Action Noether Is Time Translation Invariant

Time symmetry and energy

In classical mechanics, a symmetry of a system's action is a transformation that leaves the action unchanged. The action is a number assigned to a possible path of a system, and the principle of least action says the path Nature takes is the one that makes this number as small as possible. When the laws of physics do not change when you shift the clock forward or backward, the system is time-translation invariant. In 1915, Emmy Noether proved that such a symmetry always implies a conserved quantity, and for time translation that quantity is energy.

The Recognition Science framework's ledger, a discrete record of recognition events, models physical trajectories as real-valued functions of time. The declaration isTimeTranslationInvariant defines what it means for a J-action functional, the framework's cost assigned to a trajectory, to be invariant under such a shift: shifting the time argument of every trajectory by any amount leaves the action value unchanged. This is a formal definition, not a theorem, but it is the hypothesis that powers the framework's Noether results.

In Recognition Science, the framework's library of machine-checked formal theorems proves that if a J-action functional is time-translation invariant, then the action itself is conserved along the time-translation flow. This conserved quantity is interpreted as the total energy. The theorem time_translation_invariance_implies_energy_conservation is a direct corollary of the abstract Noether theorem noether_core, and it is proved with no unproved assumptions. The framework also proves the concrete mechanical result: for a standard Lagrangian L = ½ m q̇² - V(q) with a time-independent potential, the total energy E = T + V is conserved.

The declaration does not claim that all physical systems are time-translation invariant, nor that energy is always conserved. It establishes a conditional statement: if the symmetry holds, then the conservation follows. It also does not claim that the framework's J-action is the only action that exhibits this property, nor does it derive the specific form of the J-cost function from time symmetry alone. The declaration is a definition that sets up the framework's version of Noether's theorem, not a proof that time symmetry exists in any particular physical system.

/-- A J-action `S` on `RealAction` is time-translation invariant if
    `S(γ ∘ t-shift) = S(γ)` for every shift. -/
def isTimeTranslationInvariant (S : RealAction → ℝ) : Prop :=
  ∀ dt : ℝ, IsSymmetryOf (timeShift dt) S

/-- **Energy conservation from time-translation invariance.**

    If a J-action functional is time-translation invariant, then by
    `noether_core` it is itself conserved along the time-translation flow.
    The conserved quantity is interpreted as the total energy. -/
theorem time_translation_invariance_implies_energy_conservation
    (S : RealAction → ℝ)
    (h_inv : ∀ t, IsSymmetryOf (timeTranslationFlow.flow t) S) :
    IsConservedAlong S timeTranslationFlow.flow :=
  noether_core h_inv

/-- **The standard total energy of mechanical motion is conserved when
    the potential is time-independent.**

    This is the concrete Noether theorem for the standard mechanics
    Lagrangian `L = ½ m q̇² - V(q)`: time-translation invariance is
    automatic when `V` does not depend on `t` explicitly, and energy
    conservation `E = T + V` follows.

    Proven directly by `Action.Hamiltonian.energy_conservation`, this
    lemma packages the result in the `Noether` namespace for clarity. -/
theorem energy_conservation_of_J_action (m : ℝ) (hm : 0 < m) (V : ℝ → ℝ)
    (γ : ℝ → ℝ)
    (hV_diff : ∀ t, DifferentiableAt ℝ V (γ t))
    (hγ_diff : ∀ t, DifferentiableAt ℝ γ t)
    (hγ_diff2 : ∀ t, DifferentiableAt ℝ (deriv γ) t)
    (h_dE_eq_factored : ∀ t : ℝ,
      deriv (HamiltonianMech.totalEnergy m V γ) t =
        deriv γ t * (m * deriv (deriv γ) t + deriv V (γ t)))
    (hEL : ∀ t : ℝ, QuadraticLimit.standardEL m V γ t = 0) :
    ∀ t₁ t₂ : ℝ,
      HamiltonianMech.totalEnergy m V γ t₁ = HamiltonianMech.totalEnergy m V γ t₂ :=
  HamiltonianMech.energy_conservation m hm V γ hV_diff hγ_diff hγ_diff2 h_dE_eq_factored hEL

What this page does not claim

The declaration does not claim that time-translation invariance holds for all physical systems. The declaration does not claim that the J-action is the only action for which Noether's theorem applies. The declaration does not claim that energy conservation is derived from the J-cost function's uniqueness theorem.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Action/Noether.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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