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2 Variational Problems with Fixed Boundaries
Fig. 2.2 The geodesic line
O
x
y
z
A
B
Example 2.1.2 geodesic line problem or shortest distance problem. This problem
was posed by Johann Bernoulli in 1697. The formulation is: Given two points
A(x 0 , y 0 , z 0 ) and B(x 1 , y 1 , z 1 ) on the smooth surface f (x, y, z) = 0 (see Fig. 2.2),
on the surface find a shortest curve C connecting these two points. Like this, a shortest
curve between these known two points on the given surface is called the geodesic
line or geodesic.
Solution Let the equations of the curves can be written as
y = y(x), z = z(x) x 0 ≤ x ≤ x 1
(1)
where, both y(x) and z(x) are continuously differentiable functions. Since the curves
are on the surface f (x, y, z) = 0, so y(x) and z(x) should satisfy the constraint
condition
f (x, y, z) = 0
( 2 )
It can be known from the theory of higher mathematics that the length of the
Eq. (1) is
L =
x 1
x 0
1 + y (x) + z (x)dx
(2.1.2)
Thus, the Geodetic problem can boil down to in meeting the constraint condition (2), seek after the Eq. (1) through point A and point B, such that the integral
(2.1.2) gets the minimum. The variational problem of the geodesic line is called the
constrained extremum problem, constrained extreme-value problem or conditional extremum problem. The geodesic line on earth is the special case of the
problem. The constrant condition (2) is called the constraint equation of Eq. (2.1.2).