194
2 Variational Problems with Fixed Boundaries
Survey of Minimal Surfaces (1969, 1986), Two–dimensional calculus (1977) and
Poetry of the universe–a mathematical exploration of the cosmos (1995).
Problems 2
2.1 Let the functions F = F(x, y, y
) ∈ C
1 , y = y(x) ∈ C
2 , find
(1) The differential dF; (2) The variation δ F.
2.2 Find the first variation of the following functions, where, a, b, c and d are all
constants.
(1) F = ax + by + cy
2
+ dy
2 ; (2) F = y
1 + y 2 ; (3) F =
a + by + cy 2 .
2.3 Find the first variation of the following functionals
(1) J [y] =
x 1
x 0
(ay + by
+ cy
2
)dx;
(2) J [y] =
x 1
x 0
y
2
1 + y 2 dx;
(3) J [y] =
x 1
x 0
(y
2
− y
2
− 2y cosh x)dx;
(4) J [y] =
x 1
x 0
(ax
2 y
+ bx y
2
+ c)dx;
(5) J [y] =
x 1
x 0
(x y + y
2
− 2y
2 y
)dx.
where, a, b, c are all constants.
2.4 Find the extremal curve of the functional J [y] =
1
0 (y
2
+ 12x y)dx, the
boundary conditions are y(0) = 0, y(1) = 1.
2.5 Find the extremal curve of the functional J [y] =
x 1
x 0
(x y + y
2
)dx.
2.6 Find the extremal curve of the functional J [y] =
x 1
x 0
(2y + x y
2
)dx, the
boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .
2.7 Find the extremal curve of the functional J [y] =
2
1 (x y
2
− y)dx, the
boundary conditions are y(1) = 0, y(2) = 1.
2.8 Find the extremal curve of the functional J [y] =
e
1 (x y
2
+ yy
)dx, the
boundary conditions are y(1) = 0, y(e) = 1.
2.9 Find the extremal curve of the functional J [y] =
π
0 (4y cos x + y
2
− y
2
)dx
the boundary conditions are y(0) = 0, y(π) = 0.
2.10 Find the extremal curve of the functional J [y] =
π
0 (y
2
− 2y cos x)dx, the
boundary conditions are y(0) = y(π) = 0.
2.11 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
+ y
2
− 2y sin x)dx.
2.12 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
− y
2
− 2y sin x)dx.
2.13 Find
the
extremal
curve
of
the
functional
J [y]
=
x 1
x 0
(y
2
− y
2
− ay cosh x)dx, where, a is a constant.
2.14 Find the extremal curve of the functional J [y] =
1
0 (y
2
+ 2ye
x
)dx, the
boundary conditions are y(0) = 0, y(1) = 1.
2.15 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
+ y
2
+ 2ye
x
)dx,
and find the extremal curve when y(0) = 0, y(1) = e.
2.16 Find the functional J [y] =
x 1
x 0
[ p(x)y
2
+ 2q(x)yy
+ r (x)y
2
+ 2g(x)y
]dx
is the condition of regular problem and find the Euler equation.
2.17 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
− x
2 y
)dx, the
boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .