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PHY 7 11 Classical Mechanics and Mathematical Methods 11-11:50 AM MWF Olin 107

PHY 7 11 Classical Mechanics and Mathematical Methods 11-11:50 AM MWF Olin 107 Plan for Lecture 5: Start reading Chapter 3.17 – Introduction to the calculus of variations Example problems Brachistochrone.

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PHY 7 11 Classical Mechanics and Mathematical Methods 11-11:50 AM MWF Olin 107

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  1. PHY 711 Classical Mechanics and Mathematical Methods • 11-11:50 AM MWF Olin 107 • Plan for Lecture 5: • Start reading Chapter 3.17 – • Introduction to the calculus of variations • Example problems • Brachistochrone PHY 711 Fall 2016 -- Lecture 5

  2. PHY 711 Fall 2016 -- Lecture 5

  3. In Chapter 3, the notion of Lagrangian dynamics is developed; reformulating Newton’s laws in terms of minimization of related functions. In preparation, we need to develop a mathematical tool known as “the calculus of variation”. Minimization of a simple function local minimum global minimum PHY 711 Fall 2016 -- Lecture 5

  4. Minimization of a simple function local minimum global minimum PHY 711 Fall 2016 -- Lecture 5

  5. Functional minimization PHY 711 Fall 2016 -- Lecture 5

  6. PHY 711 Fall 2016 -- Lecture 5

  7. Calculus of variation example for a pure integral functions PHY 711 Fall 2016 -- Lecture 5

  8. After some derivations, we find PHY 711 Fall 2016 -- Lecture 5

  9. PHY 711 Fall 2016 -- Lecture 5

  10. Lamp shade shape y(x) y xiyi x xfyf PHY 711 Fall 2016 -- Lecture 5

  11. PHY 711 Fall 2016 -- Lecture 5

  12. PHY 711 Fall 2016 -- Lecture 5

  13. Another example: • (Courtesy of F. B. Hildebrand, Methods of Applied Mathematics) PHY 711 Fall 2016 -- Lecture 5

  14. Euler-Lagrange equation Alternate Euler-Lagrange equation PHY 711 Fall 2016 -- Lecture 5

  15. Brachistochrone problem: (solved by Newton in 1696) http://mathworld.wolfram.com/BrachistochroneProblem.html A particle of weight mg travels frictionlessly down a path of shape y(x). What is the shape of the path y(x) that minimizes the travel time from y(0)=0 to y(p)=-2 ? PHY 711 Fall 2016 -- Lecture 5

  16. PHY 711 Fall 2016 -- Lecture 5

  17. PHY 711 Fall 2016 -- Lecture 5

  18. Parametric equations for Brachistochrone: PHY 711 Fall 2016 -- Lecture 5

  19. Parametric plot -- plot([theta-sin(theta), cos(theta)-1, theta = 0 .. Pi y x PHY 711 Fall 2016 -- Lecture 5

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