9.6 The Hamilton Principles and Their Applications
559
Fig. 9.8 The revolution and
rolling of a circular cylinder
O
O 1
R
r
ϕ
θ
θ
ω
T =
1
2
Mv 2 +
1
2
mv 2
1 +
1
2
J ω 2 =
1
2
M R 2 ˙
ϕ 2 +
1
2
m[(R − r ) ˙
θ ] 2 +
1
4
m[R ˙
ϕ − (R − r ) ˙
θ ] 2
The potential energy of the system is
V = −mg(R − r ) cos θ
The Lagrange function is
L = T − V =
1
2
M R 2 ˙
ϕ 2 +
1
2
m[(R − r ) ˙
θ ] 2 +
1
4
m[R ˙
ϕ − (R − r ) ˙
θ ] 2
+ mg(R − r ) cos θ
Calculating the various partial derivatives
∂ L
∂ϕ
= 0,
∂ L
∂θ
= −mg(R − r ) sin θ
d
dt
∂ L
∂ ˙
ϕ
= M R
2
¨
ϕ +
1
2
m R[R ¨
ϕ − (R − r ) ¨
θ] =
1
2
(2M + m)R
2
¨
ϕ −
1
2
m R(R − r ) ¨
θ
d
dt
∂ L
∂ ˙
θ
= m(R − r ) 2 ¨
θ −
1
2
m(R − r )[R ¨
ϕ − (R − r ) ¨
θ ] =
3
2
m(R − r ) 2 ¨
θ −
1
2
m R(R − r )R ¨
ϕ
Substituting the above various equations into the Lagrange’s equation and
reducing, we give