IndisputableMonolith.Action.Noether
The Action.Noether module defines the J-action functional on real-valued trajectories together with time and space translation operators and their invariance predicates. Researchers deriving conservation laws from Recognition Science cost functionals cite it to connect the J-cost to classical Noether symmetries. The module assembles these objects by importing the Hamiltonian formulation and the general cost-stationarity Noether theorem.
claimThe J-action is the functional $\mathcal{A}_J[q] = \int J(\dot q(t))\,dt$ on trajectories $q:\mathbb{R}\to\mathbb{R}$. Time-shift and space-shift maps are defined, together with the predicates of invariance under these maps that imply conservation of energy and momentum.
background
The module operates in the setting of the J-action whose small-strain limit recovers the standard Lagrangian $L=\frac12 m\dot q^2-V(q)$. Upstream, the Hamiltonian module supplies the Legendre transform that yields $H=p^2/(2m)+V(q)$ and Hamilton's equations. The QFT.NoetherTheorem module supplies the general statement that cost stationarity implies a conserved current for every continuous symmetry of the action.
proof idea
This is a definition module, no proofs. It introduces the core objects RealAction, timeShift, spaceShift, the invariance predicates, and the two implication statements that link invariance to conservation; the actual derivations sit in the imported Hamiltonian and QFT.NoetherTheorem modules.
why it matters in Recognition Science
The module supplies the J-action and translation-invariance definitions that are imported by EnergyConservationDomainCert, the domain certificate for energy conservation along Newtonian trajectories. It thereby realizes the Noether correspondence inside Recognition Science by linking cost stationarity directly to the classical conservation laws.
scope and limits
- Does not derive the explicit J functional from the T0-T8 forcing chain.
- Does not treat relativistic or field-theoretic extensions beyond the imported Noether theorem.
- Does not contain the proofs of the conservation implications; those reside in sibling declarations.
- Does not address discrete symmetries or gauge symmetries.
used by (1)
depends on (2)
declarations in this module (11)
-
abbrev
RealAction -
def
timeShift -
def
isTimeTranslationInvariant -
def
timeTranslationFlow -
theorem
time_translation_invariance_implies_energy_conservation -
def
spaceShift -
def
isSpaceTranslationInvariant -
def
spaceTranslationFlow -
theorem
space_translation_invariance_implies_momentum_conservation -
theorem
energy_conservation_of_J_action -
def
noether_status