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3 Computer Simulation of Dynamic Systems
Fig. 3.24 Dynamic system:
elliptical pendulum
we find
v
2
= l
2 ˙
θ
2
+ ˙
y
2
+ 2l ˙
θ ˙
y cos θ
Then for kinetic energy, we get:
T = T 1 + T 2 =
1
2
(m 1 + m 2 ) ˙
y
2
+
1
2
m 2 l
2 ˙
θ
2
+ m 2 l ˙
θ ˙
y cos θ
Potential energy can be calculated as
W (q) = −m 2 gl cos θ
Thus, the Lagrangian of the system will have the form
L =
1
2
(m 1 + m 2 ) ˙
y
2
+
1
2
m 2 l
2 ˙
θ
2
+ m 2 l
˙
θ ˙
y + g
cos θ
We compose the first Lagrange equation:
∂ L
∂ ˙
y
= (m 1 + m 2 ) ˙
y + m 2 l ˙
θ cos θ
∂ L
∂ y
= 0
So,