Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 339bracket of f and g is the function{f, g} ω = −ω(X f , X g ) = −L Xf g = L Xg f.Also, for f 0 ∈ C ∞ (M, R), the map g ↦−→ {f 0 , g} ω is a derivation.connection between the Lie bracket and the Poisson bracket isThe[X f , X g ] = −X {f,g}ω ⇐⇒ dω = 0.The real vector space C ∞ (M, R) together with the Poisson bracket onit forms an infinite–dimensional Lie algebra called the algebra of classicalobservables.In canonical coordinates {q 1 , ..., q n , p 1 , ..., p n } on (M, ω) the Poissonbracket of two functions f, g ∈ C ∞ (M, R) is given byFrom this definition follows:{f, g} ω = ∂f∂q i ∂g∂p i− ∂f∂p i∂g∂q i .{q i , q j } ω = 0, {p i , p j } ω = 0, {q i , p j } ω = δ i j.Let (M, ω) be a symplectic manifold and f : M → M a diffeomorphism.Then f is symplectic iff it preserves the Poisson bracket.Let (M, ω, H) be a Hamiltonian mechanical system and φ t the flow ofX H . Then for each function f ∈ C ∞ (M, R) we have the equations of motionin the Poisson bracket notation:ddt (f ◦ φ t) = {f ◦ φ t , H} ω= {f, H} ω◦ φ t .Also, f is called a constant of motion, or a first integral, if it satisfies thefollowing condition{f, H} ω= 0.If f and g are constants of motion then their Poisson bracket is also aconstant of motion.A Hamiltonian mechanical system (M, ω, H) is said to be integrableif there exists n = 1 2 dim(M) linearly–independent functions K 1 =H, K 2 , ..., K n such that for each i, j = 1, 2, ..., n:{K i , H} ω= 0, {K i , K j } ω= 0.