Double Pendulum
The benefits of generalized coordinates become apparent with the analysis of a double pendulum. For the two masses mi, i=1, 2, let r=(xi, yi), i=1, 2 define their two trajectories. These vectors satisfy the two constraint equations,
The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates xi, yi i=1, 2 and the two Lagrange multipliers λi, i=1, 2 that arise from the two constraint equations.
Read more about this topic: Generalized Coordinates
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