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7 Lagrangian and Hamiltonian Mechanics
Equation (3) is the equation for the constancy of angular momentum about the
vertical axis.
When ˙
φ in (2) is eliminated with the aid of (3) we obtain a second-order
differential equation in θ .
7.19 Let ρ be the linear density of the rod, i.e. mass per unit length. Consider an
infinitesimal element of length of the rod
Fig. 7.21
dT = ρω
2
(l sin θ + x sin φ)
2 dx
T =
dT = ρω
2
2a
0
(l
2 sin
2
θ + 2lx sin θ sin φ + x
2 sin
2
φ)dx
= ω
2
Ml
2 sin
2
θ + 2Mla sin θ sin φ +
4
3
Ma
2 sin
2
φ
(1)
where we have substituted ρ = M/2a:
V = −Mg(l cos θ + a cos φ)
(2)
L = ω
2
Ml
2 sin
2
θ + 2Mla sin θ sin φ +
4
3
Ma
2 sin
2
φ
+ Mg(l cos θ + a cos φ)
(3)
∂ L
∂ ˙
θ
= 0,
∂ L
∂θ
= ω
2
(2Ml
2 sin θ cos θ + 2Mla cos θ sin φ) − Mgl sin θ
(4)
∂ L
∂ ˙
φ
= 0,
∂ L
∂φ
= ω
2
2Mla sin θ cos φ +
8
3
Ma
2 sin φ cos φ
− Mga sin φ (5)
The Lagrange’s equations
d
dt
∂ L
∂ ˙
θ
−
∂ L
∂θ
= 0,
d
dt
∂ L
∂ ˙
φ
−
∂ L
∂φ
= 0
( 6 )