Maple for Lagrangian Mechanics

1. Maple for Lagrangian Mechanics Frank Wang

2. Newton • Newton’s Second Law: Forces, masses, accelerations

3. Lagrange • Variational principle formulation: Kinetic energy minus potential energy is minimum.

5. 1-D Motion under Gravity x x t t

6. Fastest Path lifeguard swimmer

7. Least Action • Geometric Optics • Classical Mechanics • General Relativity • Quantum Mechanics (Feynman’s Path Integral)

8. Euler-Lagrange Equation • The only thing we need to know! • Euler-Lagrange equations gives equations of motion, which are differential equations.

9. 1-D Motion under Gravity

10. Equation of Motion

11. Calculus of Variations • Finding derivative of a function w.r.t. another function,

12. Using Maple • Substitute x(t) and v(t) with symbols, • Differentiate L w.r.t.var1 and var2 • Substitute var1 and var2 back to x(t) and v(t).

13. Result • Lagrangian and Newtonian are identical: ma F

14. Advantages • Straightforward • Only simple commands: subs, diff, dsolve • No external library • Treating x(t) and v(t) as two separate dependent variables. Maple 8 has a VariationalCalculuspackage.

15. Lagrangian in 3 Steps • Perform coordinate transformation to express KE and PE in generalized coordinates. • Employ the Euler-Lagrange equations to derive equations of motion. • Solve differential equations to find the actual path.

16. Euler-Lagrange Equation • The only thing we need to know! • Euler-Lagrange equations gives equations of motion, which are differential equations.

17. Double Pendulum q1 q2

18. Lagrangian for Double Pendulum Maple produces

19. Two Degrees of Freedom • For mass 1, • For mass 2,

20. Gyroscope • Many simple things can be deduced mathematically more rapidly than they can be really understood in a fundamental or simple sense. Feynman I-20-6

21. Polar Coordinates r f

22. Kepler Problem • Kinetic energy in polar coordinates: • Potential energy for inverse square law and a quadrupole term:

23. Symmetry • Lagrangian of Kepler problem contains no f, • For rcoordinate,

24. Planetary Motion • 18th Century: Lagrange discovered that planetary motion corresponds to least action. • 20th Century: Einstein formulated geodesic equation, i.e., the shortest “distance” in a curved space-time.

25. General Relativity • Matter tells geometry how to curve, geometry tells matter how to move. • Motion in gravitational field is none other than finding the shortest connection between two points in a curved space-time.

26. Geometry • Flat space • Curved space-time (Schwarzschild)

27. Shortest Path • The shortest path corresponds to • Lagrangian is the integrand

28. Lagrangian for GR • Flat space • Schwarzschild solution

29. Lagrangian in 3 Steps • Perform coordinate transformation to express KE and PE in generalized coordinates. • Employ the Euler-Lagrange equations to derive equations of motion. • Solve differential equations to find the actual path.

30. Euler-Lagrange Equation • The only thing we need to know! • Euler-Lagrange equations gives equations of motion, which are differential equations.

31. Application of Maple • Perform coordinate transformation which otherwise will be very tedious. • Employ chain rule and rearrange equations. • Solve differential equations (using numerical method in most cases), and graph the results.

32. Conclusions • Principle of Least Action is a powerful concept. • Maple is an ideal tool to handle this type of problems. • One can apply simple principle to elementary and sophisticated problems, and Maple does all the calculations.