16 research outputs found
The Scottish book: Mathematics from the Scottish cafe Edited by R. Daniel Mauldin. Boston (Birkhauser). 1981. xiii + 268 pp. $24.95
Exact Analytic Solution for the Rotation of a Rigid Body having Spherical Ellipsoid of Inertia and Subjected to a Constant Torque
The exact analytic solution is introduced for the rotational motion of a
rigid body having three equal principal moments of inertia and subjected to an
external torque vector which is constant for an observer fixed with the body,
and to arbitrary initial angular velocity. In the paper a parametrization of
the rotation by three complex numbers is used. In particular, the rows of the
rotation matrix are seen as elements of the unit sphere and projected, by
stereographic projection, onto points on the complex plane. In this
representation, the kinematic differential equation reduces to an equation of
Riccati type, which is solved through appropriate choices of substitutions,
thereby yielding an analytic solution in terms of confluent hypergeometric
functions. The rotation matrix is recovered from the three complex rotation
variables by inverse stereographic map. The results of a numerical experiment
confirming the exactness of the analytic solution are reported. The newly found
analytic solution is valid for any motion time length and rotation amplitude.
The present paper adds a further element to the small set of special cases for
which an exact solution of the rotational motion of a rigid body exists.Comment: "Errata Corridge Postprint" In particular: typos present in Eq. 28 of
the Journal version are HERE correcte
An asymptotic problem in renewal theory
A special problem in renewal theory is considered. The asymptotic behavior of the renewal function was studied by W. L. Smith. Here we show that his result with an exponentially small remainder term follows from a theorem of De Bruijn on Volterra integral equations