“Brachistochrone Problem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Brachistochrone Problem

释义

Brachistochrone Problem

Find the shape of the Curve down which a bead sliding from rest and Accelerated bygravity will slip (without friction ) from one point to another in the leasttime. This was one of the earliest problems posed in the Calculus of Variations. The solution, a segment of aLeibniz, L'Hospital, Newton, and the twoBernoullis.

The time to travel from a point to another point is given by the Integral

(1)

The Velocity at any point is given by a simple application of energy conservation

equating kinetic energy

to gravitational potential energy,

(2)

(3)

Plugging this into (1) then gives

(4)

The function to be varied is thus

(5)

To proceed, one would normally have to apply the full-blown Euler-Lagrange Differential Equation

(6)

However, the function

is particularly nice since

does not appear explicitly. Therefore,

, and we can immediately use the Beltrami Identity

(7)

Computing

(8)

subtracting

from

, and simplifying then gives

(9)

Squaring both sides and rearranging slightly results in

(10)

where the square of the old constant

has been expressed in terms of a new (Positive) constant

. This equation issolved by the parametric equations

			(11)
			(12)

which are--lo and behold--the equations of a Cycloid.

If kinetic friction is included, the problem can also be solved analytically, although thesolution is significantly messier. In that case, terms corresponding to the normal component of weight and the normal component of the Acceleration (present because of path Curvature) must be included. Includingboth terms requires a constrained variational technique (Ashby et al. 1975), but including the normal component of weight onlygives an elementary solution. The Tangent and Normal Vectors are