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2 Variational Problems with Fixed Boundaries
2.1 Examples of the Classical Variational Problems
The basic problem of variational methods is to find the extremal problems of functionals and the corresponding extremal functions. In order to show the research
contents of variational methods, first embark from the several classical variational
examples to raise the concept of functional.
Example 2.1.1 The brachistochrone problem, problem of brachistochrone or
problem of curve of steepest descent. This is one of the earliest appeared variational
problems in the history, which was usually considered the beginning of the history
of the variational methods, was also a symbol of the development of the variational
methods. It was first proposed by Galileo in 1630, he systematically studied the
problem again in 1638, but at that time he gave the wrong results, he thought this
was a circular arc curve. The substantial research of variational method was that
John Bernoulli wrote to his brother Jacob Bernoulli an open letter on the Leipziger
Acta Eurditorum in the June 1696 issue to ask for the solution to the problem. The
formulation of the problem was: Assuming that A and B are the two points which are
not in the same vertical straight line in a vertical plane, in all the plane curves joining
point A and point B, determine a curve, such that the time needed is the shortest when
a particle that is acted on only by gravity and the initial velocity is zero moves from
point A to point B along the curve. This problem had caused many mathematicians’
interest at that time. After Newton heard the news on January 29, 1697, he solved
this problem on the same day. Leibniz, Bernoulli brothers and L’Hospital et al. all
studied this problem, they obtained correct results in different ways, among them,
Jacob Bernoulli started from geometric intuition, he gave the more general solution,
the solution took a big step towards the direction of the variational methods. Except
Jacob Bernoulli’s method of solution, others’ methods of solution were published on
the Acta eurditorum in the May 1697 issue.
Solution The particle motion time depends not only on the length of the path, but
also is associated with the speed. In all the curves joining point A and point B,
the straight line distance AB is the shortest (see the solution of Example 2.5.11),
but it is not necessarily a particle motion time shortest path. Now to establish the
mathematical model of this problem. As shown in Fig. 2.1, taking A as the origin
of plane rectangular coordinate system, x axis is put in a horizontal position, the
direction of y axis is downward. Obviously, the brachistochrone should be in the
plane. Thus the coordinate of point A is (0, 0). Let the coordinate of point B be
(x 1 , y 1 ), the equation of a curve joining point A and point B is
y = y(x) (0 ≤ x ≤ x 1 )
(1)
At these two endpoints of the interval [0, x 1 ], they meet the following conditions
y(0) = 0, y(x 1 ) = y 1
(2)
2 Variational Problems with Fixed Boundaries
2.1 Examples of the Classical Variational Problems
The basic problem of variational methods is to find the extremal problems of functionals and the corresponding extremal functions. In order to show the research
contents of variational methods, first embark from the several classical variational
examples to raise the concept of functional.
Example 2.1.1 The brachistochrone problem, problem of brachistochrone or
problem of curve of steepest descent. This is one of the earliest appeared variational
problems in the history, which was usually considered the beginning of the history
of the variational methods, was also a symbol of the development of the variational
methods. It was first proposed by Galileo in 1630, he systematically studied the
problem again in 1638, but at that time he gave the wrong results, he thought this
was a circular arc curve. The substantial research of variational method was that
John Bernoulli wrote to his brother Jacob Bernoulli an open letter on the Leipziger
Acta Eurditorum in the June 1696 issue to ask for the solution to the problem. The
formulation of the problem was: Assuming that A and B are the two points which are
not in the same vertical straight line in a vertical plane, in all the plane curves joining
point A and point B, determine a curve, such that the time needed is the shortest when
a particle that is acted on only by gravity and the initial velocity is zero moves from
point A to point B along the curve. This problem had caused many mathematicians’
interest at that time. After Newton heard the news on January 29, 1697, he solved
this problem on the same day. Leibniz, Bernoulli brothers and L’Hospital et al. all
studied this problem, they obtained correct results in different ways, among them,
Jacob Bernoulli started from geometric intuition, he gave the more general solution,
the solution took a big step towards the direction of the variational methods. Except
Jacob Bernoulli’s method of solution, others’ methods of solution were published on
the Acta eurditorum in the May 1697 issue.
Solution The particle motion time depends not only on the length of the path, but
also is associated with the speed. In all the curves joining point A and point B,
the straight line distance AB is the shortest (see the solution of Example 2.5.11),
but it is not necessarily a particle motion time shortest path. Now to establish the
mathematical model of this problem. As shown in Fig. 2.1, taking A as the origin
of plane rectangular coordinate system, x axis is put in a horizontal position, the
direction of y axis is downward. Obviously, the brachistochrone should be in the
plane. Thus the coordinate of point A is (0, 0). Let the coordinate of point B be
(x 1 , y 1 ), the equation of a curve joining point A and point B is
y = y(x) (0 ≤ x ≤ x 1 )
(1)
At these two endpoints of the interval [0, x 1 ], they meet the following conditions
y(0) = 0, y(x 1 ) = y 1
(2)