Classical configuration, phase and reference spaces

Assume the Axiom of Countable Choice, the exact choice strength required by Tautological one-form on a cotangent bundle to construct the smooth cotangent manifold. Let $d,N\ge 1$, let $Q$ be a smooth $d$-manifold, and choose a countable smooth-chart subcover of its atlas. Such a subcover is obtained from a countable basis of the second-countable topology by choosing, for each basis element contained in a chart domain, one such chart domain; this countable selection is covered by the stated Axiom of Countable Choice. Give $Q_N:=Q^N$ the product topology and the product smooth structure generated by the charts $(U_1\times \cdots \times U_N,(q_1,\dots ,q_N)\mapsto ({\varphi }_1(q_1),\dots ,{\varphi }_N(q_N))),$ where each $(U_i,{\varphi }_i)$ belongs to the chosen countable subcover of charts of $Q$. These images are open in ${\mathbb{R}}^{dN}$, and their transition maps are componentwise smooth, so they form a smooth atlas of dimension $n=dN$; the finite product of the countable chart family is countable. Its generated maximal atlas is the product smooth structure in the convention of Smooth manifolds and their smooth charts. The induced cotangent charts are indexed by these base charts and form a countable atlas of ${\mathrm{\Gamma }}_N$. Its configuration space is $Q_N$. Its phase space is the cotangent bundle ${\mathrm{\Gamma }}_N:=T^{\ast }{Q}_N,$ whose points are pairs $(q,p)$ with $p\in T_q^{\ast }{Q}_N$. Let ${\pi }_N:T^{\ast }{Q}_N\to Q_N$ be the cotangent-bundle projection. The tautological one-form and canonical symplectic form are ${\lambda }_{(q,p)}(\xi ):=p(d{\pi }_N\xi ),{\omega }_N:=-d\lambda \mathrm{.}$ Thus in an induced cotangent chart $(q^1,\dots ,q^n,p_1,\dots ,p_n)$, $\lambda ={\sum }_{j=1}^n{p}_{j}d{q}^j,{\omega }_N={\sum }_{j=1}^{n}d{q}^j\wedge d{p}_j\mathrm{.}$ The cotangent supplier proves ${\omega }_N$ closed and nondegenerate under the same Countable Choice premise. On an open phase chart this is exactly the standard form of A phase domain, its canonical one-form and its standard symplectic two-form; The standard two-form of a phase domain is closed and nondegenerate, with contraction ${\iota }_v\omega ={\sum }_i{a}_{i}d{p}_i-{\sum }_i{b}_{i}d{q}^i$ gives its closedness, contraction and nondegeneracy coordinate calculations.

Let ${\mathcal{F}}_N:=\mathcal{B}({\mathrm{\Gamma }}_N)$ be the Borel sigma-algebra generated by the topology of ${\mathrm{\Gamma }}_N$ (The Borel sigma-algebra of a topological space). The Liouville density, denoted $d{\mathrm{\Lambda }}_N:=\mathrm{\mid }{\omega }_N^{\wedge n}\mathrm{\mid }\mathrm{/}n!,$ is the positive density whose coordinate density in every induced cotangent chart is $d{q}^1\cdots d{q}^{n}d{p}_1\cdots d{p}_n$. Define its Borel measure ${\mathrm{\Lambda }}_N$ explicitly as follows. If ${\mathrm{\Gamma }}_N=\mathrm{\varnothing }$, set ${\mathrm{\Lambda }}_N(\mathrm{\varnothing })=0$. Otherwise choose an at-most-countable induced cotangent atlas $(V_k,{\psi }_k{)}_{k\in J}$ indexed by $J\subseteq \mathbb{N}$, order $J$ by the usual order, and set $E_k:=V_k\setminus {\bigcup }_{j\in J,j<k}{V}_j(k\in J)\mathrm{.}$ For $A\in {\mathcal{F}}_N$, put ${\mathrm{\Lambda }}_N(A):={\sum }_{k\in J}{\lambda }_{2n}({\psi }_k(A\cap E_k)),$ where ${\lambda }_{2n}$ is Lebesgue measure on the chart image. Each $E_k$ is Borel, and each chart map is a homeomorphism, so the terms are defined on Borel sets. The coordinate transitions preserve ${\omega }_N$ because ${\omega }_N=-d\lambda$ is intrinsic; hence they preserve ${\omega }_N^n\mathrm{/}n!$ and have absolute Jacobian one. For two atlases, the intersections of their disjoint chart pieces form a countable common Borel refinement. On each nonempty refinement piece the coordinate transition is a $C^1$ diffeomorphism with absolute Jacobian one, so the change-of-variables theorem applied to its indicator A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions gives the same measure from either atlas. Summing over the refinement proves atlas independence. Countable additivity follows because the chart images of disjoint Borel sets remain disjoint within each $E_k$ and Lebesgue measure is countably additive. For the resulting nonnegative array $a_{k\mathrm{\ell }}$, both ${\sum }_k{\sum }_{\mathrm{\ell }}{a}_{k\mathrm{\ell }}$ and ${\sum }_{\mathrm{\ell }}{\sum }_k{a}_{k\mathrm{\ell }}$ are the supremum of the same finite subsums of $\{a_{k\mathrm{\ell }}\}$, using the nonnegative extended-sum convention of Measures on sigma-algebras. Hence the sums reorder and ${\mathrm{\Lambda }}_N$ is a measure with the displayed coordinate density. Given any finite constant $c_N>0$, define the labelled reference measure $m_N:=c_N{\mathrm{\Lambda }}_N$ on $({\mathrm{\Gamma }}_N,{\mathcal{F}}_N)$. It is sigma-finite: $Q_N$ is countable-atlased as just constructed, so it is covered by countably many cotangent chart images. Each chart image is open in ${\mathbb{R}}^{2n}$ and is covered by the countable family of bounded open boxes with rational endpoints whose closures lie in that chart image: every point of an open set lies in one such box. Each box has finite Euclidean Lebesgue measure, and its chart preimage therefore has finite $m_N$-measure because the Liouville coordinate density is the constant $c_N$. The countably many chart-box preimages cover ${\mathrm{\Gamma }}_N$ and establish sigma-finiteness directly, without a sigma-compactness supplier. The constant $c_N$ is mathematical normalization data; this definition assigns no physical unit to it.

For a partition of the particle labels into species classes of sizes $N_1,\dots ,N_s$, let $G:=S_{N_1}\times \cdots \times S_{N_s}$ act on $Q_N=Q^N$ by permuting factors within each class and on $T^{\ast }{Q}_N$ by the cotangent lift. In canonical coordinates each lift permutes corresponding $q$ and $p$ coordinates, so it preserves ${\omega }_N$, ${\mathrm{\Lambda }}_N$ and $m_N$. Let ${\mathrm{\Pi }}_N:{\mathrm{\Gamma }}_N\to {\mathrm{\Gamma }}_N\mathrm{/}G$ be the orbit map, equip the orbit set with the quotient sigma-algebra ${\mathcal{F}}_{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}}:=\{B\subseteq {\mathrm{\Gamma }}_N\mathrm{/}G:{\mathrm{\Pi }}_N^{-1}(B)\text{ is Borel in }{\mathrm{\Gamma }}_N\},$ and define the quotient reference measure by ${\bar{m}}_N(B):=\frac{1}{{\prod }_{a=1}^s{N}_a!}{m}_N({\mathrm{\Pi }}_N^{-1}(B))\mathrm{.}$ This is a measure because inverse images preserve complements and countable disjoint unions and $m_N$ is countably additive. The stated invariance makes it the normalized quotient pushforward. It is a measurable quotient; no smooth quotient structure at configurations with stabilizers is asserted. For one species the factor is $1\mathrm{/}N!$.

The quotient factor here is mathematical counting data. A physical postulate determines when labels are operationally indistinguishable and how the reference scale is interpreted; neither follows from the cotangent geometry.

Let $(Y,\mathcal{G},\nu )$ be a sigma-finite measure space, $\beta >0$, and $H:\mathbb{R}\times Y\to \mathbb{R}$ a product-measurable function whose map $z\mapsto H(z,y)$ is $C^1$ for $\nu$-almost every $y$. Set $Z:={\int }_{\mathbb{R}\times Y}{e}^{-\beta H(z,y)}d(z,y)\in (0,\mathrm{\infty }),$ where $d(z,y)$ is Lebesgue measure on $\mathbb{R}$ times $\nu$. Assume ${\int }_{\mathbb{R}\times Y}\mathrm{\mid }z{\mathrm{\partial }}_{z}H(z,y)\mathrm{\mid }{e}^{-\beta H(z,y)}d(z,y)<\mathrm{\infty }\mathrm{.}$ For $R>0$ let $B_R:={\int }_Y(R{e}^{-\beta H(R,y)}+R{e}^{-\beta H(-R,y)})d\nu (y),$ and assume $B_R\to 0$ as $R\to \mathrm{\infty }$. Define $⟨F{⟩}_{\beta }:=Z^{-1}\int F(z,y)e^{-\beta H(z,y)}d(z,y)$ whenever the integral is absolutely convergent. Then $⟨z{\mathrm{\partial }}_{z}H{⟩}_{\beta }={\beta }^{-1}\mathrm{.}$ For finite endpoints $a<b$, assume the endpoint integral below is finite and put $Z_{[a,b]}:={\int }_{[a,b]\times Y}{e}^{-\beta H}d(z,y)$. If $0<Z_{[a,b]}<\mathrm{\infty }$, define $⟨F{⟩}_{[a,b]}:=Z_{[a,b]}^{-1}{\int }_{[a,b]\times Y}F{e}^{-\beta H}d(z,y)$. Then the exact boundary identity is $⟨z{\mathrm{\partial }}_{z}H{⟩}_{[a,b]}={\beta }^{-1}(1-Z_{[a,b]}^{-1}{\int }_Y(b{e}^{-\beta H(b,y)}-a{e}^{-\beta H(a,y)})d\nu (y)),$

Let $({\mathrm{\Gamma }}_N,{\mathcal{F}}_N,m_N)$ be a sigma-finite phase-reference measure space from Classical configuration, cotangent phase and reference measures; the cotangent construction uses the stated Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). A statistical state is a probability measure $P$ on $({\mathrm{\Gamma }}_N,{\mathcal{F}}_N)$, meaning a countably additive measure with $P({\mathrm{\Gamma }}_N)=1$ (Measures on sigma-algebras). A density of $P$ relative to $m_N$ is a measurable $\rho :{\mathrm{\Gamma }}_N\to [0,\mathrm{\infty }]$ such that $P(A)={\int }_A\rho d{m}_N(A\in {\mathcal{F}}_N),{\int }_{{\mathrm{\Gamma }}_N}\rho d{m}_N=1.$ Equivalently, $P=\rho m_N$ in the definition of a measure with density (The measure with density $f$ relative to $\mu$); integration against that density has the meaning supplied by Integrating against a density agrees with integrating the product. A state need not have a density.

An observable is a real-valued measurable function $A:{\mathrm{\Gamma }}_N\to \mathbb{R}$ (Extended-real-valued measurable functions). Its expectation $⟨A{⟩}_P:=\int AdP$ is used only when $A$ is integrable, that is, $\int \mathrm{\mid }A\mathrm{\mid }dP<\mathrm{\infty }$ (Integrable real and complex functions, and their integrals). Its variance is ${\mathrm{Var}}_P(A):=\int (A-⟨A{⟩}_P{)}^{2}dP$ when $A^2$ is integrable. For square-integrable $A,B$, their covariance is ${\mathrm{Cov}}_P(A,B):=\int (A-⟨A{⟩}_P)(B-⟨B{⟩}_P)dP\mathrm{.}$ These expressions are finite under the stated square-integrability hypotheses. Indeed, $P({\mathrm{\Gamma }}_N)=1$ makes $1\in L^2(P)$ with norm $1$. By Cauchy-Schwarz inequality for $L^2$, $\int \mathrm{\mid }A\mathrm{\mid }dP\le (\int A^{2}dP{)}^{1\mathrm{/}2}$, so the mean of a square-integrable observable exists. If $a=⟨A{⟩}_P$, then $\mathrm{\mid }A-a{\mathrm{\mid }}^2\le 2A^2+2a^2$ makes $A-a$ square-integrable; the same argument applies to $B-b$. A second application of that inequality gives $\int \mathrm{\mid }(A-a)(B-b)\mathrm{\mid }dP\le \mathrm{\parallel }A-a{\mathrm{\parallel }}_2\mathrm{\parallel }B-b{\mathrm{\parallel }}_2<\mathrm{\infty }$. Thus both the variance and covariance integrals above are well-defined.

If $P=\rho m_N$ and the integrals are defined, these expectations equal $\int A\rho d{m}_N$ by Integrating against a density agrees with integrating the product.

A macrostate map is a measurable map $M:({\mathrm{\Gamma }}_N,{\mathcal{F}}_N)\to (Y,\mathcal{G})$. Its induced state on $Y$ is the pushforward $M_{\ast }P(B):=P(M^{-1}(B))$ for $B\in \mathcal{G}$. A macrocell associated with a specified measurable cell $B\in \mathcal{G}$ is the measurable phase event $C_B:=M^{-1}(B)$; its reference size is $m_N(C_B)\in [0,\mathrm{\infty }]$. A singleton fibre $M^{-1}(\{y\})$ is used as a macrocell only when $\{y\}\in \mathcal{G}$. For a specified family $\mathcal{C}\subseteq \mathcal{G}$ of cells, an entropy assignment means additional data $S:\mathcal{C}\to [-\mathrm{\infty },\mathrm{\infty }]$; no particular functional of $m_N(C_B)$ is stipulated here. Neither a cell's reference size nor a separately chosen entropy assignment is defined to be the expectation or variance of a microscopic observable.

Assume the Axiom of Choice. Let ${\mathrm{\Gamma }}_N=T^{\ast }{Q}_N$ be a cotangent phase space from Classical configuration, cotangent phase and reference measures, with $\dim Q_N=n$, and let $J\subseteq \mathbb{R}$ be open. Let $H:{\mathrm{\Gamma }}_N\to \mathbb{R}$ have a $C^2$ coordinate representative $H\circ {\psi }^{-1}$ in every canonical cotangent chart $\psi$; write $dH$ for the resulting differential. Assume $dH\mathrm{\ne }0$ on $H^{-1}(J)$ and suppose $H^{-1}(K)$ is compact for every compact $K\subset J$. Use the real canonical cotangent coordinates $z=(q^1,\dots ,q^n,p_1,\dots ,p_n)$ of Classical configuration, cotangent phase and reference measures; this mathematical theorem assigns no physical units to these coordinates.

For a compact regular energy band, take a finite smooth partition of unity $\{{\chi }_a\}$ subordinate to canonical cotangent charts. In a chart where ${\mathrm{\partial }}_{i}H\mathrm{\ne }0$, write the local energy graph as $z_i=h_a(y,E)$ and define the weighted shell measure on nonnegative Borel $f$ by ${\mu }_E(f):=c_N{\sum }_a\int f(y,h_a(y,E)){\chi }_a(y,h_a(y,E))\mathrm{\mid }{\mathrm{\partial }}_{i}H(y,h_a(y,E)){\mathrm{\mid }}^{-1}dy,$ where $c_N>0$ is the constant density of the reference measure in canonical coordinates. Then this definition is independent of the chosen finite chart partition and equals the chart expression ${\mu }_E(f)=c_N{\int }_{{\mathrm{\Sigma }}_E}f(z)\mathrm{\mid }{\mathrm{\nabla }}_{z}H(z){\mathrm{\mid }}^{-1}d{A}_z,$ with ${\mathrm{\Sigma }}_E:=H^{-1}(\{E\})$ and Euclidean area in each real canonical chart. For every nonnegative Borel $f$ compactly supported in $H^{-1}(J)$, ${\int }_{{\mathrm{\Gamma }}_N}fd{m}_N={\int }_J{\mu }_E(f)dE\mathrm{.}$ If ${\mathrm{\Sigma }}_E$ is nonempty, then $0<{\mu }_E(1)<\mathrm{\infty }$ and $P_E:={\mu }_E\mathrm{/}{\mu }_E(1)$ is a probability measure. If a Hamiltonian flow map is a $C^1$ diffeomorphism on a full compact energy band and preserves that band, then $P_E$ is invariant under that map for every such regular $E$. Critical levels and noncompact inverse bands are outside this statement. For its invariance clause, fix a compatible metric on a compact band whose interior contains the shell; such a metric exists by Topological manifolds are metrizable and paracompact.

The same shell measure can be computed after a fixed numerical coordinate normalization. Choose positive real scales ${\mathrm{\ell }}_j,p_{0j}$ with a common product ${\mathrm{\ell }}_j{p}_{0j}=A>0$, and put $q^j={\mathrm{\ell }}_j{Q}^j,p_j=p_{0j}{P}_j,F(Q,P)=(q,p),\tilde{H}=H\circ F\mathrm{.}$ The reference density becomes ${\widehat{c}}_{N}dQdP$ with ${\widehat{c}}_N=c_N{A}^n$. The graph and weighted-area formulas above then use $\tilde{H}$, its gradient and Euclidean area in $(Q,P)$, and ${\widehat{c}}_N$, with the same energy variable $E$. They describe the pullback of ${\mu }_E$, not a different shell law. The pulled-back forms are $F^{\ast }\lambda =A{\sum }_j{P}_{j}d{Q}^j,F^{\ast }{\omega }_N=A{\sum }_{j}d{Q}^j\wedge d{P}_j\mathrm{.}$ For a Hamiltonian trajectory parameterized by its original $t$, the equations in these coordinates are $\frac{d{Q}^j}{dt}=A^{-1}{\mathrm{\partial }}_{P_j}\tilde{H},\frac{d{P}_j}{dt}=-A^{-1}{\mathrm{\partial }}_{Q^j}\tilde{H}\mathrm{.}$ This chart normalization does not rescale $H$, $E$ or $t$.

For a declared classical model with $N_a$ operationally indistinguishable particles of each species $a$, label those particles by $\{(a,i):1\le i\le N_a\}$ on the auxiliary labelled cover, and let $S_{N_a}$ denote the permutation group of that species' finite label set. The product group $G={\prod }_a{S}_{N_a}$ acts on the labelled phase space ${\mathrm{\Gamma }}_N=T^{\ast }{Q}_N$ by the cotangent lifts of within-species permutations. Let ${\mathrm{\Pi }}_N:{\mathrm{\Gamma }}_N\to {\mathrm{\Gamma }}_N\mathrm{/}G$ be its measurable orbit map from Classical configuration, cotangent phase and reference measures. Choose $a_0>0$ as a fixed primitive reference parameter with action units J s; Dimensions and SI unit conventions of the non-relativistic classical mechanics framework supplies its dimension, not its value or the choice of this parameter. Here $d$ is the number of spatial dimensions and $N={\sum }_a{N}_a$. No numerical value or identification of $a_0$ with Planck's constant is imposed.

Use the weighted labelled lift $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=\frac{1}{{\prod }_a{N}_a!a_0^{dN}}d{\mathrm{\Lambda }}_N=\frac{1}{{\prod }_a{N}_a!a_0^{dN}}dqdp$ on ${\mathrm{\Gamma }}_N$, where $d{\mathrm{\Lambda }}_N$ is the Liouville density and the last expression is its coordinate density in canonical charts, with coordinates read in SI numerical units. The coordinate volume $dqdp$ has action dimension $(\mathrm{J}\mathrm{s}{)}^{dN}$, so the reference is dimensionless. The physical quotient reference is ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(B):=m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\Pi }}_N^{-1}(B))(B\in {\mathcal{F}}_{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}})\mathrm{.}$ Thus $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is on the labelled cover ${\mathrm{\Gamma }}_N$, and ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=({\mathrm{\Pi }}_N{)}_{\ast }{m}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is on ${\mathrm{\Gamma }}_N\mathrm{/}G$; with $c_N=a_0^{-dN}$, the latter is exactly the normalized quotient reference of Classical configuration, cotangent phase and reference measures.

Physical observables for indistinguishable particles are measurable $G$-invariant functions $A:{\mathrm{\Gamma }}_N\to \mathbb{R}$, equivalently measurable functions $\bar{A}$ on the quotient with $A=\bar{A}\circ {\mathrm{\Pi }}_N$. For every such observable with an integrable expectation, its integral against $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ equals the quotient integral against ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. A labelled density represents a physical state on the quotient only if it is invariant under $G$ almost everywhere with respect to $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$; an invariant representative may then be used. In particular, an indistinguishable-particle Hamiltonian must be invariant under these permutations (or agree almost everywhere with its permuted versions when only its Gibbs density is at issue). A label-specific coordinate statistic on the lift is not itself a quotient observable. Under a permutation-invariant state its marginal law can instead be interpreted as the law of a uniformly selected particle within the same species, or of a uniformly selected coordinate of the same type within that species. Symmetric sums and averages within each species are quotient observables. Calculations of labelled momentum components in Canonical momentum distribution and equipartition use this symmetric labelled lift, and their permutation-invariant kinetic sums and averages are physical observables.

If labels are operationally distinguishable, omit the corresponding permutation factor and use the labelled phase space directly. The convention is part of the preparation and reference model; it is not deduced from Hamiltonian mechanics. Whether a quantity is represented by a gauge-dependent canonical momentum or by a gauge-invariant mechanical momentum is fixed by the stated model and gauge convention. No quantum exchange law, actual preparation mechanism, or empirical indistinguishability test is asserted.

Let $(\mathrm{\Omega },\mathcal{F},m)$ be a sigma-finite measure space, let $w\ge 0$ be measurable with $m(\{w>0\})>0$, let $r\ge 1$ and let $X=(X_1,\dots ,X_r):\mathrm{\Omega }\to {\mathbb{R}}^r$ be measurable and finite, and let $\mathrm{\Theta }\subseteq {\mathbb{R}}^r$ be open. For each compact $K\subset \mathrm{\Theta }$, assume an integrable $G_K$ satisfies $w(x)e^{\theta \cdot X(x)}(1+{\sum }_a\mathrm{\mid }{X}_a(x){\mathrm{\mid }}^2)\le G_K(x)(\theta \in K)\mathrm{.}$ Then $Z(\theta )=\int w{e}^{\theta \cdot X}dm$ is finite, strictly positive and $C^2$, and the probability density ${\rho }_{\theta }=Z(\theta {)}^{-1}w{e}^{\theta \cdot X}$ defines $d{P}_{\theta }={\rho }_{\theta }dm$ as in The measure with density $f$ relative to $\mu$. For any integrable real $A$, write $⟨A{⟩}_{\theta }:=\int Ad{P}_{\theta }$; for square-integrable real $A,B$, define ${\mathrm{Cov}}_{\theta }(A,B):=\int (A-⟨A{⟩}_{\theta })(B-⟨B{⟩}_{\theta })d{P}_{\theta }$. These extend the phase-space conventions of Statistical states, densities, observables and expectations to the present measure space. Then ${\mathrm{\partial }}_a\log Z=⟨X_a{⟩}_{\theta },{\mathrm{\partial }}_a{\mathrm{\partial }}_b\log Z={\mathrm{Cov}}_{\theta }(X_a,X_b)\mathrm{.}$ In particular, the Hessian of $\log Z$ is positive semidefinite.

Let integers $d,N\ge 1$ satisfy $n=dN\ge 3$. Choose a basis $b_1,\dots ,b_d$ of ${\mathbb{R}}^d$ and the full-rank lattice $\mathrm{\Lambda }:={\sum }_{i=1}^d\mathbb{Z}{b}_i$. The one-particle flat torus is $Q:={\mathbb{R}}^d\mathrm{/}\mathrm{\Lambda }$ with its quotient Euclidean metric and translation-invariant volume $V:=\mathrm{\mid }\det (b_1,\dots ,b_d)\mathrm{\mid }$. Write ${\mathrm{\Lambda }}^N:=\mathrm{\Lambda }\times \cdots \times \mathrm{\Lambda }$ ($N$ factors) and set $Q_N:=Q^N\cong {\mathbb{R}}^n\mathrm{/}{\mathrm{\Lambda }}^N$, whose position volume is $V^N$. Use the global parallel cotangent trivialization ${\mathrm{\Gamma }}_N\cong Q_N\times {\mathbb{R}}^n$ and the product of quotient Lebesgue volume $dq$ with fibre Lebesgue measure $dp$ in the canonical reference $m_N=c_{N}dqdp$ of Classical configuration, cotangent phase and reference measures. Let $m_j>0$ for $1\le j\le n$, and let $U:Q_N\to \mathbb{R}$ be bounded with a smooth lift $\tilde{U}:{\mathbb{R}}^n\to \mathbb{R}$ satisfying $\tilde{U}(x+\lambda )=\tilde{U}(x)$ for all $x\in {\mathbb{R}}^n$ and $\lambda \in {\mathrm{\Lambda }}^N$. Set $H(q,p)={\sum }_{j=1}^n\frac{p_j^2}{2m_j}+U(q),\alpha =n\mathrm{/}2,C_m={\prod }_{j=1}^n\sqrt{2m_j}\mathrm{.}$ Let $B_n$ be the Euclidean volume of the unit ball in ${\mathbb{R}}^n$. Fix an energy $E$ that is a regular value of $H$ and whose shell ${\mathrm{\Sigma }}_E=H^{-1}(\{E\})$ is nonempty. For every continuous $g:Q_N\to \mathbb{R}$, the coarea shell measure from Coarea and invariant probability on compact regular phase shells satisfies ${\mu }_E(g\circ {\pi }_N)=c_N{C}_m\alpha B_n{\int }_{Q_N}g(q)(E-U(q){)}_{+}^{\alpha -1}dq,$ where $x_{+}=\max (x,0)$. Thus the position marginal of $P_E$ has density proportional to $(E-U{)}_{+}^{\alpha -1}$. In the scaled momentum coordinates $r_j=p_j\mathrm{/}\sqrt{2m_j}$, the coarea shell measure over each $q$ with $E>U(q)$ has fiber factor $d{A}_r\mathrm{/}(2\mathrm{\mid }r\mathrm{\mid })$ on the sphere $\mathrm{\mid }r\mathrm{\mid }=\sqrt{E-U(q)}$. Define ${\sigma }_{E,q}$ as Euclidean surface area on that sphere divided by its total area. It satisfies $\int \frac{p_j^2}{2m_j}d{\sigma }_{E,q}=\frac{E-U(q)}{n},\int {\sum }_j\frac{p_j^2}{2m_j}d{P}_E=\int (E-U(q))d{P}_E(q)\mathrm{.}$ The result concerns the full kinetic-plus-potential shell; it does not replace it by a shell of potential energy alone.