Regular Hamiltonian shell preparation data

Assume the Axiom of Choice (The Axiom of Choice). Fix an integer $n\ge 1$ and an open phase-coordinate domain $O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\subseteq {\mathbb{R}}^{2n}$ with physical canonical coordinates $q_i$ in metres and $p_i$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$. The canonical symplectic form on this chart is ${\omega }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\sum }_{i=1}^{n}d{q}_i\wedge d{p}_i$. Let ${\widehat{q}}_i=q_i\mathrm{/}(1\mathrm{m})$ and ${\widehat{p}}_i=p_i\mathrm{/}(1\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$, and let $O\subseteq {\mathbb{R}}^{2n}$ be the open image of the chart in these dimensionless coordinates. The corresponding dimensionless canonical form is $\widehat{\omega }={\sum }_{i=1}^{n}d{\widehat{q}}_i\wedge d{\widehat{p}}_i$, so ${\omega }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(1\mathrm{J}\mathrm{s})\widehat{\omega }$.

Let $\widehat{H}:O\to \mathbb{R}$ be a numerical energy function with continuous partial derivatives through order two, and let $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to \mathbb{R}\mathrm{J}$ be $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(q,p)=(1\mathrm{J})\widehat{H}(\widehat{q},\widehat{p})$. Where defined, its physical Hamiltonian vector field $X_{H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ is determined by ${\iota }_{X_{H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}}{\omega }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=d{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. For each elapsed time $t$ in its domain, the physical flow map is a function ${\varphi }_t:D_t\to O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, where $D_t\subseteq O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is the set of initial data whose trajectories exist through time $t$, and it satisfies $d{\varphi }_t\mathrm{/}dt=X_{H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\circ {\varphi }_t$. Equivalently, in the dimensionless chart, the flow map ${\widehat{\varphi }}_{\widehat{t}}:{\widehat{D}}_{\widehat{t}}\to O$ is generated by $X_{\widehat{H}}$ satisfying ${\iota }_{X_{\widehat{H}}}\widehat{\omega }=d\widehat{H}$, at dimensionless time $\widehat{t}=t\mathrm{/}(1\mathrm{s})$; here ${\widehat{D}}_{\widehat{t}}$ is the set of initial coordinates whose trajectories exist through $\widehat{t}$. These definitions specify the flow maps when they exist but impose no global-existence or completeness hypothesis on the shell preparation.

A specified open interval $J_E\subseteq \mathbb{R}$ is a regular compact-shell band when ${\widehat{H}}^{-1}(K)$ is compact in $O$ for every compact $K\subset J_E$ and $\mathrm{\nabla }\widehat{H}$ is nonzero at every point of ${\widehat{H}}^{-1}(J_E)$.

For $\widehat{E}\in J_E$, the energy shell is ${\mathrm{\Sigma }}_{\widehat{E}}:=\{x\in O:\widehat{H}(x)=\widehat{E}\}$. The graph-chart definition of the weighted shell integral is Regular energy shell and weighted area measure: in a local graph chart $x_i=h(z,\widehat{E})$, its coarea measure is $d{\mu }_{\widehat{E}}=d{A}_{\widehat{E}}\mathrm{/}\mathrm{\mid }\mathrm{\nabla }\widehat{H}\mathrm{\mid }$, with graph-area factor and gradient in the dimensionless Euclidean coordinates. Set $D(\widehat{E}):={\int }_{{\mathrm{\Sigma }}_{\widehat{E}}}d{\mu }_{\widehat{E}}$ and admit the preparation only when ${\mathrm{\Sigma }}_{\widehat{E}}\mathrm{\ne }\mathrm{\varnothing }$ and $0<D(\widehat{E})<\mathrm{\infty }$. Its normalized shell law is ${\nu }_{\widehat{E}}(A):=D(\widehat{E}{)}^{-1}{\mu }_{\widehat{E}}(A\cap {\mathrm{\Sigma }}_{\widehat{E}})$. The current draft theorem Coarea and invariant measure on compact regular energy shells proves the coarea identity, chart independence and positive finite shell mass under its stated $C^2$, compact-regular-band hypotheses; its statement and proof are present, while its acceptance remains unresolved. Multiplying the unnormalized shell measure by a fixed positive phase-volume normalization does not change ${\nu }_{\widehat{E}}$.