1. Transformations

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

3. 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

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

5. 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.

6. 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

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

8. 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
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