High School Extending the Lagrangian of a double pendulum to more complex systems

[KataruZ98]

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?

 

[anuttarasammyak]

Gravity would topple the upside down cone and make it fall down from cylinder top.

 

[KataruZ98]

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.

 

[anuttarasammyak]

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.