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Generalized Coordinates

Strange place to start looking for a minimization scheme, but:. Generalized Coordinates. Why change coordinates?. Sometimes one set of coordinates are easier to use in solving a problems than another. Can make use of conservation principles.

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Generalized Coordinates

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  1. Strange place to start looking for a minimization scheme, but: Generalized Coordinates Why change coordinates? • Sometimes one set of coordinates are easier to use in solving a problems than another. • Can make use of conservation principles. • Need to be able to write equations of motion is all coordinate systems • Constrains may be easier to handle complicated coordinate systems.

  2. Coordinate Transformations Transformation exists if:

  3. Sample Transforms No Transform: Polar Coordinates: Rotating Coordinates:

  4. Non-Orthogonal Transform y q2 q x, q1

  5. Euler’s Equations • If we can find the appropriate f to give Newton’s Laws then • We have found a minimization principle; and • We have a method of finding equations of motion in other coordinate systems

  6. Kinetic Energy Aijdiagonal ifq’s are orthogonal BjandT0are zero if coordinate system is not moving

  7. Kinetic Energy in Various Coordinate Systems No Transform: Polar Coordinates:

  8. Kinetic Energy in Various Coordinate Systems Rotating Coordinates:

  9. Kinetic Energy in Various Coordinate Systems Non-Orthogonal Coordinates:

  10. Generalized Force

  11. Working towards Euler’s Equation

  12. Working towards Euler’s Equation

  13. Working towards Euler’s Equation (2) Almost, but not quite Euler’s equation

  14. Potential If: and Define Lagrangian: Euler’s Equation: Implies integral of Lagrangian is minimized.

  15. Hamilton’s Principle (1834) Of all the possible paths along which a dynamical system may move from one point to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimized the integral of the difference between the kinetic and potential energies. Action: Equations of Motion:

  16. Extension to Quantum Mechanics Feynman and Hibbs, Quantum Mechanics and Path Integrals Just like extended Huygens’s Principle The probability to go from xa at ta to xb at tb is given by the absolute square P(b,a) = |K(b,a)|2 of the amplitude K(b,a) to go from a to b. This amplitude is the sum of the contributions for all paths where each path has equal weight and a phase given by the action:

  17. Example – Harmonic Oscillator Notice minus sign!!!

  18. Example – Harmonic OscillatorAccelerated Coordinate System

  19. Example – 1/r2 Force (2D)

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