TransformationsPoint TransformationCoordinate InvarianceContact TransformationInvariant HamiltonianDifferential TransformationTransform GenerationEquate and SolveTransformationsTransformationsPoint TransformationPoint TransformationLagrange’s method is Lagrange’s method is independent of the independent of the coordinate choice.coordinate choice.This represents a change in This represents a change in configuration space configuration space QQ..),( tqqjk0),,(21dttqqLjjttjjkkqqqq),( tqqkjkkjjqqqqyxr, Coordinate InvarianceCoordinate InvarianceHamilton’s principle depends on the variation of a Hamilton’s principle depends on the variation of a time integral.time integral.Different Lagrangians with different coordinates may Different Lagrangians with different coordinates may differ by a time derivative function of the coordinates.differ by a time derivative function of the coordinates.);,( tqqLjj);,( tqqLkk),,(),,(),,( tqqdtdtqqLtqqLkjjjkkGivenIf the two coordinate sets have matching paths, and Then the two Lagrangians describe the same systemContact TransformationContact TransformationttqqqqdtdttLqpqqLttLqpqqLkkjjjjjjkkkkjjqpjjqLpEquating the coefficients: kkqpSuggests another transformation with p, qInvariant HamiltonianInvariant HamiltonianConstruct a new Construct a new Hamiltonian.Hamiltonian.•Use Use from before from before•Use definitions of Use definitions of ppjjTransformation depends Transformation depends on the coordinate on the coordinate transformation.transformation.•Uses phase spaceUses phase spaceThis is the canonical This is the canonical transformation.transformation.LqptpqHkkkk),,(tqqqqLqqHkkjjkktqqLHjjttpqHtpqHjjkk),,(),,(),(kkjpqq),(kkkpqp),,( tqqkjDifferential TransformationDifferential TransformationThe Hamiltonian transformation can be expanded.The Hamiltonian transformation can be expanded.The function The function does not need to depend on all the does not need to depend on all the qq..•Implies a relationship between coordinate systemsImplies a relationship between coordinate systems•Independent relations Independent relations ggll can result in variable reduction can result in variable reductionddtHHqdpdqpkkjj )(0kkljjlqdqgdqqgTransform GenerationTransform GenerationTwo dimensional systemTwo dimensional system•qq11, , qq22, , pp11, , pp22Function Function only depends on only depends on qq11..Substitute for Substitute for HH.. 2121q11222112222111112211dqqdqqqdqqqpdqqqdqqqpdqpdqp11dqqd Equate and SolveEquate and SolveCoefficients of the Coefficients of the differentials must match.differentials must match.•dqdq11, , dqdq22, (, (dpdp11, , dpdp22))Solution depends on Solution depends on coordinate transformation.coordinate transformation.Assume an identity Assume an identity transformation.transformation.•Find the momentum Find the momentum transformationstransformations11221111qqqpqqpp 02222112qqpqqpp11qq 22qq 1110 qpp 022 pp111qpp 22pp
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