User:WillowW/Pendulum

Imagine a pendulum of length a in a uniform gravitational field of acceleration g.



\left( d\theta/dt \right)^{2} = \frac{2g}{a} \left[ \cos \theta - \cos \theta_{\mathrm{max}} \right] $$

Define a new time constant &tau; by the equation



\tau = \sqrt{\frac{a}{g}} $$

and a dimensionless time variable x = t/&tau; Then we have the equation



\left( d\theta/dx \right)^{2} = 2 \left[ \cos \theta - \cos \theta_{\mathrm{max}} \right] $$

Introduce the new variable &eta;



\eta = \sin \frac{\theta}{2} $$

then the equation becomes



\left( d\eta/dx \right)^{2} = \left( 1 - \eta^{2} \right) \left( \eta_{\mathrm{max}}^{2} - \eta^{2} \right) $$

Finally, making the change of variable y = &eta;/&eta;max yields the equation



\left( \frac{dy}{dx} \right)^{2} = \left( 1 - y^{2} \right) \left( 1 - \eta_{\mathrm{max}}^{2} y^{2} \right) $$

whose solution is sn, one of Jacobi's elliptic functions



y = \mathrm{sn} \left( x - x_{0}, k \right) $$

where x0 is the initial value of x and the parameter k equals &eta;max. It follows that &theta; can be expressed in terms of time by the formula



\sin \left( \frac{\theta}{2} \right) = \sin \left( \frac{\theta_{\mathrm{max}}}{2} \right) \mathrm{sn} \left( \sqrt{\frac{g}{a}} \left( t - t_{0} \right), k \right) $$

Hence, the period of the pendulum is 4&tau;K(k), where K(k) is the complete elliptic integral of the first kind



K(k) = \int_{0}^{1} \frac{dy}{\sqrt{ \left( 1 - y^{2} \right) \left( 1 - k^{2} y^{2} \right)}} $$