrick d’Arcy was sent by his parents in 1739 to an uncle in Paris to escape the oppression instituted by the penal laws. He was tutored in mathematics by Jean-Baptiste Clairaut (1680–1766) and became a friend of Clairaut’s brilliant son, Alexis-Claude (1713–1765), who, along with Voltaire (1694–1778), Gabrielle-Émilie, Marquise du Châtelet (1706–1749), and Pierre Louis Moreau de Maupertuis (1698–1759), ardently championed Newtonian mechanics in France. With a gift for mathematics and mechanics, d’Arcy soon made original contributions to dynamics. At the same time, he embarked on a career in the army. He participated in campaigns in Germany and Flanders, rose through the ranks, and eventually became a field marshal. He was accepted into the French nobility, under the title of Count. He became wealthy and was a generous patron of Irish refugees in France. Besides his work on dynamics, he conducted research on gunpowder, artillery, electricity, and the optical continuity of intermittent images. D’Arcy was elected to the Académie Royale des Sciences in 1749. He died from cholera in Paris in October 1779.
See
C. S.
Gillmor
, in
Dictionary of Scientific Biography
(
Scribner’s
, New York,
1970
), Vol.
III
, pp.
561
562
.
8.
M.
le Chevalier d’Arcy
, “
Principe géneral de dynamique, qui donne la relation entre les espaces parcourus et les temps, quel que soit le système de corps que l’on considère et quelles que soient leurs actions les uns sur les autres
,” read in 1746, Mémoires de l’Académie Royale des Sciences,
Année
1747
,
348
356
(
1751
), accessible online at gallica.bnf.fr.
9.
D’Arcy’s memoir of 1747 is published as an addition to the memoir cited in Ref. 8, pp.
356
361
.
10.
M.
le Chevalier d’Arcy
, “
Réflexions sur le principe de la moindre action de M. de Maupertuis
,” Mémoires de l’Académie Royale des Sciences,
Année
1749
,
531
538
(
1753
).
11.
J.-L.
Lagrange
,
Mécanique Analytique
(
La Veuve Desaint
, Paris,
1788
); reprinted by Editions Jacques Gabay, 1989). The fourth edition of Mécanique Analytique is published as Vols. 11 and 12 of Oeuvres de Lagrange, edited by J.-A. Serret and G. Darboux (Gauthier-Villars et Fils, Paris, 1867–1892); reprinted by Georg Olms Verlag Hildesheim and New York, 1973). An English translation of the 1811 edition of Mécanique Analytique has been edited by A. Boissonnade and V. N. Vagliente: J. L. Lagrange, Analytical Mechanics (Kluwer Academic, Dordrecht, 1997).
12.
In addition to Lagrange’s historical introduction, see also Sec. III of the Mécanique Analytique, where the equations for the conservation of angular momentum in rectangular Cartesian components may be found.
13.
Sir William
Thomson
(Lord Kelvin) and
P. G.
Tait
, Treatise on Natural Philosophy (Oxford Clarendon Press, 1867), Sec. 268; last revised edition, 1912; reprinted by Dover Publications under the (odd) title, Principles of Mechanics and Dynamics, 1962. In the revised edition, the quotation reads: “a very ill-considered designation.” E. Mach, in The Science of Mechanics: A Critical and Historical Account of its Development (Open Court Publishing, La Salle, Illinois, 1942) uses the terminology “principle of conservation of areas” (see pp.
383
386
), but unlike Lagrange, he does not distinguish between the conceptions of Daniel Bernoulli and Euler on the one hand, and those of d’Arcy on the other.
14.
Reference 1, Sommerfeld’s Mechanics, pp.
72
73
.
15.
If r×v=0 at isolated instants, we could define eN by continuity. In the trivial case when r×v=0 over an interval of time, we could take eN to be any fixed unit vector perpendicular to r for r0. If r=0 over an interval of time, we could take er, eT, eN to be a fixed right-handed orthonormal basis.
16.

The expression for L̇ in Eq. (23) is also given by L̇=δLδt+ω×L, where δLδt is the rate of change of L measured in the rotating areal basis.

17.
Let the unit vectors et, en, and eb be the tangent, principal normal, and binormal vectors, respectively, to the curve described by the particle P in motion. The Serret-Frenet formulae imply that
(ėtėnėb)=ṡ(0κ0κ0τ0τ0)(eteneb),
where s is the arc length, κ the curvature, and τ the torsion of the curve. The angular velocity of the Serret-Frenet basis is ω=(τet+κeb)ṡ. The balance of linear momentum may be written as F=d(mvet)dt=m(v̇et+κv2en), where v=vet.
See
O. M.
O’Reilly
,
Engineering Dynamics: A Primer
(
Springer-Verlag
, New York,
2001
). To take the analogy between the Serret-Frenet formulae and Eqs. (15) one step further, we could parametrize the particle’s path by the swept area A and calculate the derivatives of the areal basis vectors with respect to A. Thus, ėr=(derdA)Ȧ, etc.
18.
The reader is referred to
D. J.
Struik
,
Lectures on Classical Differential Geometry
(
Addison-Wesley
, Reading, MA,
1961
), 2nd ed.; reprinted by Dover, 1988;
E.
Kreyzsig
,
Differential Geometry
(
University of Toronto Press
, Toronto,
1959
); reprinted by Dover, 1991;
D. W.
Henderson
,
Differential Geometry: A Geometric Approach
(
Prentice Hall
, New Jersey,
1998
);
M. M.
Lipschutz
,
Schaum’s Outline of Theory and Problems of Differential Geometry
(
McGraw-Hill
, New York,
1969
);
D. W.
Henderson
,
Experiencing Geometry on Plane and Sphere
(
Prentice Hall
, New Jersey,
1996
);
J.
Casey
,
Exploring Curvature
(
Vieweg
, Braunschweig/Wiesbaden,
1996
).
19.

If SPR is a closed curve, we must first cut the conical surface along one of its generators. It is instructive to make a paper model of the sector in Fig. 4 and to bend it into a conical surface. The path of the particle in Fig. 3 may be self-intersecting at one or more points; if so, we may make a cut along a generator through each such point.

20.

The coordinate curves ψ=constant and r=constant lie along the principal directions of curvature of the conical surface (that is, they are lines of curvature). The conical surface may be represented parametrically by r(r,ψ)=rer(ψ). The length ds of the line element of the surface is given by ds2=dr2+r2dψ2. An important property of a conical surface is that along each generator the tangent plane does not vary.

21.

The tilting angular velocity can be easily identified in the paper model suggested in Ref. 19.

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