Two-dimensional body of maximum mean resistance

A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, Plakhov & Gouveia (2007, Nonlinearity, 20), shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement. Giving continuity to this study, new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier. This result, now presented, consists of a two-dimensional shape which confers on the body a resistance which is very close to its theoretical supremum value. But its interest does not lie solely in the maximization of Newtonian resistance; on regarding its characteristics, other areas of application are seen to begin to appear which are thought to be capable of having great utility. The optimal shape which has been encountered resulted from numerical studies, thus it is the object of additional study of an analytical nature, where it proves some important properties which explain in great part its effectiveness.Comment: Accepted (April 16, 2009) for publication in the journal "Applied Mathematics and Computation

On the two-dimensional rotational body of maximal Newtonian resistance

We investigate, by means of computer simulations, shapes of nonconvex bodies that maximize resistance to their motion through a rarefied medium, considering that bodies are moving forward and at the same time slowly rotating. A two-dimensional geometric shape that confers to the body a resistance very close to the theoretical supremum value is obtained, improving previous results.Comment: This is a preprint version of the paper published in J. Math. Sci. (N. Y.), Vol. 161, no. 6, 2009, 811--819. DOI:10.1007/s10958-009-9602-

Optimal control of Newton-type problems of minimal resistance

We address Newton-type problems of minimal resistance from an optimal control perspective. It is proven that for Newton-type problems the Pontryagin maximum principle is a necessary and sufficient condition. Solutions are then computed for concrete situations, including the new case when the flux of particles is non-parallel

Mathematical retroreflectors

Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object - notched angle - is a new one; a proof of its retroreflectivity is given.Comment: 32 pages, 19 figure