1. Tutorial on Nov.12

2. Remarks for Lab4 assignemnt

3. Euler method vs. Euler-Cromer method Euler methodEuler-Cromer method Sequence of the last two lines are very important! Euler-Cromer method is better than Euler method ONLY for SHM. In lab4, you are asked to implement Euler method, and marks are deducted if you implement Euler-Cromer method instead

4. Reminder for C programmer Be careful on copy-and-paste!

5. Comparison between simulation and analytical solution for SHM Distance between two particles Equilibrium distance Oscillation amplitude. Important! Angular frequency. Important!

6. Demonstration of chaotic motion

7. Simple example: double pendulum Picture from Wikipedia

8. Numerical simulation of double pendulum It look quite easy to make a numerical simulation… The simulated result looks quite messy. In fact even if we only change the initial position a little bit, the system will be in totally different state.

9. Video on YouTube  https://www.youtube.com/watch?v=PrPYeu3GRLg https://www.youtube.com/watch?v=PrPYeu3GRLg  There are actually a lot of videos on YouTube or other online resources about chaotic motion…

10. More famous chaotic motion: three-body problem  Well known chaotic system  There are some special stable solutions…  https://www.youtube.com/watch?v=jKvnn1r-9Iw https://www.youtube.com/watch?v=jKvnn1r-9Iw  But most likely it is not predictable  https://www.youtube.com/watch?v=rr0JpgKPKgg https://www.youtube.com/watch?v=rr0JpgKPKgg