2.5 Several Special Cases of the Euler Equation and Their Integrals
137
˙
y ¨
x − ˙
x ¨
y = a
2
(1 − cos t) = 2a
2 sin
2
θ
˙
x
2
+ ˙
y
2
= 2a
2
(1 − cos t) = 4a
2 sin
2
θ
(12)
Substituting Eq. (12) into Eq. (11), we obtain
k =
1
4a sin θ
(13)
The radius of curvature at any point on the cycloid is
R =
1
k
= 4a sin θ = 4a sin
t
2
= 2d sin
t
2
(14)
Historically, the cycloid had been to Helen in geometry. According to Greek
myths and legends, Helen was a daughter born by the king Zeus of the gods and
the goddess Leda, the most famous heroine in the ancient Greek epic, well known
for her beautiful and special in the world. She was the King of Sparta Menelaus’
wife, she was abducted by the Trojan prince Paris later, was taken to Troy (Some
archaeologists think that it is located in Hissarlik of Çanakkale province of now
Turkey, but it is controversial). In order to regain the unsurpassed beauty, the ancient
Greek heroes and the Trojans had a stretched 10 years of the Trojan War, later the
Greek coalition forces used of the Trojan horse offered by Odysseus, overnight broke
up Troy, obtained the victory of war. The ancient Greek epic poem the Iliad written
by the ancient Greek narrative poet Homer (about 850 BC–the 8th century BC) was
mainly to describe the story of the Trojan War last year. Karl Heinrich Marx had
said the epic was that “they still give us aesthetic pleasure and are in certain respects
regarded as a standard and unattainable ideal”.
Example 2.5.13 The problem of least (or minimal) surface of revolution. As
shown in Fig. 2.8, in all the curves passing through the endpoints A(x 0 , y 0 ) and
B(x 1 , y 1 ), find out a curve, when it revolves around the Ox axis, the resulting surface
has the smallest area.
Fig. 2.8 The least surface of
revolution
O
x
y
z
A
B
y = y(x)
137
˙
y ¨
x − ˙
x ¨
y = a
2
(1 − cos t) = 2a
2 sin
2
θ
˙
x
2
+ ˙
y
2
= 2a
2
(1 − cos t) = 4a
2 sin
2
θ
(12)
Substituting Eq. (12) into Eq. (11), we obtain
k =
1
4a sin θ
(13)
The radius of curvature at any point on the cycloid is
R =
1
k
= 4a sin θ = 4a sin
t
2
= 2d sin
t
2
(14)
Historically, the cycloid had been to Helen in geometry. According to Greek
myths and legends, Helen was a daughter born by the king Zeus of the gods and
the goddess Leda, the most famous heroine in the ancient Greek epic, well known
for her beautiful and special in the world. She was the King of Sparta Menelaus’
wife, she was abducted by the Trojan prince Paris later, was taken to Troy (Some
archaeologists think that it is located in Hissarlik of Çanakkale province of now
Turkey, but it is controversial). In order to regain the unsurpassed beauty, the ancient
Greek heroes and the Trojans had a stretched 10 years of the Trojan War, later the
Greek coalition forces used of the Trojan horse offered by Odysseus, overnight broke
up Troy, obtained the victory of war. The ancient Greek epic poem the Iliad written
by the ancient Greek narrative poet Homer (about 850 BC–the 8th century BC) was
mainly to describe the story of the Trojan War last year. Karl Heinrich Marx had
said the epic was that “they still give us aesthetic pleasure and are in certain respects
regarded as a standard and unattainable ideal”.
Example 2.5.13 The problem of least (or minimal) surface of revolution. As
shown in Fig. 2.8, in all the curves passing through the endpoints A(x 0 , y 0 ) and
B(x 1 , y 1 ), find out a curve, when it revolves around the Ox axis, the resulting surface
has the smallest area.
Fig. 2.8 The least surface of
revolution
O
x
y
z
A
B
y = y(x)