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Newton's Problem of the Body of Minimal Resistance

Imagine you're designing the nose of a high-speed train. Main design goal: to minimize air resistance. Although he wasn't thinking of trains, Sir Isaac Newton formulated this problem in his Principia Mathematica, first published in 1687. He modelled the problem by assuming that every air particle hit the body (the nose of the train) exactly once. In the collision between body and particle momentum is passed from the particle to the body, and the amount of momentum depends on the angle that the surface of the body makes at that point with the direction of the particle, as shown in the figure. The total resistance then is the sum over the body of this transfer of momentum.

If we describe the surface of the body as the graph of a function u, then we can state the problem in mathematical terms in the following way:

Statement of Newton's problem

Newton himself provided a solution to this minimization problem in the case where Omega is a ball in R², limiting himself to functions that have radial symmetry. This reduces the 2-D integral to a one-dimensional one and greatly simplifies the analysis. Here are some pictures of the solution that he found:

Newton's minimizer (M=2) Newton's minimizer (M=0.5)

The left picture is for M = 2, the right picture for M = 0.5. In both cases the radius of the ball is 1. (Thanks to Paolo Guasoni for the pictures). Although it is counterintuitive, the solution provided by Newton contains a flat zone at height M which varies in size (depending on M) but is always present.

Newton's formula for the air resistance (the integral in (1)) is found in many engineering handbooks, and his solution is presented as a simple application of the calculus of variations. Throughout the three hundred years that have passed since the first publication of the Principia, no-one seemd to ask a very simple and obvious question: if Omega is a ball, is the minimizer necessarily radially symmetric? In other words, does the solution copy the symmetry of the domain? And was Newton therefore justified in assuming that the minimizer is radially symmetric?

Intuition tells us that this is true, as does a comparison with many other minimization problems. But it is false, in fact: in 1996 a paper was published by F. Brock, V. Ferone, and B. Kawohl, where it was proved that the minimizer on the ball is not radially symmetric. This was done by showing that the minimizer could not be at the same time both strictly concave and smooth. Since Newton's solution has both these properties - apart from the top flat zone - this shows that this function does not minimize among all functions. Since it does minimize among radially symmetric functions, the `real' minimizer must be non-radially symmetric.

This result opened the witch-hunt for the answer of the immediate question:

What does the minimizer look like, then?

Not much is known about the real minimizer. For instance, it need not be smooth; the concaveness constraint provides a limited amount of regularity, but as the functional in (1) is neither convex nor coercive there is no reason to expect more regularity than that. As an example, one can think of a convex polyhedron, which clearly has corners. This observation also means that the result of Brock et al. mentioned above doesn't apply to the real minimizer (because the proof uses the regularity assumption in an essential way).

We managed to prove a couple of things, though.

T. Lachand-Robert and Mark A. Peletier, An Example of Non-convex Minimization and an Application to Newton's Problem of the Body of Least Resistance, Annales de l'Institut Henri Poincaré, Analyse non linéaire 18, pp. 179-198 (2001)

As a first step towards a characterization we generalized the result mentioned above to functions wth general regularity, i.e.
If u is the minimizer of Newton's problem, then for any open subet of Omega, u is not strictly convex on that subset.
Or, said differently, the graph of u must contain a dense set of straight line segments.

I called this result a generalization, but I mean to stress that the proof is not a simple extension of the proof given by Brock et al. Because of the lack of regularity, choosing an appropriate class of perturbations of the minimizer is highly non-trivial, since most perturbations make the function non-concave and therefore take it outside of the set of admissible functions.

Alternative function, with less resistanceThe fact that u (the minimizer) is non-strictly concave does not give a lot information about its actual form, however. Whilst trying to imagine possible forms, it might be useful to think of the function here on the right, discovered by Guasoni (he called it the screwdriver). For large values of M (M >= 2) the screwdriver has a lower resistance than Newton's function. Note that the screwdriver is indeed non-strictly concave, everywhere: from every point on the circular base there runs at least one straight line up to the top.

T. Lachand-Robert and Mark A. Peletier, Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions (930 kB), Mathematische Nachrichten, Vol. 226, pp. 153-176 (2001).

One way of describing the screwdriver is as the convex hull, or envelope, of two sets in 3-space: the circular base and the line segment at the top. In this paper we considered the following question: supposing the minimizer is of this form, then what should it look like? In this class the functions are fully characterized by the form of the top zone, and it turns out that this set must be a regular polygon. The screwdriver corresponds to the simplest polygon, with two sides, and it is the optimal one when M is large; as we decrease M, optimality jumps from one regular polygon to the next:

T. Lachand-Robert and Mark A. Peletier, Extremal Points of a Functional on the Set of Convex Functions, Proceedings of the AMS, Vol. 127, pp. 1723-1727 (1999).

As a spin-off of the work on the Newton's problem, which involves a non-convex functional, we also came across some new results on the minimization of convex functionals on the set of convex (or concave) functions. Provided the boundary values are restricted this paper gives an interesting, and surprisingly simple, characterisation of extremal points of such a functional.

T. Lachand-Robert and Mark A. Peletier, The Minimum of Quadratic Functionals of the Gradient on the Set of Convex Functions (72 kB), Calc. Var. Partial Differential Equations, Vol. 15, pp. 289-297 (2002).

Another spin-off result concerns minimizing sequences for the minimization problem

under the additional constraint that the functions u are convex. Without the convexity constraint, the minimum is equal to the smallest eigenvalue of the matrix A; under the convexity constraint, however, the situation is different, and depends on the boundary of the set Omega. See the paper itself for the details.

Some of these results are summarized in the Comptes Rendus paper T. Lachand-Robert and Mark A. Peletier,Minimisation de fonctionnelles dans un ensemble de fonctions convexes, C. R. Acad. Sci. Paris, t. 325, Série I, p. 851-855 (1997).

Last modified on March 27, 2010 by Mark Peletier