How to Find Turning Points of Particle Motion on a Smooth Cone?

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SUMMARY

The discussion focuses on finding the turning points of a particle moving on the inside surface of a smooth cone with a half-angle α, influenced by gravitational force. The motion is described by a cubic equation derived from the Hamiltonian: mgr³cotα - Er² + (l²)/(2m) = 0, where l represents generalized angular momentum and E denotes the system's energy. The participant seeks assistance in solving this cubic equation, noting that while they can find three roots, only two are real. They suggest using Maple for computational assistance and reference Cardano's method for solving cubic equations.

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Homework Statement



Consider the motion of particle moving on the inside surface of a smooth cone of half-angle α, subject to the gravitational force. Although this problem does not involve a central force, certain aspects of the motion are the same as for a central-force motion.

Show that the turning points of the motion can be found from the solution of a cubic
equation in r.

Homework Equations



After solving the Hamiltonian I have found the cubic equation.

mgr^{3}cot\alpha - Er^{2} + frac{l^{2}}{2m} =0

where l is the generalized angular momentum and E is the energy of the system.

*** the third term should read (l^2)/(2m)

The Attempt at a Solution



I know that if I assume a E=const I can find 3 roots, 2 of which are real. However I have never taken a course in nonlinear physics and only know how to solve analytical cubic equations. I'm sure Maple is the easiest way to do this, but my Googling has turned up nothing but analytics. Can someone help me in solving this?
 
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