1. The Calculus of Variations! A Primer by Chris Wojtan

2. Variational Calculus: What’s in Store What are all these big words? What are all these big words? Simple applications Simple applications Euler-Lagrange equation Euler-Lagrange equation Hamilton’s Principle Hamilton’s Principle Noether’s theorem Noether’s theorem Graphics applications Graphics applications

3. Variational Calculus: What’s in Store What are all these big words? What are all these big words? Simple applications Simple applications Euler-Lagrange equation Euler-Lagrange equation Hamilton’s Principle Hamilton’s Principle Noether’s theorem Noether’s theorem Graphics applications Graphics applications

4. What is the Calculus of Variations? Please interrupt me if I go too fast! Please interrupt me if I go too fast! No point in giving this talkNo point in giving this talk if nobody gets anything out of it

5. What is the Calculus of Variations? Calculus uses functions Calculus uses functions Function: maps real numbers to real numbersFunction: maps real numbers to real numbers Variational Calculus uses functionals Variational Calculus uses functionals Functional: maps functions to real numbersFunctional: maps functions to real numbers 2.0-213.60.99

6. Optimization Review Input: scalar or vector Input: scalar or vector Output: scalar Output: scalar Extremize that output! Extremize that output! Zero first derivative at local extremum Zero first derivative at local extremum 25.1 -4.0 500 6.2 9 -1.1 -0.5 100 8.2 10.17.42.9 Input Vector Cost Function

7. Variational Calc as Optimization Input: function Input: function Output: scalar Output: scalar Extremize that output! Extremize that output! Zero first variation at local extremum Zero first variation at local extremum 10.17.42.9 Cost Functional Input Function

8. One Dimensional Calculus Input Scalar Input Value

9. Three Dimensional Calculus Input Vector Input Values

10. Ten Dimensional Calculus Input Vector Input Values

11. 50 Dimensional Calculus Input Vector Input Values

12. Infinite Dimensional Calculus Input Function Input Value Variational Calculus Input Vector

13. Variational Calculus: What’s in Store What are all these big words? What are all these big words? Simple applications Simple applications Euler-Lagrange equation Euler-Lagrange equation Hamilton’s Principle Hamilton’s Principle Noether’s theorem Noether’s theorem Graphics applications Graphics applications

14. Plane Geodesics Geodesics Find the curve that minimizes arclength Find the curve that minimizes arclength Shortest distance is a straight line Plane

15. Geodesics Geodesics Find the curve that minimizes arclength Find the curve that minimizes arclength Sphere Shortest distance is a great circle

16. Catenary What shape does a hanging rope form? What shape does a hanging rope form? Resting shape minimizes potential energy Resting shape minimizes potential energy Catenary forms a Hyperbolic Cosine

17. Catenoid What shape does a bubble form? What shape does a bubble form? Resting shape minimizes surface area Resting shape minimizes surface area Photograph3d Plot

18. Variational Calculus: What’s in Store What are all these big words? What are all these big words? Simple applications Simple applications Euler-Lagrange equation Euler-Lagrange equation Hamilton’s Principle Hamilton’s Principle Noether’s theorem Noether’s theorem Graphics applications Graphics applications

19. Gradients and Extrema Input vector Input vector Cost function Cost function We want to find that extremizes We want to find that extremizes Zero gradient at extremum Zero gradient at extremum

20. Variations and Extrema Input function Input function Cost function Cost function We want to find that extremizes We want to find that extremizes Euler-Lagrange equation satisfied at extrememum Euler-Lagrange equation satisfied at extrememum

21. Euler-Lagrange Equation Analogous to gradient Analogous to gradient To extremize, find function that satisfies this equation To extremize, find function that satisfies this equation

22. Example: Geodesics in the Plane Arclength Arclength Minimize Minimize Euler-Lagrange Euler-Lagrange Shortest distance between 2 points is a line! Shortest distance between 2 points is a line!

23. Variational Calculus: What’s in Store What are all these big words? What are all these big words? Simple applications Simple applications Euler-Lagrange equation Euler-Lagrange equation Hamilton’s principle Hamilton’s principle Noether’s theorem Noether’s theorem Graphics applications Graphics applications

24. The Lagrangian Kinetic minus Potential Energy Kinetic minus Potential Energy Principle of Least Action: Principle of Least Action: Equations of motion resultEquations of motion result from extremizing Lagrangian

25. Example: Ballistic Motion

26. Variational Calculus: What’s in Store What are all these big words? What are all these big words? Simple applications Simple applications Euler-Lagrange equation Euler-Lagrange equation Hamilton’s Principle Hamilton’s Principle Noether’s theorem Noether’s theorem Graphics applications Graphics applications

27. Variational Symmetry Study the change in Lagrangian with respect to perturbations Study the change in Lagrangian with respect to perturbations Invariance (symmetry) in the Lagrangian creates conservation laws Invariance (symmetry) in the Lagrangian creates conservation laws CONSERVATION OF ENERGY

28. Noether’s Theorem Variational symmetry = conservation law Variational symmetry = conservation law Invariance wrt time conserves energy Invariance wrt time conserves energy Invariance wrt translation conserves linear momentum Invariance wrt translation conserves linear momentum Invariance wrt rotation conserves angular momentum Invariance wrt rotation conserves angular momentum All of these properties are implicit in a single function! (the Lagrangian) All of these properties are implicit in a single function! (the Lagrangian)

29. Variational Calculus: What’s in Store What are all these big words? What are all these big words? Simple applications Simple applications Euler-Lagrange equation Euler-Lagrange equation Hamilton’s Principle Hamilton’s Principle Noether’s theorem Noether’s theorem Graphics applications Graphics applications

30. Variational Calc In Graphics Two main approaches Two main approaches Discretize problem, reduce to optimizationDiscretize problem, reduce to optimization Solve problem analytically, plug variables into resulting equationSolve problem analytically, plug variables into resulting equation Many applications in graphics! Many applications in graphics!

31. Variational Integrators Break integral into discrete time steps Break integral into discrete time steps Explicitly set gradient to zero Explicitly set gradient to zero Principle of Least Action says Lagrangian extremizesPrinciple of Least Action says Lagrangian extremizes Automatically conserves momentum and energy! Automatically conserves momentum and energy!

32. Useful Variational Integrators Symplectic Euler Symplectic Euler Newmark Newmark Others… Others…

33. Variational Integrators

35. Variational Tetrahedral Meshing Find the mesh that minimizes energy Find the mesh that minimizes energy

36. Shape Transformation Using Variational Implicit Functions Interpolate between 3D shapes Interpolate between 3D shapes Scattered data interpolation Scattered data interpolation Find shape that minimizes energy Find shape that minimizes energy

37. Variational Solid-Fluid Coupling Pressure minimizes kinetic energy Pressure minimizes kinetic energy Solve Euler-Lagrange equation to derive Navier-Stokes Solve Euler-Lagrange equation to derive Navier-Stokes Replace density term with accurate mass matrix Replace density term with accurate mass matrix

38. Variational Eulerian Geometry Processing Surface operations with Eulerian grids Surface operations with Eulerian grids Mean curvature flow presented as minimization of surface area Mean curvature flow presented as minimization of surface area Mass conserved explicitly Mass conserved explicitly

39. The End Have fun!