B Extending the Lagrangian of a double pendulum to more complex systems

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The discussion focuses on extending the Lagrangian formulation of a double pendulum to include more complex shapes, specifically a cylinder and an inverted cone. The total kinetic energy of the original double pendulum is established, but the challenge lies in adapting this to the new system where the cylinder and cone are interconnected and can tip and swing. The angles θ1 and θ2 are defined for the cylinder and cone, respectively, as they move under the influence of gravity. The impact of gravity on the stability of the cone is emphasized, particularly its tendency to topple from the cylinder. The conversation seeks to derive a suitable Lagrangian that accounts for the dynamics of this modified system.
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The total kinetic energy of a double pendulum can be calculated through the formule reported in the following article: https://dassencio.org/33

This works if the double pendulum in question is formed by two masses connected to each other and — one of them — to the point of origin by a "massless" rod. However, I'm interested in expanding this formula to cover systems where two bodies of more complex shapes are the swinging parts of the "double pendulum", if it possible that is.

Say I have a Cartesian plane, and at the origin point there's a cylinder of height H and radius R with the y-axis passing through the center of mass. On top of it is a cone placed upside down, with the center of mass also passing through the y-axis. Said cone has radius r and height h.

The cylinder is then tipped to the right, forming now an angle θ1 with the x-axis. As a consequence, the cone swings as well, producing an angle θ2 respect to the horizontal. It all happens within time t. How can I work out the Lagrangian from the data given?
 
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Gravity would topple the upside down cone and make it fall down from cylinder top.
 
Last edited:
anuttarasammyak said:
Gravity would topple the upside down cone and make it fall down from cylinder top.
Suppose for the sake of the argument that the two are connected, not simply placed one below the other.
 
Then we may regard it as double pendulum around its upright position of
\theta_1,\theta_2=\piforgetting surface contact of cylinder and cone. Cylinder mass center position should be considered for in detail.
 
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