[ \fracddt \left( \frac\partial L\partial \dotq_i \right) - \frac\partial L\partial q_i = 0 ]

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: Use the fundamental equation to derive the equations of motion for each coordinate:

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L=12mṙ2+12mr2ω2cap L equals one-half m r dot squared plus one-half m r squared omega squared

[ (m_1+m_2)\ddotx = (m_1 - m_2)g ]

Evaluate the derivatives for each coordinate to obtain the differential equations of motion. Practice Problems and Detailed Solutions Problem 1: The Simple Pendulum hangs from a massless string of fixed length under uniform gravity . Find the equation of motion.