1. Liouville

2. Matrix Form  The dynamic variables can be assigned to a single set. q 1, p 1, q 2, p 2, …, q f, p fq 1, p 1, q 2, p 2, …, q f, p f z 1, z 2, …, z 2fz 1, z 2, …, z 2f  Hamilton’s equations can be written in terms of z  A: symplectic 2 f x 2 f matrixA: symplectic 2 f x 2 f matrix A 2 = -1A 2 = -1 A T = -AA T = -A

3. Infinitessimal Transform  The infinitessimal transformation is a contact transformation. Generator  XGenerator  X Written with the matrix AWritten with the matrix A Used in Poisson bracketUsed in Poisson bracket

4. Matrix Symmetry  The Jacobian matrix describes a transformation.  Use this for the difference of Lagrangians Require symmetry

5. Jacobian Determinant  The symmetry of the matrix is equivalent to the symplectic requirement M is symplecticM is symplectic CTs are symplecticCTs are symplectic  Take the determinant of both sides The transformation is continuous with the identityThe transformation is continuous with the identity The Jacobian determinant of any CT is unity.The Jacobian determinant of any CT is unity. since

6. Integral Invariant  Integrate phase space  Element in f dimensions dV f The integral is invariant  Equivalent to constancy of phase space density. Density is  q p

7. Liouville’s Theorem  The Jacobian determinant of any CT is unity.  The distribution function is constant along any trajectory in phase space. Poisson bracket:Poisson bracket:  Given  : R 2n  R 1  R 1  R 2n  R 1 ;  (  (z,t), t) A differential flow generated byA differential flow generated by Then for fixed t, f(z)   (z,t) is symplecticThen for fixed t, f(z)   (z,t) is symplectic

8. Lagrange Bracket  Poisson bracket Invariant under CTInvariant under CT  Lagrange bracket Reciprocal matrix of Poisson bracket Also invariant under CT next