applied
plain-language theorem explainer
Energy conservation along Newtonian trajectories follows from time-translation symmetry of the J-action. Researchers deriving classical mechanics within Recognition Science cite this certificate to justify the standard Hamiltonian in the small-strain limit. The statement bundles the conservation law H(γ(t), p(t)) constant with the forced Hamilton equations under explicit regularity hypotheses on V and γ. The proof is a domain certificate that invokes the upstream Hamiltonian construction without additional tactics shown in this module.
Claim. Let V be differentiable along the image of a twice-differentiable trajectory γ satisfying the Euler-Lagrange equation of the J-cost Lagrangian. Then the Hamiltonian H(γ(t), p(t)) = p(t)^2/(2m) + V(γ(t)) is independent of t, and the pair of equations γ̇ = p/m, ṗ = -V'(γ) holds.
background
The module supplies a domain certificate for energy conservation in the Action domain. It rests on the shifted cost H(x) = J(x) + 1 = ½(x + x^{-1}) from CostAlgebra, under which the Recognition Composition Law reduces to the d'Alembert equation H(xy) + H(x/y) = 2 H(x) H(y). The local theoretical setting is the small-strain J-action whose Lagrangian L = ½ m γ̇² - V(γ) yields Newtonian mechanics via the Euler-Lagrange equation, as stated in the module header.
proof idea
one-line wrapper that applies the energy conservation result from Action.Hamiltonian together with the Noether application in Action.Noether under the regularity hypotheses hV_diff, hγ_diff, hγ_diff2 and h_dE_factored.
why it matters
This declaration places the classical Hamiltonian limit inside the Recognition framework, feeding downstream results such as bandEnergy in Chemistry.PeriodicTable and the Duhamel kernel rewrite in ClassicalBridge.Fluids.ContinuumLimit2D. It fills the Hamiltonian Formulation as a Corollary in Paper A, §Hamiltonian Formulation as a Corollary, by applying Noether's theorem to time-translation symmetry of the J-action. The placement is consistent with the eight-tick octave and phi-ladder construction in the foundation.
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