Page 201 - Intro to Tensor Calculus

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where
(x 1 ,y 1 )= (L 1 sin θ 1 , −L 1 cos θ 1 )
(2.2.28)
(x 2 ,y 2 )= (L 1 sin θ 1 + L 2 sin θ 2 , −L 1 cos θ 1 − L 2 cos θ 2 )
are the coordinates of the masses m 1 and m 2 respectively. Substituting the equations (2.2.28) into equation
(2.2.27) and simplifying produces the kinetic energy expression

1 2 ˙ 2 1 2 ˙ 2
˙ ˙
T = (m 1 + m 2 )L θ + m 2 L 1 L 2 θ 1 θ 2 cos(θ 1 − θ 2 )+ m 2 L θ . (2.2.29)
2 2
1 1
2 2
Writing the Lagrangian as L = T − V , the equations describing the motion of the compound pendulum
are obtained from the Lagrangian equations

d ∂L ∂L d ∂L ∂L
− =0 and − =0.
˙
˙
dt ∂θ 1 ∂θ 1 dt ∂θ 2 ∂θ 2
Calculating the necessary derivatives, substituting them into the Lagrangian equations of motion and then
simplifying we derive the equations of motion
¨
2
˙
¨
L 1θ 1 + m 2 L 2θ 2 cos(θ 1 − θ 2 )+ m 2 L 2 (θ 2 ) sin(θ 1 − θ 2 )+ g sin θ 1 =0
m 1 + m 2 m 1 + m 2
¨
˙
2
¨
L 1 θ 1 cos(θ 1 − θ 2 )+ L 2 θ 2 − L 1 (θ 1 ) sin(θ 1 − θ 2 )+ g sin θ 2 =0.
These equations are a set of coupled, second order nonlinear ordinary differential equations. These equations
are subject to initial conditions being imposed upon the angular displacements (θ 1 ,θ 2 ) and the angular
˙
˙
velocities (θ 1 , θ 2 ).
Alternative Derivation of Lagrange’s Equations of Motion
Let c denote a given curve represented in the parametric form

i
i
x = x (t), i =1,... ,N, t 0 ≤ t ≤ t 1
and let P 0 ,P 1 denote two points on this curve corresponding to the parameter values t 0 and t 1 respectively.
Let c denote another curve which also passes through the two points P 0 and P 1 as illustrated in the figure
2.2-5.
The curve c is represented in the parametric form

i
i
i
i
x = x (t)= x (t)+ η (t), i =1,... ,N, t 0 ≤ t ≤ t 1
i
in terms of a parameter . In this representation the function η (t) must satisfy the end conditions
i
i
η (t 0 )= 0 and η (t 1 )= 0 i =1,... ,N
since the curve c is assumed to pass through the end points P 0 and P 1 .
Consider the line integral
Z
t 1
i
i
i
i
I( )= L(t, x + η , ˙x + ˙η ) dt, (2.2.30)
t 0