LiouvilleMatrix FormInfinitessimal TransformMatrix SymmetryJacobian DeterminantIntegral InvariantLiouville’s TheoremLagrange BracketLiouvilleLiouvilleMatrix FormMatrix FormThe dynamic variables can The dynamic variables can be assigned to a single set.be assigned to a single set.•qq11, , pp11, , qq22, , pp22, …, , …, qqff, , ppff•zz11, , zz22, …, , …, zz22ffHamilton’s equations can be Hamilton’s equations can be written in terms of written in terms of zz•AA: symplectic 2: symplectic 2f f xx 22ff matrix matrix•AA22 = -1 = -1•AATT = - = -AA0100100000010010JztzHJz),(jjjjqpdtdtzHqptzHzJztzzL21),(),(21),,(Infinitessimal TransformInfinitessimal TransformThe infinitessimal The infinitessimal transformation is a contact transformation is a contact transformation.transformation.•Generator Generator XX •Written with the matrix Written with the matrix AA•Used in Poisson bracketUsed in Poisson bracket zXJzYXY,jjqtpqXp),,(jjptpqXq),,(ztzXJz),( jjjjpYqXpXqYXY ,Matrix SymmetryMatrix SymmetryThe Jacobian matrix describes a transformation.The Jacobian matrix describes a transformation.Use this for the difference of LagrangiansUse this for the difference of LagrangianszzMtzzMz  ddtHHdtzJzzJzkjkjkjkj21 ddttzJzHHdzMJzJzljljklkjljjkj2121   lijljjijklkjljjkjiMJzJzzMJzJzzRequire symmetryJacobian DeterminantJacobian DeterminantThe symmetry of the matrix The symmetry of the matrix is equivalent to the is equivalent to the symplectic requirementsymplectic requirement•M is symplecticM is symplectic•CTs are symplecticCTs are symplecticTake the determinant of both Take the determinant of both sidessides•The transformation is The transformation is continuous with the identitycontinuous with the identity•The Jacobian determinant The Jacobian determinant of any CT is unity.of any CT is unity.  JMMJMJMJJMMJJMMJTTTTTTT 1det2M1det M JMMJMJzJzzTlkjljjkjiJJMMTsinceMMJTdetdet1detIntegral InvariantIntegral InvariantIntegrate phase space Integrate phase space •Element in Element in ff dimensions dimensions dVdVff•The integral is invariantThe integral is invariantEquivalent to constancy of Equivalent to constancy of phase space density.phase space density.•Density is Density is 2tt ffdVI)(0zzt0zzfffffIdVMMdVVdIdet1tt qpzzzzt 0dtdzzt0Liouville’s TheoremLiouville’s TheoremThe Jacobian determinant of any CT is unity.The Jacobian determinant of any CT is unity.The distribution function is constant along any The distribution function is constant along any trajectory in phase space.trajectory in phase space.•Poisson bracket: Poisson bracket: Given Given : : RR22nn  RR11  RR11  RR22nn  RR11; ; ((((zz,,tt), ), tt))•A differential flow generated byA differential flow generated by•Then for fixed Then for fixed tt, , ff(z) (z)  ((zz,,tt)) is symplectic is symplectic Ht,Lagrange BracketLagrange BracketPoisson bracketPoisson bracket•Invariant under CTInvariant under CTLagrange bracketLagrange bracket•Reciprocal matrix of Reciprocal matrix of Poisson bracketPoisson bracket•Also invariant under CTAlso invariant under CTnext jjjjpXqYpYqXYX , zYJzXYX, XpYqYpXqYXjjjj,