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Analytical Mechanics

Analytical Mechanics. -An Introduction for Chemists. Christian Hedegaard Jensen. Outline. Motivation Generalised Coordinates Potential Energy Kinetic Energy Lagrange’s Equations Hamilton’s Equations Test. Outline. Motivation Generalised Coordinates Potential Energy Kinetic Energy

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Analytical Mechanics

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  1. Analytical Mechanics -An Introduction for Chemists Christian Hedegaard Jensen

  2. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  3. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  4. Motivation • Part I. In understanding Molecular Dynamics. • Better understanding of Mechanics • You do not have to think so much!

  5. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  6. Generalised Coordinates • Any set of coordinates that describe the system. • Choose coordinates that fits the problem the best.

  7. l  Example – Pendulum • The simple choice is to use polar coordinates, where we have a fixed l. • The system has one degree of freedom. I.e. the variable 

  8. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  9. l  Potential Energy • Specify the potential energy as function of the generalised coordinates. V = 0 ( = 0)

  10. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  11. Kinetic Energy • Specify the kinetic energy as function of the generalised coordinates.

  12. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  13. Lagrange’s Equations • Plug it all into the equations • + Initial conditions

  14. Lagrange’s Equations • Pendulum • I.e.

  15. Lagrange’s Equations • Small angle oscillations • This gives an angular frequency of

  16. Lagrange’s Equations • Particles moving in potential V

  17. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  18. Hamilton’s Equations

  19. Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test

  20. l1 1 l2 2 Test • Double Pendulum • Taking 2 to be constant. What is the angular frequency for small angle vibrations (of 1)? • Hint:

  21. Answer

  22. References • P. G. Hjorth; A few Concepts from Analytical Mechanics; Lecture Notes; Sep 2003 • H. Goldstein, C. Poole and J. Safko; Classical Mechanics; Addison Wesley 2002

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