Jos Coenen's blog

Sunday, November 11, 2007

Life is complicated

See a simple thing called Pendulum

An animation of a pendulum showing the velocity and acceleration vectors (v and A).

The formula for the period, T, is

$T \approx 2\pi \sqrt\frac{\ell}{g}\,$
The resulting period is given by an infinite series:: $T = 2\pi \sqrt{\frac{l}{g}} \left ( 1 + \frac{1}{4} \cdot \sin^2 \frac{\theta_{max}}{2} + \frac{9}{64} \cdot \sin^4 \frac{\theta_{max}}{2} + \cdots \right )$
The first formula, OK. The second more precize one, is far too complex.
A more complex example is the double pendulum:

The Lagrangian is given by:
$L = \frac{1}{2} m \left ( v_1^2 + v_2^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right )$
Resulting in:
${\dot p_{\theta_1}} = \frac{\partial L}{\partial \theta_1} = -\frac{1}{2} m \ell^2 \left [ {\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + 3 \frac{g}{\ell} \sin \theta_1 \right ]$

That's life: starting simple and getting complexer by the minute.

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