Transformations.

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Presentation transcript:

Transformations

Point Transformation Lagrange’s method is independent of the coordinate choice. This represents a change in configuration space Q. y r, q x

Coordinate Invariance Hamilton’s principle depends on the variation of a time integral. Different Lagrangians with different coordinates may differ by a time derivative function of the coordinates. Given If the two coordinate sets have matching paths, and Then the two Lagrangians describe the same system

Contact Transformation Equating the coefficients: Suggests another transformation with p, q

Invariant Hamiltonian Construct a new Hamiltonian. Use f from before Use definitions of pj Transformation depends on the coordinate transformation. Uses phase space This is the canonical transformation.

Differential Transformation The Hamiltonian transformation can be expanded. The function f does not need to depend on all the q. Implies a relationship between coordinate systems Independent relations gl can result in variable reduction

Transform Generation Two dimensional system q1, q2, p1, p2 Function f only depends on q1. Substitute for H.

Equate and Solve next Coefficients of the differentials must match. dq1, dq2, (dp1, dp2) Solution depends on coordinate transformation. Assume an identity transformation. Find the momentum transformations next