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Maple for Lagrangian Mechanics

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  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.

  4. 1-D Motion under Gravity x x t t

  5. Fastest Path lifeguard swimmer

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

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

  8. 1-D Motion under Gravity

  9. Equation of Motion

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

  11. 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).

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

  13. 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.

  14. 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.

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

  16. Double Pendulum q1 q2

  17. Lagrangian for Double Pendulum Maple produces

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

  19. 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

  20. Polar Coordinates r f

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

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

  23. 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.

  24. 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.

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

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

  27. Lagrangian for GR • Flat space • Schwarzschild solution

  28. 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.

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

  30. 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.

  31. 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.