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.

## REFERENCES

*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).

*Mécanique Analytique*, where the equations for the conservation of angular momentum in rectangular Cartesian components may be found.

*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.

The expression for $L\u0307$ in Eq. (23) is also given by $L\u0307=\delta L\u2215\delta t+\omega \xd7L$, where $\delta L\u2215\delta t$ is the rate of change of $L$ measured in the rotating areal basis.

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.

The coordinate curves $\psi =constant$ 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 $r(r,\psi )=rer(\psi )$. The length $ds$ of the line element of the surface is given by $ds2=dr2+r2d\psi 2$. An important property of a conical surface is that along each generator the tangent plane does not vary.

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

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