5.5 Introduction to the Famous Scientists
361
Problems 5
5.1 Find the extremal curve and minimum of the functional J [y] =
π
0 y
2 d x
under the isoperimetric condition
π
0 y
2 d x = 1, the boundary conditions are
y(0) = 0, y(π) = 0.
5.2 Find the extremal curve of the functional
J [y, z]
=
1
0 (y
2
+ z
2
− 4xz
− 4z)dx
under
the
isoperimetric
condition
1
0 (y
2
− x y
− z
2
)dx = 2, the boundary conditions are y(0) = z(0) = 0
and y(1) = z(1) = 1.
5.3 In the smooth closed curves x = x(t), y = y(t) (0 ≤ t ≤ 2π) enclosing the
area π, find a curve passing through point (−1, 0), such that its length is the
shortest, the boundary conditions are x(0) = x(2π) = 1, y(0) = y(2π) = 0.
5.4 Under the isoperimetric condition
1
0 y
2 dx = 2, find the extremal curve of
the functional J [y] =
1
0 (y
2
+ x
2
)dx, the boundary conditions are y(0) =
y(1) = 0.
5.5 Find the geodesic line from point A(R, 0, 0) to point B(0, R, R) on the circular
cylindrical surface of r = R.
Prompt: It is more convenient to solve by cylindrical coordinates r, θ , z.
5.6 Under the isoperimetric condition
x 1
x 0
ydx = a, find the extremal curve of the
functional J [y] =
x 1
x 0
y
2 dx, there a is a constant.
5.7 Under the isoperimetric condition
x 1
x 0
r (x)y
2 dx = 1 and the boundary conditions y(x 0 ) = y(x 1 ) = 0, write the differential equation of the extremal
curve for the functional J [y] =
x 1
x 0
[ p(x)y
2
+ q(x)y
2
]dx, where, p(x), q(x),
r (x) ∈ C
1
[x 0 , x 1 ] are all the known functions.
5.8 Find the extremal function u = u(t) of the functional J =
t 1
t 0
u
2 dt, the fixed
boundary conditions are
x(0) = x 0 , x(t 1 ) = x 1 , y(0) = y 0 , y(t 1 ) = y 1
the constraint conditions are
x
= y, y
= ku
and find the constraint conditions and the extremal function under the following
two groups of cases
x(0) = x 0 , x(t 1 ) = y(0) = y(t 1 ) = 0
x(0) = y(0) = 0, x(1) = y(1) = 1
5.9 Find the geodesic line on the circular cylindrical surface z =
√
1 − x 2 .
5.10 Find the shortest distance between two points A(1, −1, 0) and B(2, 1, −1) on
the surface 15x − 7y + z − 22 = 0.
361
Problems 5
5.1 Find the extremal curve and minimum of the functional J [y] =
π
0 y
2 d x
under the isoperimetric condition
π
0 y
2 d x = 1, the boundary conditions are
y(0) = 0, y(π) = 0.
5.2 Find the extremal curve of the functional
J [y, z]
=
1
0 (y
2
+ z
2
− 4xz
− 4z)dx
under
the
isoperimetric
condition
1
0 (y
2
− x y
− z
2
)dx = 2, the boundary conditions are y(0) = z(0) = 0
and y(1) = z(1) = 1.
5.3 In the smooth closed curves x = x(t), y = y(t) (0 ≤ t ≤ 2π) enclosing the
area π, find a curve passing through point (−1, 0), such that its length is the
shortest, the boundary conditions are x(0) = x(2π) = 1, y(0) = y(2π) = 0.
5.4 Under the isoperimetric condition
1
0 y
2 dx = 2, find the extremal curve of
the functional J [y] =
1
0 (y
2
+ x
2
)dx, the boundary conditions are y(0) =
y(1) = 0.
5.5 Find the geodesic line from point A(R, 0, 0) to point B(0, R, R) on the circular
cylindrical surface of r = R.
Prompt: It is more convenient to solve by cylindrical coordinates r, θ , z.
5.6 Under the isoperimetric condition
x 1
x 0
ydx = a, find the extremal curve of the
functional J [y] =
x 1
x 0
y
2 dx, there a is a constant.
5.7 Under the isoperimetric condition
x 1
x 0
r (x)y
2 dx = 1 and the boundary conditions y(x 0 ) = y(x 1 ) = 0, write the differential equation of the extremal
curve for the functional J [y] =
x 1
x 0
[ p(x)y
2
+ q(x)y
2
]dx, where, p(x), q(x),
r (x) ∈ C
1
[x 0 , x 1 ] are all the known functions.
5.8 Find the extremal function u = u(t) of the functional J =
t 1
t 0
u
2 dt, the fixed
boundary conditions are
x(0) = x 0 , x(t 1 ) = x 1 , y(0) = y 0 , y(t 1 ) = y 1
the constraint conditions are
x
= y, y
= ku
and find the constraint conditions and the extremal function under the following
two groups of cases
x(0) = x 0 , x(t 1 ) = y(0) = y(t 1 ) = 0
x(0) = y(0) = 0, x(1) = y(1) = 1
5.9 Find the geodesic line on the circular cylindrical surface z =
√
1 − x 2 .
5.10 Find the shortest distance between two points A(1, −1, 0) and B(2, 1, −1) on
the surface 15x − 7y + z − 22 = 0.