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A linear two-shell potential fails first-class closure

Example

Assume ${A C}_{ω}$ for the cotangent/constraint supplier. Use the product cotangent domain $P = T^{*} (R^{4} \times R^{4})$ with coordinates $(x_{i}^{a}, p_{i a})$ for $i = 1, 2$ , Minkowski metric $η = diag (- 1, 1, 1, 1)$ , canonical form $ω = \sum_{i = 1}^{2} \sum_{a = 0}^{3} d x_{i}^{a} \land d p_{i a}$ , and induced bracket ${F, G} = \sum_{i = 1}^{2} \sum_{a = 0}^{3} (F_{x_{i}^{a}} G_{p_{i a}} - F_{p_{i a}} G_{x_{i}^{a}})$ . Take $x_{i}^{a}$ in metres, $p_{i a}$ in kg m/s, $P_{i}^{a} : = η^{a b} p_{i b}$ , $p_{i}^{2} : = η^{a b} p_{i a} p_{i b}$ , $r^{a} : = x_{1}^{a} - x_{2}^{a}$ , and $r^{2} : = η_{a b} r^{a} r^{b}$ . Use constraints

$C_{1} = \frac{1}{2} (p_{1}^{2} + m_{1}^{2} c^{2}) + V (r^{2}), C_{2} = \frac{1}{2} (p_{2}^{2} + m_{2}^{2} c^{2}) + V (r^{2}) .$

Here $m_{i} > 0$ in kg, $c > 0$ in m/s, and $V : R \to R$ is smooth with units of momentum squared. Their canonical bracket is

${C_{1}, C_{2}} = - 2 V^{'} (r^{2}) r \cdot (P_{1} + P_{2}) .$

For $V (z) = λ z$ , with $λ > 0$ in momentum squared per square metre, this bracket is nonzero at explicit points on the joint surface $C_{1} = C_{2} = 0$ where $d C_{1}, d C_{2}$ are independent. Thus, by First-class closure for a regular constraint family, the pair fails first-class closure on the local regular surface around each such point, and arbitrary independent multipliers need not preserve both constraints there. Additional consistency conditions are required; no universal obstruction for all interacting relativistic models is claimed.

Facts & Assumptions

Given: Two canonical spacetime cotangent factors, the stated constraints, smooth $V$ , and the canonical bracket convention.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume countable choice ${A C}_{ω}$ only for the referenced cotangent/constraint supplier; no full axiom of choice is used.

[F1]

First-class closure for a regular constraint family: on a regular common zero surface, “first-class” means every pairwise Poisson bracket vanishes at every point of that surface. This is the exact local definition used by the conclusion.

[F2]

Canonical Poisson brackets and local Hamiltonian flow: in induced cotangent coordinates, ${F, G} = F_{q^{i}} G_{p_{i}} - F_{p_{i}} G_{q^{i}}$ and $ι_{X_{H}} ω = d H$ . The H25 Step 1 record is ready/hash-current, but the carrier remains draft/unbound and its ai-altered proof is not independently approved.

[L1]

Constraint consistency for a regular first-order model: the current H2 draft assumes an open parameter interval, a symplectic manifold, smooth moving constraints with independent differentials on each $N_{λ}$ , a smooth ambient $H_{c}$ , paths in a Darboux chart on a compact subinterval, and smooth multipliers. It states $\partial_{λ} χ_{a} + {χ_{a}, H_{c} (λ, \cdot)} + u^{b} (λ) {χ_{a}, χ_{b}} = 0$ on $N_{λ}$ as the tangency condition; for autonomous constraints the first term vanishes. Its H25 Step 1 record is ready/hash-current, but the carrier remains draft/unbound with ai-altered proof provenance and no independent proof approval. The autonomous $H_{c} = 0$ specialization used here matches its current Statement.

Verification

technique · direct differentiation and an on-surface witness

Given: Canonical coordinates, $C_{1}, C_{2}$ , the bracket convention [F2], the local first-class definition [F1], and the autonomous tangency condition [L1].

1.1F2A1algebra

The potential derivatives are $\partial_{x_{1}^{a}} V = 2 V^{'} (r^{2}) r_{a}$ and $\partial_{x_{2}^{a}} V = - 2 V^{'} (r^{2}) r_{a}$ . The kinetic terms have $\partial_{p_{i a}} C_{i} = P_{i}^{a}$ . In ${C_{1}, C_{2}}$ , the first particle contributes $- P_{1}^{a} (2 V^{'} r_{a})$ and the second contributes $(- 2 V^{'} r_{a}) P_{2}^{a}$ ; all other cross terms vanish. Their sum is the displayed bracket.

2.1F1L1A1step 1.1algebra∎

Choose $r = (0, R, 0, 0)$ with $R > 0$ , $V (z) = λ z$ , and $λ > 0$ . Let $P_{i}^{1} = u_{i} > 0$ , other spatial components zero, and $P_{i}^{0} = \sqrt{m_{i}^{2} c^{2} + 2 λ R^{2} + u_{i}^{2}}$ . Then $p_{i}^{2} = - (P_{i}^{0})^{2} + u_{i}^{2} = - m_{i}^{2} c^{2} - 2 λ R^{2}$ , so both $C_{i} = 0$ , while $r \cdot (P_{1} + P_{2}) = R (u_{1} + u_{2}) > 0$ . Each constraint differential has nonzero momentum component $P_{i}$ in its own cotangent factor, so $d C_{1}, d C_{2}$ are independent at this witness. Independence persists on a sufficiently small neighborhood, making the local joint level regular. The bracket is ${C_{1}, C_{2}} = - 2 λ R (u_{1} + u_{2}) \neq 0$ there, so [F1] says the pair is not first-class on that local regular surface. With autonomous constraints and $H_{c} = 0$ , [L1] gives ${\dot{C}}_{1} = u^{2} {C_{1}, C_{2}}$ and ${\dot{C}}_{2} = u^{1} {C_{2}, C_{1}}$ ; hence preserving both constraints at this point forces $u^{1} = u^{2} = 0$ . Thus arbitrary independent nonzero multipliers need not preserve the joint surface.

Depends on

Dependency tree · 2 levels

Sources

Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., complete book PDF; §12.1 Theorem 12.1.4 pp. 393–395; §12.4 Theorem 12.4.1 pp. 412–413; §§14.2–14.3 pp. 466–473; §7.9 pp. 208–211.