Double pendulum, time question

[fluidistic]

I'm probably going to learn about the double pendulum in a few weeks, however I have a question that I can't get rid off from my head.
Is there a time (I imagine it to be very large) where the pendulum reach the initial position/configuration? In another words, a time where it moves as it has moved. Maybe we can call this a period, but I'm not really sure.
If I remember well, Poincaré's recurrence theorem implies the existence of such a time.

Mathematically I must have the motion equation under my eyes and set $$t=0$$. I do the same but setting $$t=t_1$$. And lastly I equal both equation and I solve for $$t_1$$. I'm guessing it's very hard to solve for $$t_1$$ since I never heard of a period of a double pendulum.
Do someone has something to say?

 

[Dale]

In general, no. A double pendulum is a chaotic system and will not generally return to the same configuration as before. Also, you can usually only solve the equations of motion numerically and they are numerically unstable over long periods of time.

 

[fluidistic]

In general, no. A double pendulum is a chaotic system and will not generally return to the same configuration as before. Also, you can usually only solve the equations of motion numerically and they are numerically unstable over long periods of time.

Ok thanks a lot for the information.

 

[brad_M22]

A double pendulum executes simple harmonic motion (two normal modes) when displacements from equilibrium are small. However, when large displacements are imposed, the non-linear system becomes dramatically chaotic in its motion and demonstrates that deterministic systems are not necessarily predictable.

http://www.fas.harvard.edu/~scdiroff/lds/MathamaticalTopics/ChaoticPendulum/ChaoticPendulum002.gif
________________
http://makersmarket.com/products/91-chaotic-pendulum

 

[PJC01]

I can confirm that your question about the time it takes for a double pendulum to reach its initial position is a valid one. The concept of a period, which is the time it takes for a system to repeat its motion, applies to the double pendulum as well. However, determining the exact period of a double pendulum is a complex and challenging task due to the nonlinear nature of its motion.

Poincaré's recurrence theorem does imply the existence of a time where the double pendulum will return to its initial configuration, but it does not provide a way to calculate this time. As you mentioned, it would require solving the motion equations for t=t_1, which is a difficult task.

There have been studies and simulations done to estimate the period of a double pendulum, but it ultimately depends on the initial conditions and the length and mass of the pendulum. It is also worth noting that the period of a double pendulum is not a constant value, as it can change depending on the initial conditions.

In conclusion, while the concept of a period applies to the double pendulum, it is not easy to determine its exact value due to the complexity of its motion. Further research and calculations may provide more insight into this question, but for now, we can appreciate the beauty and unpredictability of the double pendulum's motion.