Newton's minimal resistance problem is a problem of finding a solid of revolution which experiences a minimum resistance when it moves through a homogeneous fluid with constant velocity in the direction of the axis of revolution, named after Isaac Newton, who studied the problem in 1685 and published it in 1687 in his Principia Mathematica.
[4][5] Even though Newton's model for the fluid was wrong as per our current understanding, the fluid he had considered finds its application in hypersonic flow theory as a limiting case.
Following this proposition is a scholium containing the famous condition that the curve which, when rotated about its axis, generates the solid that experiences less resistance than any other solid having a fixed length, and width.
In modern form, Newton's problem is to minimize the following integral:[7][8] where
represents the curve which generates a solid when it is rotated about the x axis, and
I is the reduction in resistance caused by the particles impinging upon the sloping surface DNG, formed by rotating the curve, instead of perpendicularly upon the horizontal projection of DNG on the rear disc DA from the direction of motion, in Fig.
Note that the front of the solid is the disc BG, the triangles GBC and GBR are not part of it, but are used below by Newton to express the minimum condition.
This integral is related to the total resistance experienced by the body by the following relation: The problem is to find the curve that generates the solid that experiences less resistance than any other solid having a fixed axial length L and a fixed width H. Since the solid must taper in the direction of motion, H is the radius of the disc forming the rear surface of the curve rotated about the x axis.
Let y = h when x = L. When the curve is the horizontal line DK, so the solid is a cylinder,
The simplest way to apply the Euler–Lagrange equation to this problem is to rewrite the resistance as where
The equation of x as a function of p is obtained from the minimum condition (1), and an equivalent of it was first found by Newton.
can be determined in terms of H, h and L. Because y from equation (1) can never be zero or negative, the front surface of any solid satisfying the minimum condition must be a disc (GB in Fig.
As this was the first example of this type of problem, Newton had to invent a completely new method of solution.
Also, he went much deeper in his analysis of the problem than simply finding the condition (1).
increases the radius, Bg = h, of the disc at x = L decreases and the curve becomes steeper.
Directly before the minimum resistance problem, Newton stated that if on any elliptical or oval figure rotated about its axis, p becomes greater than unity, one with less resistance can be found.
Whiteside supplies a proof and contends that Newton would have used the same reasoning.
3 where the overall resistance of the solid varies against the radius of the front surface disc, the minimum occurring when h = BG, corresponding to p = 1 at G. In the Principia, in Fig.
1 the condition for the minimum resistance solid is translated into a geometric form as follows: draw GR parallel to the tangent at N, so that
In Fig 4, assume DNSG is the curve that when rotated about AB generates the solid whose resistance is less than any other such solid with the same heights, AD = H, BG = h and length, AB = L. Fig.
(2) Let the minimum resistance solid be replaced by an identical one, except that the arc between points I and K is shifted by a small distance to the right
The resistance of the arcs of the curve DN and SG are unchanged.
Also, the resistance of the arc IK is not changed by being shifted, since the slope remains the same along its length.
The 2 displacements have to be equal for the slope of the arc IK to be unaffected, and the new curve to end at G. The new resistance due to particles impinging upon NJ or Nj, rather that NI is:
This is the original 1685 derivation where he obtains the above result using the series expansion in powers of o.
In his 1694 revisit he differentiates (2) with respect to w. He sent details of his later approach to David Gregory, and these are included as an appendix in Motte’s translation of the Principia.
The approximation of taking straight lines for the finite arcs, NI and KS becomes exact in the limit as HN and OS approach zero.
was to vary along the curve, it would be possible to find 2 infinitesimal arcs NI and KS such that (2) was false, and the coefficient of o in the expansion of
As noted above, Newton went further, and claimed that the resistance of the solid is less than that of any other with the same length and width, when the slope at G is equal to unity.
Therefore, in this case, the constant in (3) is equal to one quarter of the radius of the front disc of the solid,