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.
Why change coordinates?
Transformation exists if:
Aijdiagonal ifq’s are orthogonal
BjandT0are zero if coordinate system is not moving
Almost, but not quite Euler’s equation
Euler’s Equation: Implies integral of Lagrangian is minimized.
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.
Equations of Motion:
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:
Notice minus sign!!!