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Examples of superlatives using the calculus of variations
1 Straight line - the shortest distance between two given points A and B which lie in a fixed plane. The problem resolves itself into that of finding the curves for which the distance, the integral I = ∫ ds, is least. Using Euler equation for F(x,y,y'), Fy d(Fy')/dx = 0 we find the answer
y = mx + b.
2 Straight line in Polar coordinates
Find the shortest distance between two points A and B using polar coordinates instead of Cartesian coordinates. With the usual notation for polar coordinates we have ds2 = dr2 + r2 dq2. and so the problem becomes that of finding the curve which minimizes the integral resulting in the polar form of a straight line. Using Euler equation for F(r,q,q'), Fq d(Fq')/dr = 0, we find the answer
c = r sin(q + a )
Figures # 7 to #123 The catenary is the shape taken for a hanging chain to be in equilibrium (see figure #7)
A thin heavy uniform flexible rope, with its ends fastened at C and D, passes over two pegs, A and B and hangs between them in equilibrium under gravity. Determine the form of the curve in which the rope hangs.
The potential energy of the vertical parts of the rope, AC and BD, is constant however the curve between the pegs is varied. If m is the mass per unit length of the rope and y is the height of the elementary arc ds above the table, then the potential energy of the part of the rope hanging between the pegs is I = ∫ mgy ds So in a position of equilibrium this integral must be a minimum. On using ds2 = dz2+dy2 and ignoring the constant factor my it follows that we must minimize the integral I = ∫ y(1+y'2 )1/2 dx. Using Euler equation for F(x,y,y'), Fy d(Fy')/dx = 0 we find the answer
y = a Cosh (x +c) 4 Paths of minimum time
Reconsider the problem of finding the path of a ray of light (or of a particle) which passes from A to B in minimum time. If P is a point in an isotropic medium, then the physical properties of the medium at P are the same in all directions from P and are therefore functions of the coordinates (x,y,z) of P.
Let v be the velocity and ds the element of arc at the is point P. then the time taken to traverse ds is ds/v. The problem then resolves itself into that of finding the path for which I = ∫ ds/v is a minimum. On using ds2 = dx2 +dy2 we must then minimize f the integral I = ∫ ds/v.
5 Parabolaic trajectory of a particle.
To find the trajectory of a particle moving under the earth's gravitational field. On writing T = mv2 /2 and v dt = (ds/dt) dt = ds and ignoring the constant factor m, we have to minimize the integral I = ∫ v ds Using Euler equation for F(x,y,y'), Fy d(Fy')/dx = 0 we find the answer is a parabolaic trajectory.
6 Brachistochrone (see figure #8)
The following problem, first solved by Bernoulli in 1696, led to the foundation of the Calculus of variations in its modern form.
A particle slides under gravity from rest along a smooth vertical curve joining two points A and B. To find the curve in which the time from A to B is a minimum. A curve of minimum time in dynamics is known as a brachistochrone. Taking the upper point A as the origin and measuring y vertically downwards, the velocity at a depth y is (2gy)1/2 and the integral I = ∫ ds/v Using Euler equation for F(x,y,y'), Fy d(Fy')/dx = 0 we find the answer is a cycloid in the usual parametric form, with 2f as the angular parameter and c as the radius of the generating circle. The generating circle rolls on the horizontal line through A which lies in the vertical plane through AB.
7 Minimal surfaces (see figure #9)
For in surfaces the Differential equation is Rr + Ss + Tt = U where R. S. T. and U are functions of x, y, z, p, and q. This nonlinear second order partial differential equation, called the equation of Monge, is not easy to solve; equations of this form had been the subject of research from the days of Euler. In the case of the minimal surface problem, the integral becomes I = ∫ ∫ ( 1 + p2 + q2)1/2 dx dy and for this special class of problems the partial differential equation becomes (1 + q2)r - 2pqs + (l + p2)t = 0.
The catenoidCatenoid as a minimal surface
Given two points A and B and a line I which i intersects AB produced. Let w denote the plane through A, B and L. To find the curve joining A and B which lies in the plane w and which, on rotation about I through four right angles, generates a surface of minimum area. If ds is the element of arc of a curve joining A to B and y is its distance from L, then the area generated by rotation about L is 2p ∫ y ds. This is minimized by the curve whose equation is
y = c cosh (x+b)/c
where b and c are arbitrary constants. This is a catenary whose directrix coincides with the axis of rotation.
Helicoid as a minimal surface
Helicoids have a very interesting property of being minimal surfaces, which means that they locally minimize. Minimal surfaces satisfy nature's uncompromising demand for efficiency, and this make them extra strong and stable. Helicoids are the only surfaces in three-dimensional space which are at the same time ruled and minimal. They often occur in nature, e.g. in cell membranes, Since they are also aesthetically pleasing, they catch the interest of architects and engineers some of whom have transferred many of lightweight constructions of roofs directly from models of minimal surfaces.
Principle of least action
Euclid and Heron showed that the path PRQ which light actually takes, is shorter than any other path, such as PR'Q, which it could conceivably take. Since the light takes the shortest path, if the medium on the upper side of the line RR' is homogeneous, then the light travels with constant velocity and so takes the path requiring least time. Heron applied this principle of shortest path and least time to problems of reflection from concave and convex spherical mirrors.
Basing their case on this phenomenon of reflection and on philosophic, theological, and aesthetic principles, philosophers and scientists after Greek times propounded the doctrine that nature acts in the shortest possible way or, as Olympiodorus (6th cent. A.D.) said in his Catoptrica, "Nature does nothing superfluous or any unnecessary work." Leonardo da Vinci said "Nature is economical and her economy is quantitative", and Robert Grosseteste believed that nature always acts in the mathematically shortest and best possible way. In medieval times it was commonly accepted that nature behaved in this manner.
Even Newton's first law of motion, which states that the straight line or shortest distance is the natural motion of a body, showed nature's desire to economize. These examples suggested that there might be some more general principle. The search for such a principle was undertaken by Maupertuis.
Pierre-Louis Moreau de Maupertuis (1698 1759), while working with the theory of light in 1744, propounded his famous Principle of Least Action: "Action is the integral of the product of mass, velocity, and distance traversed, and any changes in nature are such as to make the action least." This is somewhat vague because Maupertuis failed to specify the time interval over which the product of m, v, and s was to be taken and because he assigned a different meaning to action in each of the applications he made to optics and some problems of mechanics.
Though he had some physical examples to support his Principle, Maupertuis advocated it also for theological reasons. The laws of behavior of matter had to possess the perfection worthy of God's creation; and the least action principle seemed to satisfy this criterion because it showed that nature was economical. Maupertuis proclaimed his principle to be a universal law of nature and the first scientific proof of the existence of God.
Euler, who had corresponded with Maupertuis on this subject between 1740 and 1744, agreed with Maupertuis that God must have constructed the universe in accordance with some such basic principle and that the existence of such a principle evidenced the hand of God. He observed that he can reduce all other laws of statics to the principle of least action: "Nature, in the ' production of its effects, acts always by the simplest means" . . . and "the path is that along which the quantity of action is the least."
The Elliptical orbit of a planet8 Principle of least action proves that The Elliptical orbit of a planet
A particle of mass m is attracted towards a fixed point O by a force of magnitude mm /r2, where m is a constant and r is its distance from O. Show that the orbit of the particle is an ellipse, one of whose foci is at O. On using the principle of least action, we must minimize the integral
2 ∫ T ds where T (the kinetic energy) = mv2/2.
On writing v dt = ds the integral is transformed to m ∫ v ds
We first find v in terms of r by using the potential energy. This is the work done by the field in displacing a unit particle to some convenient standard position. At a point distant r from O the potential energy is - m /r So by the conservation of energy we have
mm /r2- mm /r = constant.
Using Euler equation for F(r,q ,q '), Fq d(Fq' dq = 0
we find the answer in polar coordinates:
r = c/[1 + (1-c/a)1/2 cos (q +b )]
which is elliptical since the eccentricity (1-c/a)1/2 <1 when a>c
9 Fluid motion of a liquid rotating inside a cylinder (see figure #10)
A uniform perfect liquid rotates inside a cylindrical container with constant angular velocity U about a vertical axis. Show that the free surface is a paraboloid of revolution.
Let r and z be the distances of any point P of the liquid from the axis of rotation and the bottom of the container respectively. Consider the liquid as made up of a number of elementary particles of which one, of mass m, is situated at P and apply D'Alembert's principle, which states that the external forces and the reversed effective forces are in equilibrium.
For the particle at P the external force is my vertically downwards. The reversed effective force
(i) is of magnitude mr2w,
(ii) lies m the horizontal line which passes through P and intersects the axis of revolution,
(iii) is directed away from the axis.
The potential function for such a system of forces is mgz mr2w/2 + a, where a is constant. We now sum this function for all particles of the liquid situated on a thin cylinder.
The potential energy is the integral I = ∫ (r rgy2/2 r x2yw /2 + ay)2pd xdx Using Euler equation for F(x,y,y'), Fy d(Fy'.)/dx = 0 we find the answer is a parabola.
10 Geodesics on a sphere (Figures #11 and #12)
Consider the family of curves lying wholly on a given sphere S, and passing through two given points, A and B. both lying on S. Among these curves there will be one for which the length of the arc AB is a minimum. Such curves are known as geodesics. They are most easily determined by the methods of the calculus of variations, as the following example will show. Let (x, y, z) be the coordinates of a point P on a sphere whose center is at the origin and whose radius is a. Then in spherical coordinates
x = a sin q cos f ,
y = a sin q sin f
z = a cos f ,
where q is the latitude and f the longitude or azimuth.
Evidently ds2 = dx2 + dy2 + dz2 = a2(dq 2 + sin2q df 2)
so we are required to minimize the integral
I = a I = ∫ ( l + f 2 sin2q )1/2 dq ,
Using Euler equation we find the answer to be the plane (in x,y,z coordinates) y cos b + x sin b = z tan a which passes through the center of the sphere.
AN INTRODUCTION TO THE HISTORY OF MATHEMATICS by Howard Eves Holt, New York: 1976
THE CALCULUS by Otto Toeplitz The University of Chicago Press, Chicago: 1963
AN INTRODUCTION TO THE CALCULUS OF VARIATION by Charles Fox Oxford University Press London: 1960
WHAT IS MATHEMATICS? by Richard Courant and Herbert Robbins Oxford University Press: London 1963
MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN TIMES by Morris Kline OXFORD New York: 1972