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PHY 7 11 Classical Mechanics and Mathematical Methods 9-9:50 AM MWF Olin 107

In this lecture, we will finish reading Chapter 6 and discuss the Virial theorem, canonical transformations, and the Hamilton-Jacobi formalism in classical mechanics. Examples of the Virial theorem will be provided, along with an introduction to canonical transformations and the relations between old and new variables. The concept of "Canonical" transformations will be explained, followed by a discussion on the Hamiltonian formalism and the canonical equations of motion. We will also explore the possibilities of finding all constants of motion using the Hamilton-Jacobi theory. The lecture will conclude with a recap of the Lagrangian and Hamiltonian pictures in classical mechanics.

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PHY 7 11 Classical Mechanics and Mathematical Methods 9-9:50 AM MWF Olin 107

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  1. PHY 711 Classical Mechanics and Mathematical Methods • 9-9:50 AM MWF Olin 107 • Plan for Lecture 13: • Finish reading Chapter 6 • Virial theorem • Canonical transformations • Hamilton-Jacobi formalism PHY 711 Fall 2017 -- Lecture 13

  2. PHY 711 Fall 2017 -- Lecture 13

  3. Virial theorem (Clausius ~ 1860) Note that this implies that the motion is bounded PHY 711 Fall 2017 -- Lecture 13

  4. Examples of the Virial Theorem PHY 711 Fall 2017 -- Lecture 13

  5. Examples of the Virial Theorem PHY 711 Fall 2017 -- Lecture 13

  6. Hamiltonian formalism and the canonical equations of motion: PHY 711 Fall 2017 -- Lecture 13

  7. Notion of “Canonical” transformations PHY 711 Fall 2017 -- Lecture 13

  8. Some relations between old and new variables: PHY 711 Fall 2017 -- Lecture 13

  9. PHY 711 Fall 2017 -- Lecture 13

  10. Note that it is conceivable that if we were extraordinarily clever, we could find all of the constants of the motion! Possible solution – Hamilton-Jacobi theory: PHY 711 Fall 2017 -- Lecture 13

  11. PHY 711 Fall 2017 -- Lecture 13

  12. 0 0 0 PHY 711 Fall 2017 -- Lecture 13

  13. 0 0 0 PHY 711 Fall 2017 -- Lecture 13

  14. Differential equation for S: PHY 711 Fall 2017 -- Lecture 13

  15. Continued: PHY 711 Fall 2017 -- Lecture 13

  16. Continued: PHY 711 Fall 2017 -- Lecture 13

  17. Another example of Hamilton Jacobi equations PHY 711 Fall 2017 -- Lecture 13

  18. PHY 711 Fall 2017 -- Lecture 13

  19. Check action: PHY 711 Fall 2017 -- Lecture 13

  20. Recap -- Lagrangian picture Hamiltonian picture PHY 711 Fall 2017 -- Lecture 13

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