Repositório Institucional da Universidade de Aveiro > Departamento de Matemática > MAT - Artigos > Two-dimensional Newton's problem of minimal resistance
 Please use this identifier to cite or link to this item `http://hdl.handle.net/10773/4109`

 title: Two-dimensional Newton's problem of minimal resistance authors: Silva, C.J.Torres, D.F.M. keywords: Calculus of variationsDimension twoNewton's problem of minimal resistanceOptimal control issue date: 2006 publisher: Polish Academy of Sciences abstract: Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at a first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and asked at least as many experts on the problem, and we have not been able to identify a single source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the two-dimensional problem is richer than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classical three-dimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem does not admit a local minimum), we prove that in dimension two the unrestricted problem is also well-posed when the ratio of height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, we show that in the restricted two-dimensional problem the minimizer is not always unique - when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions. URI: http://hdl.handle.net/10773/4109 ISSN: 0324-8569 publisher version/DOI: http://control.ibspan.waw.pl:3000/contents/export?filename=Silva-Torres.pdf source: Control and Cybernetics appears in collections MAT - Artigos

files in this item

file description sizeformat