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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

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.
coordinate transformations
Coordinate Transformations

Transformation exists if:

sample transforms
Sample Transforms

No Transform:

Polar Coordinates:

Rotating Coordinates:

euler s equations
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
kinetic energy
Kinetic Energy

Aijdiagonal ifq’s are orthogonal

BjandT0are zero if coordinate system is not moving

kinetic energy in various coordinate systems
Kinetic Energy in Various Coordinate Systems

No Transform:

Polar Coordinates:

working towards euler s equation 2
Working towards Euler’s Equation (2)

Almost, but not quite Euler’s equation

potential
Potential

If:

and

Define Lagrangian:

Euler’s Equation: Implies integral of Lagrangian is minimized.

hamilton s principle 1834
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:

extension to quantum mechanics
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: