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Canonical Poisson brackets and local Hamiltonian flow

Physical scope

Mathematical cotangent phase space; physical dimensions are interpreted only when a model assigns units to coordinates and observables.

Statement

Let $Q$ be a finite-dimensional smooth manifold as in Cotangent phase data, observables and action dimensions, and write its cotangent phase space in induced coordinates $(q^{i}, p_{i})$ . With $ω = d q^{i} \land d p_{i}$ , define $X_{H}$ by $ι_{X_{H}} ω = d H$ and define the Poisson bracket by ${F, G} : = X_{G} (F)$ . Then

$X_{H} = \frac{\partial H}{\partial p_{i}} \frac{\partial}{\partial q^{i}} - \frac{\partial H}{\partial q^{i}} \frac{\partial}{\partial p_{i}}, {F, G} = F_{q^{i}} G_{p_{i}} - F_{p_{i}} G_{q^{i}} .$

The bracket is coordinate independent and makes smooth functions a Poisson algebra. If $H_{t}$ is smooth on an open time-dependent domain $D \subset R \times T^{*} Q$ , each solution of $\dot{z} = X_{H_{t}} (z)$ has local uniqueness and smooth dependence on initial data; on its actual interval,

$\frac{d}{d t} F (t, z (t)) = \partial_{t} F + {F, H_{t}} .$

For parameter dependence, let $Λ \subseteq R^{ℓ}$ be open, with $ℓ \geq 0$ , and let $H : D_{Λ} \to R$ be jointly smooth on an open $D_{Λ} \subseteq R \times T^{*} Q \times Λ$ . With $μ \in Λ$ fixed along each solution, the field is $X_{H (t, \cdot, μ)}$ . Near any base data $(t_{0}, z_{0}, μ_{0}) \in D_{Λ}$ , its solutions through nearby $(z, μ)$ exist on a common compact interval about $t_{0}$ and depend smoothly on $(t, z, μ)$ . The unparameterized case is $ℓ = 0$ . No completeness or global flow is asserted. If $H$ is smooth, each local time- $t$ flow map preserves ω.

Facts & Assumptions

Given: $Q$ , its canonical symplectic cotangent phase space, a smooth $H$ , and where used a smooth time-dependent Hamiltonian $H_{t}$ , or the jointly smooth family $H (t, z, μ)$ on $D_{Λ}$ with fixed $μ \in Λ$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Assume countable choice ACω, meaning every sequence of nonempty sets has a choice function; no full Axiom of Choice is assumed.

[F1]

Cotangent phase data, observables and action dimensions: in induced coordinates $ω = \sum_{i} d q^{i} \land d p_{i}$ ; observables are smooth real functions.

[F2]

Smooth functions form a Poisson algebra: the symplectic bracket is coordinate independent, skew, bilinear, satisfies Leibniz and Jacobi.

[F3]

Smooth dependence of ODE solutions on parameters: a jointly $C^{\infty}$ field on an open time-state-parameter domain has locally unique solutions on a common compact interval for nearby initial states and parameters, with smooth solution dependence. Its proof also gives joint smoothness in time and the variational equation.

Proof

1.1givenA1F1algebra

Write $X_{H} = A^{i} \partial_{q^{i}} + B_{i} \partial_{p_{i}}$ . Contraction gives $ι_{X_{H}} ω = A^{i} d p_{i} - B_{i} d q^{i}$ . Equating with $d H = H_{q^{i}} d q^{i} + H_{p_{i}} d p_{i}$ yields $A^{i} = H_{p_{i}}$ , $B_{i} = - H_{q^{i}}$ .

2.1givenF1F2step 1.1algebra

Substitution into $X_{G} (F)$ gives the displayed coordinate bracket. Its intrinsic definition from ω makes the formula invariant under cotangent coordinate changes; [F2] supplies the Poisson identities.

2.2givenF2step 1.1algebra

Along a solution, the chain rule gives $d F / d t = F_{t} + d F (X_{H_{t}})$ ; the definition of the bracket and skewness give $d F (X_{H_{t}}) = X_{H_{t}} (F) = {F, H_{t}}$ .

2.3givenF3step 1.1

In a cotangent chart the jointly smooth family has field $(H_{p_{i}}, - H_{q^{i}}) (t, z, μ)$ on an open time-state-parameter domain. Thus [F3] gives unique solutions on a common compact local interval and smooth dependence on nearby initial data and $μ$ ; its time-regularity conclusion gives joint smoothness in $(t, z, μ)$ . With no extra parameters take $ℓ = 0$ . Chart transitions preserve these uniquely determined solutions, and finitely many local solution maps compose to give the same dependence near any compact segment contained in the actual solution domain. This does not assert extension past that domain.

3.1givenF1F3step 1.1algebra∎

For smooth autonomous $H$ , let $z = (q, p)$ and let $J = (\begin{array}{cc} 0 & I \\ - I & 0 \end{array})$ , so that $ω (u, v) = u^{T} J v$ . By step 1.1, $X_{H} = J \nabla H$ and $D X_{H} = J K$ , where $K = D^{2} H$ is symmetric. Hence $(D X_{H})^{T} J + J D X_{H} = K^{T} J^{T} J + J^{2} K = K - K = 0$ . For a local flow map $ϕ_{t}$ , its derivative $A (t) = D ϕ_{t} (z)$ exists by [F3] and satisfies $\dot{A} = D X_{H} (ϕ_{t} (z)) A$ . Therefore $\frac{d}{d t} (A^{T} J A) = A^{T} ((D X_{H})^{T} J + J D X_{H}) A = 0$ . Since $A (0) = I$ , $A (t)^{T} J A (t) = J$ , which is exactly $ϕ_{t}^{*} ω = ω$ on the actual local flow domain.

Canonical Poisson brackets and local Hamiltonian flow

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources