The problem posed by Hammersley (1983) of finding the shortest path along which a sphere can roll from one prescribed state to another is formulated by using quaternion calculus of variations and optimal control theory. This leads to a system of coupled nonlinear differential equations with prescribed end conditions. From the resulting expression for the curvature, it is shown that the differential equation of the required path in intrinsic coordinates is the same as the equation of motion of a simple pendulum, giving a solution in terms of elliptic integrals.
We present a geometrical method for the solution of a certain class of nonlinear boundary value problems. The results generalize those of the standard hypercircle method for linear problems. Two illustrative examples are described.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.