SeparabilityPrinicipal FunctionHamiltonian SeparationStaeckel ConditionsCombined PotentialsParabolic CoordinatesEnergy and MomentumSeparation of VariablesGenerator SeparationSeparabilitySeparabilityPrinicipal FunctionPrinicipal FunctionIn some cases Hamilton’s principal function can be In some cases Hamilton’s principal function can be separated.separated.•Each Each WW depends on only one coordinate. depends on only one coordinate.•This is totally separable.This is totally separable.EtqWtqSjkNkkjj),(),,(1),,(),(),,(1tqWqWtqSjmjmjkmkkjjFunction can be partially separable.Hamiltonian SeparationHamiltonian SeparationSimpler separability occurs Simpler separability occurs when when HH is a sum of is a sum of independent parts.independent parts.The Hamilton-Jacobi The Hamilton-Jacobi equation separates into equation separates into NN equations.equations.kkkkqWqH,NkkE1),(),,(1kkNkkjjpqHtpqHStaeckel ConditionsStaeckel ConditionsSpecific conditions exist for separability.Specific conditions exist for separability.•HH is conserved. is conserved.•LL is no more than quadratic in is no more than quadratic in dqdqjj//dtdt, so that in matrix form: , so that in matrix form: HH=1/2=1/2((p p aa))TT-1-1((p p  aa)+)+VV((qqjj))•The coordinates are orthogonal, so The coordinates are orthogonal, so TT is diagonal. is diagonal.•The vector The vector aa has has aajj = = aajj ( (qqjj))•The potential is separable.The potential is separable.•There exists a matrix There exists a matrix  with with ijij = = ijij((qqii)) jjjjTT11jjjjTqVV)( jjjT111Combined PotentialsCombined PotentialsParticle under two forcesParticle under two forces•Attractive central forceAttractive central force•Uniform field along zUniform field along zEg: charged particle with Eg: charged particle with another fixed point charge in another fixed point charge in a uniform electric field.a uniform electric field.gzrkzrV ),(222zyxr XYZParabolic CoordinatesParabolic CoordinatesSelect coordinatesSelect coordinates•Constant value Constant value   describe describe paraboloids of revolutionparaboloids of revolution•Other coordinate is Other coordinate is •Equate to cartesian systemEquate to cartesian systemFind differentials to get Find differentials to get velocity.velocity.zr cosxzr siny2/)(zddddx sincos21ddddy cossin212/)(dddz Energy and MomentumEnergy and Momentum22dtdsmT22222221141ddddzdydxds  14mddTp 14mddTpmddTp  mpppmT2)(2222)(2)(2gkVSubstituting for the new variables:Separation of VariablesSeparation of VariablesHamiltonian is not directly separable.Hamiltonian is not directly separable.•Set Set E = T + VE = T + V•Multiply by Multiply by   There are parts depending just on There are parts depending just on ..There is a cyclic coordinate There is a cyclic coordinate ..•Constant of motion Constant of motion pp•Reduce to two degrees of freedomReduce to two degrees of freedomkEmpgmpEmpgmp244244222222Generator SeparationGenerator SeparationSet Hamilton’s function.Set Hamilton’s function.•Use momentum definitionUse momentum definition•Expect two constants Expect two constants Find one variableFind one variableWppWWW 244222Empgmp222221mpgmEdWDo the same for the other Do the same for the other variable.variable.And get the last constant.And get the last constant.244222Empgmpk