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⇔ vG(ψt(x))F = 0 ⇔ {G, F }(ψt(x)) = 0 ⇔ {F, G}(ψt(x)) = 0 ⇔ {F, G}(φt(x)) = 0 ⇔ vF (φt(x))F = 0 ⇔ dG(vF (φt(x))) = 0 ⇔ dG( (^) dt dφt(x)) = 0 ⇔ (^) dt dG(φt(x)) = 0 ⇔ G(φt(x)) = G(x) ⇔ F generates symmetries of G
Moral: the antisymmetry of the Poisson bracket is crucial!
Theorem 2 F generates symmetries of F.
(If F is called the “Hamiltonian” this says: energy is conserved!)
Proof:
F generates symmetries of F ⇔ F (φt(x)) = F (x) ⇔ (^) dt dF (φt(x)) = 0 ⇔ dF
( (^) d dt φt(x)
⇔ dF (vF (φt(x))) = 0 ⇔ vF (F φt(x)) = 0 ⇔ {F, F }(φt(x)) = 0
but {F, F } = −{F, F } so {F, F } = 0. Again, the antisymmetry of the Poisson bracket is crucial!
Given F such that vF is integrable, let A = {G ∈ C∞(X)|F generates symmetries of G} = {G ∈ C∞(X)|G(φt(x)) = G(x), ∀t, x} = {G ∈ C∞(X)|{F, G} = 0}
If F is called the “Hamiltonian”, elements of A are called bf conserved quantities.
Theorem 3 A is a Poisson subalgebra of C∞(X), i.e. it is closed under:
- linear combinations
- multiplication
- Poisson bracket
2
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Proof: Suppose G, H ∈ A.
- αG + βH ∈ A, (α, β ∈ R), since: {F, αG + βH} = α{F, G} + β{F, H} = 0 since {·, ·} is bilinear.
- GH ∈ A, since: {F, GH} = {F, G}H + G{F, H} = 0 since {·, ·} satisfies the Leibniz law.
- {G, H} ∈ A, since: {F, {G, H}} = {{F, G}, H} + {G, {F, H}} = 0 since {·, ·} satisfies the Jacobi identity. What we are doing is laying the groundwork for an axiomatic approach to classical mechanics. The key “axioms” would be:
- observables form a commutative algebra
- sufficiently nice observables generate “flows”
- any observable generates a flow that leaves itself constant (generates symmetriew of itself). (I.e., energy is always conserved!) From axioms like this, we would like to derive the existence of a Poisson algebra of observables. 3 would give the antisymmetry.
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