Homework Help: Local Minimum of Potential Function of Spherical Pendulum

1. Apr 14, 2012

ovidiupetre

1. The problem statement, all variables and given/known data
http://img13.imageshack.us/img13/5793/84188411.jpg [Broken]

2. Relevant equations
Find a condition on b such that x = 0 is a local minimum of the potential function.

3. The attempt at a solution
To find local minimum, potential function (V) of the system should be written. V must be positive definite and derivative of V must be negative semi definite. I tried to write hundreds of potential functions that provide local minimum constraints but i can't get rid of sin and cos terms from derivative of V so i couldn't find a condition for b to show x = 0 is a local minimum of the potential function.

Last edited by a moderator: May 5, 2017
2. Apr 14, 2012

fzero

Which derivative of V must be negative? Also remember that if you write the equation of motion in terms of the force, you can get a nice expression for the potential.

3. Apr 14, 2012

ovidiupetre

First derivative of course. I couldn't get a nice expression for the potential, my equations are not simplifiable.

4. Apr 14, 2012

praharmitra

The equations of motion are of the form
$$\ddot{x} = -\frac{\partial V}{\partial x}$$
From this you can read off V. Now that you have the function V(x), what are the conditions for such a function to have a minimum at x=0 ?

5. Apr 14, 2012

ovidiupetre

V is a candidate Lyapunov function and it must be positive definite. Moreover, first derivative of V must be negative semi definite in order to x = 0 be local minimum.

6. Apr 14, 2012

fzero

The first derivative vanishes at an extremum. The second derivative is used to distinguish between local maxima and minima. You don't need to solve for V explicitly to do this problem, but it's not that hard to do so.

7. Apr 14, 2012

ovidiupetre

Problem is not about solving V, the problem is "construction of V". If i construct V, i can find a condition to make x = 0 local minimum by looking negative definiteness of the derivative of V.

8. Apr 14, 2012

fzero

You're given the first derivative of V. This is enough information to compute the second derivative and find the condition.

9. Apr 14, 2012

ovidiupetre

First derivative of potential function is not given. Construction of potential function (V) is the aim or the step that is needed to pass. When you construct potential function, the rest is easy.

10. Apr 14, 2012

fzero

As I explained in post #2, you can relate the acceleration to the potential. praharmitra gave you the formula up to a factor of the mass of the particle in post #4.