11.15** Write down the exact Lagrangian (good for all angles) for the double pendulum of Figure 11.9 and find the corresponding equations of motion. Show that they reduce to Equations (11.41) and (11.42) if both angles are small. 4₁₁ 402 L2 $2 m2 Figure 11.9 A double pendulum. The velocity of m2 is the vector sum of the two velocities shown, separated by an angle 2-1. = ас 1 Эф Эф and dt a 10 ог ас = афг (m₁ + m²)L² + m₂L₁L₁₂ = −(m₁ + m2)gL₁₁ (11.41) == or m₂L₁₂+m₂L2=-m2g L202 (11.42)