554
9 Variational Principles in Mechanics and Their Applications
O
x
u
A
B
l
u
u
u(x, t)
x
x x
Fig. 9.5 The vibrating string diagram
=
1
2
ρρxu
2
t
where, u t is the displacement velocity of the string. The total kinetic energy of the
string is
T =
ρ
2
l
0
u
2
t dx
The potential energy of the string is
V = τ
l
0
1 + u 2
x dx − l
Using the Taylor formula, there is
1 + u 2
x ≈ 1 +
1
2
u
2
x , thus
V =
τ
2
l
0
u
2
x dx
According to the Hamilton’s principle, the variational of the functional
J =
1
2
t 1
t 0
l
0
(ρu
2
t − τ u
2
x )dxdt
is equal to zero, or δ J = 0, the Euler equation can be obtained
9 Variational Principles in Mechanics and Their Applications
O
x
u
A
B
l
u
u
u(x, t)
x
x x
Fig. 9.5 The vibrating string diagram
=
1
2
ρρxu
2
t
where, u t is the displacement velocity of the string. The total kinetic energy of the
string is
T =
ρ
2
l
0
u
2
t dx
The potential energy of the string is
V = τ
l
0
1 + u 2
x dx − l
Using the Taylor formula, there is
1 + u 2
x ≈ 1 +
1
2
u
2
x , thus
V =
τ
2
l
0
u
2
x dx
According to the Hamilton’s principle, the variational of the functional
J =
1
2
t 1
t 0
l
0
(ρu
2
t − τ u
2
x )dxdt
is equal to zero, or δ J = 0, the Euler equation can be obtained