328
7 Lagrangian and Hamiltonian Mechanics
which yield the equations of motion
(M + m) ¨
x − mr ¨
θ = 0
(15)
¨
x − r ¨
θ − g θ = 0
(16)
Equations (15) and (16) constitute the equations of motion. Eliminating ¨
x
we obtain
¨
θ +
M + m
M
g
r
θ = 0
(17)
which is the equation for simple harmonic motion with frequency ω =
(M + m)
M
g
r
and time period
T =
2π
ω
= 2π
M
(M + m)
r
g
(18)
On comparing (18) with (8) it is observed that the period of oscillation is
smaller by a factor [M/(M + m) 1/2 ] as compared to the case where the
bowl is fixed.
7.28 Take the differential of the Lagrangian
L(q 1 , . . . , q n , ˙
q 1 , . . . , ˙
q n , t)
dL =
n
r =1
∂ L
∂q r
dq r +
∂ L
∂ ˙
q r
d ˙
q r
+
∂ L
∂t
dt
(1)
Now the Lagrangian equations are
d
dt
∂ L
∂ ˙
q r
−
∂ L
∂q r
= 0
( 2 )
and the generalized momenta are defined by
∂ L
∂ ˙
q r
= p r
(3)
Using (3) in (2) we have
d
dt
∂ L
∂ ˙
q r
= ˙
p r
(4)
7 Lagrangian and Hamiltonian Mechanics
which yield the equations of motion
(M + m) ¨
x − mr ¨
θ = 0
(15)
¨
x − r ¨
θ − g θ = 0
(16)
Equations (15) and (16) constitute the equations of motion. Eliminating ¨
x
we obtain
¨
θ +
M + m
M
g
r
θ = 0
(17)
which is the equation for simple harmonic motion with frequency ω =
(M + m)
M
g
r
and time period
T =
2π
ω
= 2π
M
(M + m)
r
g
(18)
On comparing (18) with (8) it is observed that the period of oscillation is
smaller by a factor [M/(M + m) 1/2 ] as compared to the case where the
bowl is fixed.
7.28 Take the differential of the Lagrangian
L(q 1 , . . . , q n , ˙
q 1 , . . . , ˙
q n , t)
dL =
n
r =1
∂ L
∂q r
dq r +
∂ L
∂ ˙
q r
d ˙
q r
+
∂ L
∂t
dt
(1)
Now the Lagrangian equations are
d
dt
∂ L
∂ ˙
q r
−
∂ L
∂q r
= 0
( 2 )
and the generalized momenta are defined by
∂ L
∂ ˙
q r
= p r
(3)
Using (3) in (2) we have
d
dt
∂ L
∂ ˙
q r
= ˙
p r
(4)