Energy of a pendulum (variable length, Lyapunov)

[Mugged]

Hello, question about the energy of a variable length pendulum.

Suppose you have a pendulum in the standard sense where θ is the angle, and we let the length r be a function of time r = r(t). What is the energy of the pendulum?

So far, I have determined that kinetic energy is = (1/2)m(r*dθ/dt)^2 + (1/2)m*(dr/dt)^2
and the potential energy is = mgr(1-cosθ).

In my homework problem i need to come up with a suitable Lyapunov function to study the stability of the pendulum and the typical approach approach is to set the lyapunoc function V = E (energy).

But the problem is that this is a time varying lyapunov function, i.e. V = V(θ,t). And i have to satisfy a positive definite constraint on V that is V(0,t) = 0 for all t. the problem is that dr/dt term.

Is there another lyapunov function i can choose here? or am i misrepresenting the energy?
Thank you

 

[BvU]

Hello robbed,

I don't see an answer here, but I do have a question: what can be a possible equilibrium point if you don't know anything about r(t) ?

 

[Mugged]

So I already solved this question myself. Turns out that I did have the proper Lyapunov function.

Also BvU, in this homework problem there were some bounds given for r(t), and its derivatives which I omitted because it was irrelevant to my question. To answer your question though, there aren't any equilibrium points unless r(t) is constant. homework problem was about stability anyways,

 

[BvU]

Very good ! Thanks for the extra info.