Finding Roots of cubic equation

In summary, the conversation discusses the motion of a particle on the inside surface of a cone, subject to gravitational force. The turning points of this motion can be found by solving a cubic equation in r, which involves the generalized angular momentum and energy of the system. The solution can be found using Cardano's method, which involves finding three roots, two of which are real.
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misterpickle
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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.

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

where [tex]l[/tex] 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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Related to Finding Roots of cubic equation

What is a cubic equation?

A cubic equation is a polynomial equation in which the highest power of the variable is three. It can be written in the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and x is the variable.

Why is it important to find the roots of a cubic equation?

Finding the roots of a cubic equation helps us understand the behavior of a function and solve real-world problems. It also allows us to graph the function and make predictions about its values.

How do you find the roots of a cubic equation?

To find the roots of a cubic equation, we can use various methods such as factoring, the rational root theorem, or the cubic formula. These methods involve manipulating the equation to solve for the variable x.

What is the rational root theorem?

The rational root theorem states that if a polynomial equation has rational roots, they will be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem can be used to narrow down the possible rational roots of a cubic equation.

Are there any real-world applications of finding roots of a cubic equation?

Yes, there are many real-world applications of finding roots of a cubic equation. For example, it can be used to calculate the maximum or minimum values of a function, determine the break-even point in business, or model the trajectory of a projectile. It is a valuable tool in fields such as physics, engineering, and economics.

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