Effective potential and stability

[LHarriger]

I am asked to check the stability at theta=0 of the pendulum system shown in the attachment.
I set up the energy for the system and found it to be
$E=\frac{1}{2}m(L\sin(\theta))^{2}+\frac{1}{2}m(L\frac{d\theta}{dt})^{2}-mgL\cos(\theta)$
which is in agreement with the books answer "A guide to physics problems, part 1"
To check stability for theta=0, I have to take the second deriv of the effective potential energy. This leads me to my first question.
1) What conditions require me to include a kinetic energy term as a ficticious potential energy term.
I know I should include the first term in my effective potential, but not the second. I kind of expected that this was correct but I lack the ability to give a physical or mathematical justification for discerning this. Does it have to do with the fact that theta shows up in both the potential energy term and the first term, or does it have to do with the fact that the first term is cyclic in theta dot or something else altogether?
At any rate I have that:
$V_{eff}=\frac{1}{2}m(L\sin(\theta))^{2}-mgL\cos(\theta)$
The book did not do this, instead they used the lagrangian:
$L=\frac{1}{2}m(L\sin(\theta))^{2}+\frac{1}{2}m(L\frac{d\theta}{dt})^{2}+mgL\cos(\theta)$
rewrote it as:
$L=\frac{1}{2}m(L\frac{d\theta}{dt})^{2}-\left(-\frac{1}{2}m(L\sin(\theta))^{2}-mgL\cos(\theta)\right)$
and since $L=T-U$ they identified the effective potential as:
$V_{eff}=-\frac{1}{2}m(L\sin(\theta))^{2}-mgL\cos(\theta)$
My effective potential is different than theirs by a sign on the first term. However, this makes all the difference. My plus sign results in stability for all values W, while their result yeilds W^2<g/l as the condition for stability.
My next two questions are:
2) Why do they use the lagrangian to determine their effective potential?
3) Moreover, why is it apparently wrong in this case to use the eff. pot. from the equation for energy.
These last two questions are very disturbing because in every problem I have ever worked or seen worked in a text, they have always used the energy and not the lagrangian to find the effective potential.

 

[Jhero]

it is important to understand the underlying principles and concepts behind the calculations and equations used in a problem. In this case, the stability of a pendulum system at theta=0 can be determined by analyzing the effective potential energy. However, it is crucial to understand the reasoning behind the equations and the differences in approaches.

To answer your first question, the inclusion of a kinetic energy term as a fictitious potential energy term is necessary when the system has non-conservative forces acting on it. In this case, the pendulum system experiences friction and air resistance, which are non-conservative forces. These forces dissipate energy from the system, causing the pendulum to eventually come to a stop. Therefore, in order to accurately analyze the stability of the system, the kinetic energy term must be included to account for the effects of these non-conservative forces.

In regards to your second question, using the Lagrangian to determine the effective potential is a common approach in classical mechanics. The Lagrangian is a mathematical function that describes the dynamics of a system and is a fundamental concept in the study of mechanics. It is derived from the principle of least action, which states that a system will follow a path that minimizes the action (the integral of the Lagrangian over time). By using the Lagrangian, we can derive the equations of motion for a system and determine its stability.

As for your third question, it is not necessarily wrong to use the equation for energy to find the effective potential. However, it may not always give the most accurate or complete result. The Lagrangian approach takes into account all forces and energies in the system, while the energy equation may only consider the conservative forces. In the case of the pendulum system, the energy equation does not account for the dissipative forces, leading to a different effective potential and stability condition.

In conclusion, it is important to understand the principles and concepts behind the equations and methods used in a problem. Different approaches may yield different results, and it is crucial to carefully consider all factors in order to accurately analyze a system's stability.