Lagrangian for Conical Pendulum: T-V

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SUMMARY

The Lagrangian for a conical pendulum is defined as L = 1/2 I ω² - mgy, where I represents the moment of inertia, ω is the angular frequency, and y is the vertical displacement given by y = r sin(θ). The pendulum operates in a cylindrical coordinate system with a constant angle θ and angular velocity φ(dot). This formulation assumes two-dimensional motion and does not account for spin, which would introduce precessional effects in three-dimensional scenarios.

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What is the Lagrangian L=T-V for a conical pendulum? This is a pendulum with length l and bob mass m that rotates in a horizontal circle with theta (angle l makes with z axis) and phi(dot) (angular velocity-omega) are constant (cylindrical coordinate system).
 
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L =1/2 I w^2 - mgy

I is the moment of inertia
and w is the angular frequency w =d(theta)/dt and y =r sin(theta)

cos could be sin depending where you take your angle. and r is the distance from the fulcrum to the center of mass of the cone.
This doesn't include spin motion, and assumes the motion is two dimensional. If the motion were three dimensional and there was spin, you could get some precesional effects.
 
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