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.

 

[paranormal]

for your question. It is clear that you have put a lot of thought into understanding the energy of a variable length pendulum. Your approach is correct in considering both the kinetic and potential energy terms. However, in order to use the Lyapunov function to study the stability of the pendulum, it is important to choose a function that is time-invariant, i.e. does not change with time.

One potential solution could be to consider the total energy of the pendulum, which is the sum of kinetic and potential energy. This would give you a time-invariant Lyapunov function, as the total energy remains constant throughout the motion of the pendulum.

Alternatively, you could also consider a Lyapunov function that is a combination of the kinetic and potential energy terms, but with a time-varying coefficient that accounts for the change in length of the pendulum. This would also give you a time-invariant function, as the coefficient would cancel out the time-varying term.

Ultimately, the choice of the Lyapunov function will depend on the specific problem and the stability analysis you are trying to perform. It may be helpful to consult with your instructor or a colleague to determine the most appropriate approach for your particular case.

I hope this helps and wish you success in your studies. Keep up the great work!