Understanding Uniqueness of solutions of differential equations – nonlinear ODEs – pendulum example
I am trying to understand If the nonlinear ODE of the classical equation for the pendulum with friction:
$$\ddot{\theta}+a\,\dot{\theta}+b\,\sin(\theta) = 0,\quad (\theta_0,\,\dot{\theta_0}) = \left(\frac{\pi}{2},\,0\right)$$
fulfills the conditions of Uniqueness of solutions ($a$ and $b$ are real valued constants, both different from zero).
Arguing on another question, other user tell me that the equations should stand an unique solution, but I believe this user is mistaken but I cannot prove it.
My intuition is the following: Since the equation model a "realistic" pendulum without introducing approximations, and the "physical pendulum" due it stop moving in reality after a period of time, the "exact solutions" of the equation should become exactly zero after an "ending time" and remain zero forever after, so "there is non-uniqueness of solutions through zero in backwards time" as is mentioned in this paper.
I am trying to understand Uniqueness through Wikipedia but my background is not enough (I don´t fully understand the mentioned paper either), so I would like to someone to explain briefly if the equation presented here has or not uniqueness of solutions and why (references to more deep insights are welcome).
Beforehand, thanks you very much.
Best Answer
You are mixing contexts with your assumptions and thus get to absurd conclusions.
"the equation models a "realistic" pendulum without introducing approximations" is a statement on the combination of physical laws into a mathematical model. The approximations like linearization of the sine for small angles are of mathematical nature.
The laws of kinematics themselves are only approximations, even inside physics. They neglect the elasticity of the materials used, turbolence in the air, down to atomic structures.
"the "physical pendulum" due it stop moving in reality": Here you switch context to an actual physical pendulum and what can be measured with actual physical instruments. This is not the mathematical model.
If you want to, you can consider the infinitely prolonged oscillations as an artifact of the simplifications in the mathematical model. Every mathematical model is an imperfect reflection of the real reality. Most often the mathematical simulation is too "clean".
On the other hand, the idea of "coming to rest" or "stop moving" is also a fiction, perhaps bringing the apparatus to zero Kelvin might count. Otherwise there will always be small movement, thermal oscillations, air currents, electro-magnetic fields, changes in gravity,... All incredibly small, but still present.