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Liouville

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

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Liouville

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  1. Liouville

  2. The dynamic variables can be assigned to a single set. q1, p1, q2, p2, …, qf, pf z1, z2, …, z2f Hamilton’s equations can be written in terms of za A: symplectic 2f x2f matrix A2 = -1 AT = -A Matrix Form

  3. The infinitessimal transformation is a contact transformation. Generator eX Written with the matrix A Used in Poisson bracket Infinitessimal Transform

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

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

  6. Integral Invariant • Integrate phase space W • Element in f dimensions dVf • The integral is invariant • Equivalent to constancy of phase space density. • Density is r p q

  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: • Given F: R2nR1R1R2nR1; F(f(z,t), t) • A differential flow generated by • Then for fixed t, f(z) f(z,t) is symplectic

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

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