1. Discrete Geometric Mechanics for Variational Time Integrators Ari Stern Mathieu Desbrun Geometric, Variational Integrators for Computer Animation L. Kharevych Weiwei Y. Tong E. Kanso J. E. Marsden P. Schröder M. Desbrun

2. Time Integration Interested in Dynamic Systems Analytical solutions usually difficult or impossible Need numerical methods to compute time progression

3. Local vs. Global Accuracy Local accuracy (in scientific applications) In CG, we care more for qualitative behavior Global behavior > Local behavior for our purposes A geometric approach can guarantee both

4. Simple Example: Swinging Pendulum

5. Discretizing the Problem

6. Taylor Approximation

7. Explicit Euler Method

8. Implicit Euler Method

9. Symplectic Euler Method

10. Symplecticity

11. Geometric View: Lagrangian Mechanics

12. Euler-Lagrange Equation

13. Lagrangian Example: Falling Mass

14. The Discrete Lagrangian

15. The Discrete Action Functional

16. Discrete Euler-Lagrange Equation

17. Discrete Lagrangian Example: Falling Mass

18. More General: Hamilton-Pontryagin Principle

19. Discrete Hamilton-Pontryagin Principle

20. Faster Update via Minimization

21. Minimization: The Lilyan

22. Results http://tinyurl.com/n5sn3xq