584
9 Variational Principles in Mechanics and Their Applications
Solution The compound pendulum has one degree of freedom. Choosing the angle
of intersection ϕ of OC and the vertical line as generalized coordinate, when ϕ = 0
let the level surface through the center of mass be the zero position of potential
energy, the potential energy of the system at any position is
V = mga(1 − cos ϕ)
(1)
The Lagrange function of the system is
L =
J O
2
˙
ϕ
2
− mga(1 − cos ϕ)
(2)
The generalized momentum is
p =
∂ L
∂ ˙
ϕ
= J O ˙
ϕ
(3)
The Hamiltonian function is
H = p ˙
ϕ −
J O
2
˙
ϕ 2 + mga(1 − cos ϕ) =
J O
2
˙
ϕ 2 + mga(1 − cos ϕ) =
p 2
2J O
+ mga(1 − cos ϕ) (4)
Substituting Eq. (4) into the canonical equations, we get
˙
ϕ =
∂ H
∂ p
=
p
J O
˙
p = −
∂ H
∂ϕ
= −mga sin ϕ
(5)
After deriving Eq. (3) with respect to time, substituting it into the second equation of the canonical Eqs. (5), the differential equation of motion of the compound
pendulum cab be obtained
¨
ϕ +
mag
J O
sin ϕ = 0
( 6 )
The canonical equations for solving the problem of general mechanics seem to
be no special superiority, but it is easy to make a more general study for complex
mechanical problems, such as celestial mechanics, statistical mechanics and quantum
mechanics etc.