GeneratorsPhase SpaceHarmonic OscillatorCyclic VariableHarmonic MotionCanonical TransformationIndependent VariablesFour GeneratorsSpherical TransformationGeneratorsGeneratorsPhase SpacePhase SpaceA contact transformation acts on a Hamiltonian.A contact transformation acts on a Hamiltonian.•Transform on Transform on TT**QQ..The function The function  does not always give a point does not always give a point transformation.transformation.•Would transform only on Would transform only on QQ..ddtHHqdpdqpkkjj )(Harmonic OscillatorHarmonic OscillatorThe 1-D harmonic oscillator The 1-D harmonic oscillator has a simple Hamiltonian.has a simple Hamiltonian.•Make a contact Make a contact transformationtransformation•Generator is Generator is ..Find the associated Find the associated momenta.momenta. 22221qpHpqE = Hqq2cot212qqqp2cotqqqp2csc22Cyclic VariableCyclic VariableWrite one system in terms of Write one system in terms of the other.the other.•Find the new HamiltonianFind the new HamiltonianThe new generalized The new generalized position is cyclic.position is cyclic.•Simplified HamiltonianSimplified Hamiltonian•Frequency dependence is Frequency dependence is explicitexplicitqpq2sin21qpp2cos21 pHHqpH221222Harmonic MotionHarmonic MotionThe equations of motion The equations of motion follow from the new follow from the new HamiltonianHamiltonian•Angular momentum and Angular momentum and angleangleThe equations of motion for The equations of motion for the original system follow.the original system follow.Ep2pqJ = 2E/)(20ttq dtqdpH)(sin202ttEq = t)(cos20ttEp Canonical TransformationCanonical TransformationThe set of contact transformations is a group.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 The form of the canonical equations is invariant with respect to the group of contact transformations.respect to the group of contact transformations.AkkjjddtHHqdpdqp )(BmmkkddtHHqdpqdp )~(~~BAmmjjdddtHHqdpdqp )~(~~Independent VariablesIndependent VariablesUse is limited when new Use is limited when new coordinates coordinates QQ are functions are functions of of qq but not but not pp..•dQdQkk not independent of not independent of dqdqjjTransform the generator to Transform the generator to new independent variables.new independent variables.•““type 3” transformationtype 3” transformationjjkjqptqq ),,(jjjjkkjjdpqdqpddtHHqdpdqp)(),,()(tqpddtHHqdpdpqkjkkjjjjpqkkqpFour GeneratorsFour Generators),,( tqqkjjjqpkkqpjjkjkjqptqptqq  ),,(),,(kkqpjjpqjjpqkkpqjjjjkjkjqpqptpptqq  ),,(),,(jjkjkjqptpqtqq  ),,(),,(jjqpkkpqType 1 Type 2 Type 3 Type 4Spherical TransformationSpherical Transformationrppprqppzyxkkrcossinsincossincossin1rxq sinsin2ryq cos3rzq xpp 1ypp 2zpp 3cossinsincossin rprprppqzyxkkLet  sinsincoscoscos rprprpqppzyxkk cossinsinsin rprpqppyxkkCoordinate transformations may be represented as contact