Calculus of variations and optimal control; optimization Classical problems Problem

Newton's minimal resistance problem

Builds on Integral functional with a Lagrangian density Convex function Surface of revolution

Problem

A solid body moves with constant speed along its axis through a rarefied medium. The particles of the medium do not interact with each other and bounce off the body elastically. Among convex bodies of revolution with base radius $R > 0$ and height $h > 0$ , which shape has the least resistance?

In Newton's model, the body is $\set{(x, z) \in \R^2 \times \R : \abs{x} \le R, \ 0 \le z \le u(\abs{x})}$ . Its profile $u \colon [0, R] \to [0, h]$ is concave and nonincreasing. The resistance is proportional to the integral functional $F(u) = \int_0^R \frac{r}{1 + u'(r)^2} \, dr.$ The problem is to minimise $F$ over such profiles.

Remarks

Newton stated the answer in the Principia. The optimal profile is not pointed: it has a flat disc of positive radius at the front, followed by a curved part that meets the disc at a nonzero angle. The convexity condition matters. Without it, profiles with steep zigzags make $F(u)$ as small as one likes. But then particles would hit the body more than once, and the model no longer describes the resistance. Later work showed that convex bodies without rotational symmetry can have smaller resistance than Newton's body of the same radius and height.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Convex function concept
Integral functional with a Lagrangian density concept
Surface of revolution example
Newton's minimal resistance problem problem

Tag	`TGWM8G5R0C`
URI	`m2://TGWM8G5R0C`
Kind	problem
Writing	W0 stub
MSC	49Q10, 49K05, 49-03
Status	This copy of the site has no build data. Run `m2 status TGWM8G5R0C`.
About	Calculus of variations
Uses	Integral functional with a Lagrangian density, Convex function, Surface of revolution
Same as	Wikidata Q39087306 Wikipedia Newton's minimal resistance problem

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)
a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.