9. Canonical transformations 9.5 Poisson brackets and other canonical invariants Z Concepts z Poisson bracket z fundamental Poisson brackets z canonical invariants z Lagrange bracket z integral invariants of Poincaré z correspondence principle (quantum mechanics) z Jacobi identity, Lie algebra, Leibniz property z Lagrange brackets, integral invariants of Poincaré Z Results z Poisson brackets are canonical invariants z the functions on phase space form a Lie algebra with respect to the Poisson bracket z phase space volumes are canonical invariants JF 12 10 18 – p. 11/?? 9. Canonical transformations 9.5 Poisson brackets and other canonical invariants Z Formulas ∂u ∂v ∂u ∂v z (9.67) {u, v}q,p = − X ∂q ∂p ∂p ∂q i i i i i ∂u ∂v z (9.68) {u, v} = J ~η ∂~η ∂~η z (9.69) {qi, pj }q,p = δi,j {pi, pj }q,p = 0 = {qi, qj }q,p z (9.70) {~η, ~η}~η = J JF 12 10 18 – p. 11/?? 9. Canonical transformations 9.6 Equations of motion, infinitesimal canonical transformations, conservation laws Z Concepts z total time derivative Z Results z Poisson’s theorem: u and v conserved =⇒ {u, v} conserved z time evolution as continuous sequence of canonical transformations z existence of canonical transformation to constant canonical variables z constant of motion as generating function of canonical transformation Z Formulas du ∂u z (9.94) = {u, H} + dt ∂t z (9.95) ~η˙ = {~η, H} ∂u z (9.97) {H, u} = (u conserved) ∂t z (9.100) δ~η = ε {~η, G} JF 12 10 18 – p. 12/?? 9. Canonical transformations 9.7 The angular momentum Poisson brackets Z Concepts z system quantity, system vector z Lie algebra so(3), vector representation of so(3) Z Results z any set of canonical variables can contain at most one component of L~ Z Formulas z (9.123) {F,~ L~ · ~n} = ~n × F~ 3 z (9.128) {L ,L } = ǫ L α β X αβγ γ γ=1 2 z (9.129) {L ,Lα} = 0 JF 12 10 18 – p. 13/?? 9. Canonical transformations 9.8 Some other symmetries of mechanical systems Z Concepts z Lie group SU(2), Lie algebra su(2) z Pauli matrix, spinor representation z Lie algebras so(4), so(3, 1), iso(3) z Lie algebras su(n) Z Results z description of rotations in terms of su(2) z extended Lie algebra of symmetries of the Kepler problem z Lie algebra of symmetries of the isotropic harmonic oscillator JF 12 10 18 – p. 14/?? 9. Canonical transformations 9.8 Some other symmetries of mechanical systems Z Concepts z Lie group SU(2), Lie algebra su(2) z Pauli matrix, spinor representation z Lie algebras so(4), so(3, 1), iso(3) z Lie algebras su(n) Z Formulas 3 z (9.134) {D ,L } = ǫ D α β X αβγ γ γ=1 3 ǫ L for E < 0 X αβγ γ γ=1 z = (9.135) {Dα,Dβ} 3 − ǫαβγ Lγ for E > 0 X γ=1 JF 12 10 18 – p. 14/?? 9. Canonical transformations 9.9 Liouvilles theorem Z Concepts z identical systems, ensemble z isolated system z statistically distributed initial conditions z density in phase space z time reversal symmetry Z Results z trajectories in phase space do not cross z the density in phase space is constant (Liouville’s theorem) z {D, H} is zero in statistical equilibrium Z Formulas dD ∂D z (9.149) = {D, H} + dt ∂t ∂D z (9.150) = − {D, H} ∂t JF 12 10 18 – p. 15/??.