Poisson BracketsMatrix FormDynamical VariableAngular MomentumPoisson BracketBracket PropertiesPoisson PropertiesPoisson Bracket TheoremNot HamiltonianPoisson BracketsPoisson BracketsMatrix FormMatrix FormThe dynamic variables can The dynamic variables can be assigned to a single set.be assigned to a single set.•qq11, , qq22, …, , …, qqnn, , pp11, , pp22, …, , …, ppnn•zz11, , zz22, …, , …, zz22nnHamilton’s equations can be Hamilton’s equations can be written in terms of written in terms of zz•Symplectic 2Symplectic 2n n xx 22nn matrix matrix•Return the LagrangianReturn the Lagrangian00IIJztzHJz),(jjjjqpdtdtzHqptzHzJztzzL21),(),(21),,(Dynamical VariableDynamical VariableA dynamical variable A dynamical variable FF can can be expanded in terms of the be expanded in terms of the independent variables.independent variables.This can be expressed in This can be expressed in terms of the Hamiltonian.terms of the Hamiltonian.The Hamiltonian provides The Hamiltonian provides knowledge of knowledge of F F in phase in phase space.space.S1tFzHJzFdtdFttzFzztzFdtdF),(),(Angular MomentumAngular MomentumExampleExampleThe two dimensional The two dimensional harmonic oscillator can be harmonic oscillator can be put in normalized put in normalized coordinates.coordinates.•mm = = kk = 1 = 1Find the change in angular Find the change in angular momentum momentum ll..•It’s conservedIt’s conserved   222121222121qqppH 1221pqpql 021122112qqqqppppdtdlqHplpHqldtdlzHJzldtdliiiiPoisson BracketPoisson BracketThe time-independent part of The time-independent part of the expansion is the Poisson the expansion is the Poisson bracket of bracket of FF with with HH..This can be generalized for This can be generalized for any two dynamical variables.any two dynamical variables.Hamilton’s equations are the Hamilton’s equations are the Poisson bracket of the Poisson bracket of the coordinates with the coordinates with the Hamitonian.Hamitonian.S1 zHJzFHF, zBJzABA, iiiiqApBpBqABA, HzzHJzzz ,Bracket PropertiesBracket PropertiesThe Poisson bracket defines The Poisson bracket defines the Lie algebra for the the Lie algebra for the coordinates coordinates qq, , pp..•BilinearBilinear•AntisymmetricAntisymmetric•Jacobi identityJacobi identityS1{A + B, C} ={A, C} + {B, C}{A, B} = {A, B}{A, B} = {B, A}{A, {B, C}} + {B, {C, A}} + {C, {A , B}} = 0Poisson PropertiesPoisson PropertiesIn addition to the Lie algebra In addition to the Lie algebra properties there are two properties there are two other properties.other properties.•Product ruleProduct rule•Chain ruleChain ruleThe Poisson bracket acts The Poisson bracket acts like a derivative.like a derivative. },{},{, CABCBABCA  Jzz , zBJzAtBtA),(),,( BzJzABA,  BABA ,,Poisson Bracket TheoremPoisson Bracket TheoremLet Let zz((tt)) describe the time describe the time development of some development of some system. This is generated by system. This is generated by a Hamiltonian if and only if a Hamiltonian if and only if every pair of dynamical every pair of dynamical variables satisfies the variables satisfies the following relation:following relation:   BABABAdtd,,,       BAtHBABAdtd,,,,        HBABHAHBA ,,,,,,  tBABtAtzBJzAzBJtzAzBJzAtBAt,,,22Not HamiltonianNot HamiltonianEquations of motion must Equations of motion must follow standard form if they follow standard form if they come from a Hamiltonian.come from a Hamiltonian.Consider a pair of equations Consider a pair of equations in 1-dimension.in 1-dimension.            qppqqppqqpqpqppqpqpqdtd,,,,,,,pqqHp pqpHq pqq pqp pqpH2qpqH2CtCteqpCeqq000CteqpCpp000qpqpC 00Not consistent with motionNot consistent with H 0,