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5 Variational Problems of Conditional Extrema
on her family’s property, Pygmalion killed her husband. Then Dido was forced to
leave with her followers the Tyre of Phoenicia, fled to Tunisia, one of the Barbary
States in north Africa. Dido wanted to purchase the land, The King Iarbas promised
only sold her a piece of land which a piece of leather can cover, unusually smart
princess cut the bull leather into thin strips, and formed a rope more than 4 km in
length, then selected the land near the coast formed a semicircle with the rope, the
land became her possessions, she called this place Byrsa, which means the hide of
bull. Later she established the City of Carthage in the land (The City of Carthage
is today near Tunis, the capital of Tunisia, the city was built about 853 BC, was
destroyed by the Romans in 146 BC), and became the Queen of Carthage. In the
Phoenician, Carthage means a new town. In ancient Greek and Roman mythology,
Dido was usually associated with Aeneas. In Homeric epic the Iliad, Aeneas was a
both intelligent and courageous, famous warrior in the Trojan coalition. The ancient
Roman poet Virgil (or Vergil, Publius Vergilius Maro, 70.10.15 BC–19.9.15 BC) in
one of his epics “Aeneid”, described the tortuous experience that the Trojan war hero
Aeneas, after the fall of Troy, under the guard of gods escaped from there, led a
wandering life, and finally to build Rome in Italy, in which the love story of Dido
and Aeneas was woven. This problem is also called the Dido(’s) problem.
Solution Take the straight line through fixed points A, B as the x axis, and let the
area surrounded by the curve y = y(x) be above the x axis, it can be expressed as
J [y] =
x 1
x 0
ydx
(5.1)
The constraint condition is
L =
x 1
x 0
1 + y 2 dx
(5.2)
The boundary conditions are
y(x 0 ) = 0, y(x 1 ) = 0
(5.3)
The problem is to find the maximum of the functional (1) under the constraint
condition (2) and the boundary condition (3). Making the auxiliary functional
J
∗
=
x 1
x 0
(y + λ
1 + y 2 )dx
(5.4)
Since H = y + λ
1 + y 2 does not contain x, there is the first integral
y + λ
1 + y 2 − λy
y
1 + y 2
= c 1
(5.5)
Simplifying the above equation, we give
5 Variational Problems of Conditional Extrema
on her family’s property, Pygmalion killed her husband. Then Dido was forced to
leave with her followers the Tyre of Phoenicia, fled to Tunisia, one of the Barbary
States in north Africa. Dido wanted to purchase the land, The King Iarbas promised
only sold her a piece of land which a piece of leather can cover, unusually smart
princess cut the bull leather into thin strips, and formed a rope more than 4 km in
length, then selected the land near the coast formed a semicircle with the rope, the
land became her possessions, she called this place Byrsa, which means the hide of
bull. Later she established the City of Carthage in the land (The City of Carthage
is today near Tunis, the capital of Tunisia, the city was built about 853 BC, was
destroyed by the Romans in 146 BC), and became the Queen of Carthage. In the
Phoenician, Carthage means a new town. In ancient Greek and Roman mythology,
Dido was usually associated with Aeneas. In Homeric epic the Iliad, Aeneas was a
both intelligent and courageous, famous warrior in the Trojan coalition. The ancient
Roman poet Virgil (or Vergil, Publius Vergilius Maro, 70.10.15 BC–19.9.15 BC) in
one of his epics “Aeneid”, described the tortuous experience that the Trojan war hero
Aeneas, after the fall of Troy, under the guard of gods escaped from there, led a
wandering life, and finally to build Rome in Italy, in which the love story of Dido
and Aeneas was woven. This problem is also called the Dido(’s) problem.
Solution Take the straight line through fixed points A, B as the x axis, and let the
area surrounded by the curve y = y(x) be above the x axis, it can be expressed as
J [y] =
x 1
x 0
ydx
(5.1)
The constraint condition is
L =
x 1
x 0
1 + y 2 dx
(5.2)
The boundary conditions are
y(x 0 ) = 0, y(x 1 ) = 0
(5.3)
The problem is to find the maximum of the functional (1) under the constraint
condition (2) and the boundary condition (3). Making the auxiliary functional
J
∗
=
x 1
x 0
(y + λ
1 + y 2 )dx
(5.4)
Since H = y + λ
1 + y 2 does not contain x, there is the first integral
y + λ
1 + y 2 − λy
y
1 + y 2
= c 1
(5.5)
Simplifying the above equation, we give