330
7 Lagrangian and Hamiltonian Mechanics
Equations (11) are called Hamilton’s canonical equations. These are 2n is
number. For a system with n degrees of freedom the n Lagrangian equations
(2) of the second order are replaced by 2n Hamiltonian equations of the first
order. We note from the second equation of (11) that if any coordinate q l is not
contained explicitly in the Hamiltonian function H , the conjugate momentum
p l is a constant of motion. Such coordinates are called ignorable coordinates.
7.29 The generalized momentum p r conjugate to the generalized coordinate q r
is defined as
∂ L
∂ ˙
q r
= p r . If the Lagrangian of a dynamical system does not
contain a certain coordinate, say q s , explicitly then p s is a constant of motion.
(a) The kinetic energy arises only from the motion of the particle P on the
table as the particle Q is stationary. The potential energy arises from the
particle Q alone.
When P is at distance r from the opening, Q will be at a depth l–x from
the opening:
T =
1
2
mv
2
p =
1
2
m(˙ r
2
+ r
2 ˙
θ
2
)
(1)
V = −mg(l − r )
(2)
L = T − V =
1
2
m(˙ r
2
+ r
2 ˙
θ
2
) + mg(l − r )
(3)
For the two coordinates r and θ , Lagrange’s equations take the form
d
dt
∂ L
∂ ˙
r
−
∂ L
∂r
= 0,
d
dt
∂ L
∂ ˙
θ
−
∂ L
∂θ
= 0
( 4 )
Equations (4) yield
¨
r = r ˙
θ
2
− g
(5)
d
dt
(mr
2 ˙
θ) = 0
∴ r
2 ˙
θ = C = constant
(6)
Equations (5) and (6) constitute the equations of motion.
(b) Initial conditions: At r = a, r ˙
θ =
√
ag
∴ ˙
θ =
g
a
(7)
Using (7) in (6) with r = a, we obtain
C
2
= a
3 g
(8)
7 Lagrangian and Hamiltonian Mechanics
Equations (11) are called Hamilton’s canonical equations. These are 2n is
number. For a system with n degrees of freedom the n Lagrangian equations
(2) of the second order are replaced by 2n Hamiltonian equations of the first
order. We note from the second equation of (11) that if any coordinate q l is not
contained explicitly in the Hamiltonian function H , the conjugate momentum
p l is a constant of motion. Such coordinates are called ignorable coordinates.
7.29 The generalized momentum p r conjugate to the generalized coordinate q r
is defined as
∂ L
∂ ˙
q r
= p r . If the Lagrangian of a dynamical system does not
contain a certain coordinate, say q s , explicitly then p s is a constant of motion.
(a) The kinetic energy arises only from the motion of the particle P on the
table as the particle Q is stationary. The potential energy arises from the
particle Q alone.
When P is at distance r from the opening, Q will be at a depth l–x from
the opening:
T =
1
2
mv
2
p =
1
2
m(˙ r
2
+ r
2 ˙
θ
2
)
(1)
V = −mg(l − r )
(2)
L = T − V =
1
2
m(˙ r
2
+ r
2 ˙
θ
2
) + mg(l − r )
(3)
For the two coordinates r and θ , Lagrange’s equations take the form
d
dt
∂ L
∂ ˙
r
−
∂ L
∂r
= 0,
d
dt
∂ L
∂ ˙
θ
−
∂ L
∂θ
= 0
( 4 )
Equations (4) yield
¨
r = r ˙
θ
2
− g
(5)
d
dt
(mr
2 ˙
θ) = 0
∴ r
2 ˙
θ = C = constant
(6)
Equations (5) and (6) constitute the equations of motion.
(b) Initial conditions: At r = a, r ˙
θ =
√
ag
∴ ˙
θ =
g
a
(7)
Using (7) in (6) with r = a, we obtain
C
2
= a
3 g
(8)