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Liouville's theorem for a potential presentation: the phase-volume form $ω^{3 N} / (3 N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains

Physical scope

Phase-volume conservation for the declared potential presentation: the top form $ω^{3 N} / (3 N)!$ is nowhere zero and is preserved by every time slice of the $C^{2}$ branch Hamiltonian evolution; the induced Lebesgue measure is preserved on bounded Borel sets of the actual flow domain, and measure comparison between charts is asserted only for genuine equal-dimensional chart transitions. The phase-volume unit is action to the power $3 N$ . The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics, and in its measure-theoretic part it uses the Axiom of Countable Choice exactly where the cited change-of-variables theorem requires it; no empirical content is asserted.

Statement

$Ω : = \frac{1}{n!} ω^{\land n}$

be the phase-volume form of the presentation. Its dimension is $[Ω] = [ω]^{n} = (μ ℓ^{2} τ^{- 1})^{3 N}$ , namely action to the power $3 N$ ; the Lebesgue measure $λ_{6 N}$ in a canonical phase chart has the same phase-volume unit. By Symplectic manifolds have a canonical orientation and volume form applied to the symplectic phase domain of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: $Φ_{X}^{*} λ = \sum P_{a} d q^{a}$ and $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ , $Ω$ is a nowhere-zero top form on the $6 N$ -dimensional Cartesian phase domain. Pointwise compatibility of pullback with wedge products shows that it is preserved by every equal-dimensional canonical transformation, not by a lower-dimensional symplectic immersion. Let $t_{0} \in I_{0}$ , $z_{0} \in W$ and let $Φ_{t} = Φ (t; \cdot)$ be the time- $t$ solution map of the Hamiltonian system on the neighbourhood $U_{0}$ supplied by The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time, with image $V_{t} : = Φ_{t} (U_{0})$ . Then:

(form preservation) For every $t$ on the existence interval, $Φ_{t}^{*} Ω = Ω$ on $U_{0}$ ; the time slices of the model's Hamiltonian evolution preserve the phase-volume form.
(measure preservation) For every $t$ on the existence interval and every bounded Borel set $A \subseteq U_{0}$ with $\overline{A} \subseteq U_{0}$ , the sets $A$ and $Φ_{t} (A)$ are Lebesgue measurable and

$λ_{6 N} (Φ_{t} (A)) = λ_{6 N} (A),$

where $λ_{6 N}$ is the Lebesgue measure of the chart (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ , The Borel sigma-algebra of a topological space, Integrable real and complex functions, and their integrals). 3. (chart independence) The measure statement is independent of the full-dimensional canonical chart: for two genuine $6 N$ -dimensional phase charts with a $C^{1}$ diffeomorphism transition $Θ$ on an open overlap and a Borel set $B$ of finite measure in its domain, $λ_{6 N} (Θ (B)) = λ_{6 N} (B)$ because this equal-dimensional canonical transformation has $∣ \det D Θ ∣ = 1$ (Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts, A $C^{1}$ map between phase domains preserves the standard two-form iff $D C^{T} J D C = J$ , and a canonical diffeomorphism carries $X_{H \circ C}$ to $X_{H}$ ). 4. (potential branch) On the potential branch ( $d \equiv 0$ ) the time slices are the momentum-phase maps of the model's motions (Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential), so the model's Hamiltonian evolution conserves phase volume on its actual domains; no completeness and no all-time statement is claimed.

Facts & Assumptions

Given: The presentation with its two-form and canonical charts, a $C^{2}$ branch Hamiltonian $H$ , base data $(t_{0}, z_{0})$ , the solution map $Φ : [t_{0} - h, t_{0} + h] \times U_{0} \to W$ of The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time, and the phase-volume form $Ω$ .

[F1]

Flow regularity and canonicity (The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time, Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests, A $C^{1}$ map between phase domains preserves the standard two-form iff $D C^{T} J D C = J$ , and a canonical diffeomorphism carries $X_{H \circ C}$ to $X_{H}$ , The canonical one-form and the two-form of a potential presentation: $λ = p \cdot d x$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts, The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: $Φ_{X}^{*} λ = \sum P_{a} d q^{a}$ and $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ ): for each $t$ the map $Φ_{t}$ is $C^{1}$ on $U_{0}$ , is a canonical map with $Φ_{t}^{*} ω = ω$ , and satisfies $∣ \det D Φ_{t} (z) ∣ = 1$ for every $z \in U_{0}$ ; at every point of $U_{0}$ it is a local diffeomorphism, so its image $V_{t}$ is open, and the inverse assignment on $V_{t}$ is the backward solution map $Ψ_{t}$ , which is $C^{1}$ by $C^{1}$ dependence of solutions on initial data applied with base time $t$ and the common existence interval of Continuous dependence of ODE solutions on initial data and parameters.

[F2]

Change of variables (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions, The Axiom of Countable Choice ( ${A C}_{ω}$ )): assume ${A C}_{ω}$ ; for open $V, W \subseteq R^{m}$ and a $C^{1}$ diffeomorphism $T : V \to W$ , $\int_{W} f d λ_{m} = \int_{V} (f \circ T) ∣ \det D T ∣ d λ_{m}$ for every $f \in L^{1} (W)$ .

[F3]

Borel bookkeeping (A continuous map has Borel preimages of Borel sets, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, The Borel sigma-algebra of a topological space, Integrable real and complex functions, and their integrals, Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ): for Borel $A \subseteq U_{0}$ the image $Φ_{t} (A) = Ψ_{t}^{- 1} (A) \cap V_{t}$ is Borel, because $Ψ_{t}$ is continuous and $V_{t}$ is open; if $A$ is bounded with $\overline{A} \subseteq U_{0}$ , then $\overline{A}$ is compact, $Φ_{t} (\overline{A})$ is compact and hence bounded, and the indicator of $Φ_{t} (A)$ is a bounded measurable function of compact support, so it lies in $L^{1}$ .

[F4]

Top forms and pointwise pullback (Symplectic manifolds have a canonical orientation and volume form, A smooth differential $k$ -form, The wedge product of differential forms, The pullback of a differential form, Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests): $Ω = ω^{\land n} / n!$ is a nowhere-zero top form on the $2 n = 6 N$ dimensional phase domain. For an equal-dimensional $C^{1}$ map, the pointwise pullback is compatible with wedges by its alternating multilinear definition, so $C^{*} (ω^{\land n}) = (C^{*} ω)^{\land n}$ and a canonical map preserves $Ω$ ; a different-dimensional source has a different form degree and its own degree-matched phase-volume form.

Proof

technique · direct. Pull the volume form back along the symplectic time-t map, and convert the measure statement into an integral identity covered by the change-of-variables theorem with unit Jacobian determinant

1.1F1F4algebra

By [F4] and the canonicity of [F1], $Φ_{t}^{*} Ω = Φ_{t}^{*} (ω^{\land n}) / n! = (Φ_{t}^{*} ω)^{\land n} / n! = ω^{\land n} / n! = Ω$ ; this proves clause 1.

1.2F1F3

By [F1] the map $Φ_{t} : U_{0} \to V_{t}$ is a $C^{1}$ diffeomorphism onto the open set $V_{t}$ with inverse $Ψ_{t}$ , and $∣ \det D Φ_{t} ∣ = 1$ ; by [F3], for a bounded Borel $A \subseteq U_{0}$ with $\overline{A} \subseteq U_{0}$ both $A$ and $Φ_{t} (A)$ are Borel and the indicator $f : = 1_{Φ_{t} (A)}$ lies in $L^{1}$ .

1.3F2F3F4algebra

Clause 3: let $Θ : U_{1} \to U_{2}$ be the genuine transition between two full-dimensional canonical charts on an open overlap. Write the two charts as $θ_{1} : E \to U_{1}$ and $θ_{2} : E \to U_{2}$ on the same overlap, with $θ_{j}^{*} ω_{j}^{s t} = σ$ by Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts. Then $Θ = θ_{2} \circ θ_{1}^{- 1}$ , and pointwise pullback and the chain rule give $θ_{1}^{*} (Θ^{*} ω_{2}^{s t}) = θ_{2}^{*} ω_{2}^{s t} = σ = θ_{1}^{*} ω_{1}^{s t}$ . Since $D θ_{1}$ is invertible, its pullback on two-forms is injective; hence $Θ^{*} ω_{2}^{s t} = ω_{1}^{s t}$ . Thus this $C^{1}$ diffeomorphism is canonical by Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests, and its matrix test gives $∣ \det D Θ ∣ = 1$ , independently of the flow-specific fact [F1]. For a Borel set $B \subseteq U_{1}$ of finite measure, $Θ (B)$ is Borel because $Θ$ is a homeomorphism. For each integer $j \geq 1$ , set $E_{j} : = Θ (B) \cap [- j, j]^{6 N}$ . Each $E_{j}$ is bounded Borel, so $1_{E_{j}} \in L^{1} (U_{2})$ . Applying [F2] with $T = Θ$ and $f = 1_{E_{j}}$ gives $λ_{6 N} (E_{j}) = λ_{6 N} (B \cap Θ^{- 1} ([- j, j]^{6 N}))$ , since $\det D Θ$ has absolute value one. As $j \to \infty$ , both sets increase respectively to $Θ (B)$ and $B$ ; continuity from below of Lebesgue measure yields $λ_{6 N} (Θ (B)) = λ_{6 N} (B)$ . This proves chart-measure independence for genuine full-dimensional phase charts; a lower-dimensional immersion instead has its own degree-matched volume form $ω_{X}^{\land r} / r!$ .

2.1step 1.2F2algebra

Applying the change-of-variables identity of [F2] with $T = Φ_{t}$ , $V = U_{0}$ , $W = V_{t}$ and $f = 1_{Φ_{t} (A)}$ : since $f \circ T = 1_{A}$ and $∣ \det D T ∣ = 1$ , the identity $\int_{W} f d λ = \int_{V} (f \circ T) ∣ \det D T ∣ d λ$ reads $λ_{6 N} (Φ_{t} (A)) = \int_{U_{0}} 1_{A} d λ_{6 N} = λ_{6 N} (A)$ , which is clause 2; this is the point at which ${A C}_{ω}$ is used, through [F2].

3.1step 1.1step 2.1step 1.3F1∎

Clause 4 is the identification of the time slices with the momentum-phase maps of the model's motions on the potential branch, which is [F1] combined with Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential clause 1; collecting steps 1.1, 2.1 and 1.3 proves the theorem.

Remarks

Where the choice assumption is used. ${A C}_{ω}$ enters only through the change-of-variables corollary [F2], in the measure-theoretic clauses 2 and 3; the form-level clause 1 and the autonomy-free determinant identity are choice-free.
Volume form versus coordinate measure. On a full-dimensional canonical chart, $Ω$ and the coordinate top form $d x^{1} \land \dots \land d x^{n} \land d p_{1} \land \dots \land d p_{n}$ differ by the constant sign $(- 1)^{n (n - 1) / 2}$ : expand $ω^{\land n} / n!$ and reorder the paired differentials. Thus the absolute density of $Ω$ is the coordinate Lebesgue density, explaining the equivalence of the form and measure conclusions. An arbitrary nonconstant rescaling of this density need not be preserved. A lower-dimensional generalized-coordinate immersion has its own degree-matched phase-volume form $ω_{X}^{\land r} / r!$ .
Source caveat. The cited treatments argue by the infinitesimal Jacobian and state that the conclusion holds for time-dependent Hamiltonians as well; the present theorem proves the corresponding statement with the hypotheses $\det D Φ_{t} = 1$ and the actual two-time solution domains, and makes no all-time or global claim.

Liouville's theorem for a potential presentation: the phase-volume form $ω^{3 N} / (3 N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Liouville's theorem for a potential presentation: the phase-volume form ω3N/(3N)!ω3N/(3N)! and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Liouville's theorem for a potential presentation: the phase-volume form $ω^{3 N} / (3 N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains