1. Generators

2. Phase Space  A contact transformation acts on a Hamiltonian. Transform on T* Q.Transform on T* Q.  The function  does not always give a point transformation. Would transform only on Q.Would transform only on Q.

3. Harmonic Oscillator  The 1-D harmonic oscillator has a simple Hamiltonian. Make a contact transformationMake a contact transformation Generator is .Generator is .  Find the associated momenta. p q E = H

4. Cyclic Variable  Write one system in terms of the other. Find the new HamiltonianFind the new Hamiltonian  The new generalized position is cyclic. Simplified HamiltonianSimplified Hamiltonian Frequency dependence is explicitFrequency dependence is explicit

5. Harmonic Motion  The equations of motion follow from the new Hamiltonian Angular momentum and angleAngular momentum and angle  The equations of motion for the original system follow. p q J = 2  E/   =  t

6. Canonical Transformation  The set of contact transformations is a group. Two successive contact transformations is also oneTwo successive contact transformations is also one Associative property holdsAssociative property holds Identity transformation is a contact transformationIdentity transformation is a contact transformation Every transform has an inverse (consider –  )Every transform has an inverse (consider –  )  The form of the canonical equations is invariant with respect to the group of contact transformations.

7. Independent Variables  Use is limited when new coordinates Q are functions of q but not p. dQ k not independent of dq jdQ k not independent of dq j  Transform the generator to new independent variables. “type 3” transformation“type 3” transformation

8. Four Generators Type 1 Type 2 Type 3 Type 4

9. Spherical Transformation Let Coordinate transformations may be represented as contact transformations. next