9.6 The Hamilton Principles and Their Applications
567
Fig. 9.11 Particle and
spring system diagram
O
k
m
p
r
θ
ωt
U =
1
2
k(r − r 0 )
2
The Lagrange function is
L =
1
2
ma
2
ω
2
+
1
2
m(˙ r
2
+ r
2 ˙
θ
2
) + maω[˙ r sin(θ − ωt) + r ˙
θ cos(θ − ωt)]
−
1
2
k(r − r 0 )
2
The functional established by the Hamilton’s principle is
J =
t 1
t 0
1
2
ma
2
ω
2
+
1
2
m(˙ r
2
+ r
2 ˙
θ
2
) + maω[˙ r sin(θ − ωt) + r ˙
θ cos(θ − ωt)]
−
1
2
k(r − r 0 )
2
dt
The Euler equations of the functional namely the differential equations of motion
of the system are