Motion inside a cone

A point mass moves on the interior surface of an upside-down cone. Friction is negligible. The simulation allows displaying the mass motion, as well of that of the equivalent one-dimensional problem.

The unit mass is the particle mass m.
The unit length is u_L = (L²/m²g)^1/3, where L is the angular momentum of particle m around the cone vertical axis.
The unit time is u_T = (L/mg²)^1/3.
In these units we have m = L = g = 1 and the total mass is 1+M/m.
Gravitational energy is 0 when m is in the hole.

On the right the effective potential energy m g r cos α + 1/2 r² sin² α is displayed in red (α is the cone half-opening angle). You may use the mouse (or the controls below) to select the mechanical energy E and the initial polar distance r.
The point of view of the three-dimensional projection can be changed with the mouse.
Each whole image can be moved with the mouse while pressing Ctrl.
To change the zoom in the projections, press Shift when moving up or down the mouse pointer.
Put the mouse pointer over an element to get the corresponding tooltip.

Activities

Use the spherical polar coordinates r and φ of the mass m to write the Lagrangian.
Check that φ is cyclic and that the corresponding constant of motion is L. Use the latter to write the one-dimensional problem describing the radial evolution. Check the expression of the effective potential energy.
How many circular orbits there exist for each value of L? Are they stable? Use the simulation to check your answers.
Is there any unbounded orbit? May the mass go to the cone apex?
Find a non-circular periodic orbit near E = 2.74.

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.