Hamilton-JacobiTime-Dependent GeneratorIndependent HamiltonianHamilton-Jacobi EquationPrincipal FunctionPrincipal Function and ActionPath EquationsHamilton-JacobiHamilton-JacobiTime-Dependent Time-Dependent GeneratorGeneratorA generator determines a canonical transformation.A generator determines a canonical transformation.•The transform generally changes the form of H.The transform generally changes the form of H.•If time-dependent, the value of H also changes.If time-dependent, the value of H also changes.ttqqtpqHtpqHkjjjkk),,(),,(),,(jjqpkkqpdtqdpHjjdtqdpHjjIndependent HamiltonianIndependent HamiltonianThe easiest integration for The easiest integration for HH is if is if HH is independent of all is independent of all variables.variables.•Select Select to give that result to give that resultCoordinates and momenta Coordinates and momenta are constants of the motion.are constants of the motion.•22ff constantsconstants tqkj,,kkpkkqjjqpkkttqtqqHkjjj),,(),,(0Hamilton-Jacobi EquationHamilton-Jacobi EquationHamilton-Jacobi is a partial differential equation.Hamilton-Jacobi is a partial differential equation.•First order, generally second degreeFirst order, generally second degree•ff+1+1 independent variables: independent variables: qqjj, , tt•ff+1+1 constants, one additive, others are constants, one additive, others are kk..ttqtqqHkjjj),,(),,(0Principal FunctionPrincipal FunctionThe Lagrangian is directly related to the generator.The Lagrangian is directly related to the generator.•Generator Generator is Hamilton’s principal function is Hamilton’s principal function0 HqpLkk0HLdtdtdLdtdLL0qsince ttjjdqpHdtd1dtHHqdpdqpdkkjj)( setting one set at t and the other at t1:Principal Function and Principal Function and ActionActionThe action is defined when The action is defined when HH does not involve time.does not involve time.• is the principal functionis the principal function•Additive constant is possibleAdditive constant is possibleSubsitute to simplify HJ Subsitute to simplify HJ equation.equation.•Time-independent Time-independent HH•Time variable separatesTime variable separates•EE is not independent is not independent)()(1221ttEdtLESttjkjjqqSp),(EqSqHjj,SEt )(kEEPath EquationsPath EquationsChoose energy Choose energy symmetricallysymmetrically•Simplifies action relationsSimplifies action relations•Gives Gives ff parameterized parameterized equationsequationsOne can pick one for One can pick one for EE..•Path described without timePath described without time•One equation for location on One equation for location on the paththe pathfkkE1kkSt1t2tfEfkSkk
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