# Cubic polynomial for the motion of a heavy symmetric top

• Happiness
In summary: The statement about the LHS and RHS of (5.62') is simply showing that ##u=\pm 1## is always a root to ##f(u)=0##, regardless of whether it is a real or complex root.
Happiness
In the second paragraph after the expression of ##f(u)## below, it wrote "there are three roots to a cubic equation and three combinations of solutions". However, the combination of having three equal real roots was not mentioned. Why?

In the next paragraph, in the second sentence, it wrote "at points ##u=\pm1##, ##f(u)## is always negative, except for the unusual case where ##u=\pm1## is a root". But LHS of (5.62') ##=\dot{u}^2=\dot{\theta}^2\sin^2\theta=0## when ##u=\cos\theta=\pm1##,i.e., ##\theta=0## or ##\pi##. Thus RHS should also be 0. That means ##u=\pm1## is always a root to ##f(u)=0##. What's wrong?

The expression of ##f(u)## is given by:$$f(u)=Au^3+Bu^2+Cu+D$$The statement that there are three roots to a cubic equation and three combinations of solutions is referring to the fact that, for a cubic equation with real coefficients, there are three possible cases:1) all three roots are real and distinct 2) two of the three roots are equal and real 3) all three roots are equal and real. The statement is not mentioning the third case because it is already being handled in the next paragraph. The next paragraph is talking about the unusual case where one or both of the roots at ##u=\pm 1## is equal to 0. In this situation, the root at ##u=\pm 1## is not necessarily a real number, as the equation may have complex roots.

## 1. What is a cubic polynomial for the motion of a heavy symmetric top?

A cubic polynomial is a mathematical equation that describes the motion of a heavy symmetric top, which is a physical object with a single point of support and three axes of symmetry. It is used to model the rotation of the top as it moves through space.

## 2. What are the variables in a cubic polynomial for the motion of a heavy symmetric top?

The variables in a cubic polynomial for the motion of a heavy symmetric top include time, angular velocity, and the moments of inertia along the three axes of symmetry. These variables are used to calculate the position, velocity, and acceleration of the top at any given point in time.

## 3. How is a cubic polynomial for the motion of a heavy symmetric top derived?

A cubic polynomial for the motion of a heavy symmetric top is derived using the Euler-Lagrange equations, which relate the motion of a system to its potential and kinetic energy. By solving these equations, a cubic polynomial can be obtained that describes the motion of the top.

## 4. What are the applications of a cubic polynomial for the motion of a heavy symmetric top?

A cubic polynomial for the motion of a heavy symmetric top has many applications in physics and engineering. It can be used to model the motion of objects such as gyroscopes, satellites, and spacecraft. It is also useful in predicting the stability and maneuverability of these objects.

## 5. How accurate is a cubic polynomial for the motion of a heavy symmetric top?

The accuracy of a cubic polynomial for the motion of a heavy symmetric top depends on the accuracy of the input parameters and the complexity of the system being modeled. In general, it provides a good approximation of the motion of a heavy symmetric top, but more sophisticated models may be needed for highly complex systems or precise measurements.

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