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For a simple pendulum of mass ( m ) with a length ( l(t) ) varying with time, we begin by defining the generalized coordinate ( \theta ), which measures the angular displacement from the vertical. The position of the mass in Cartesian coordinates can be expressed as: [ x = l(t) \sin \theta, \quad y = -l(t) \cos \theta ] The velocity components are obtained by differentiating with respect to time: [ \dot{x} = \dot{l} \sin \theta + l \cos \theta , \dot{\theta}, \quad \dot{y} = -\dot{l} \cos \theta + _________ __________ __________ ___ __________ ___ ___ ___.
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