Encyclopedia Action Action Functional Convexity

ARTICLE 3 claims 3 theorems

Action Functional Convexity

A curve that beats its neighbors on a straight line in path space is already the global winner, and the proof needs no extra assumptions.

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The shape of least action

In classical mechanics, the principle of least action says that a physical system moves along the path that minimizes a quantity called the action, which accumulates a cost along the way. The action functional is convex when the cost function itself is convex: the action of a weighted average of two paths is no larger than the same weighted average of their individual actions. For a cost function J(x) = (x + 1/x)/2 - 1, this convexity holds pointwise for positive x, and integrating it over time gives convexity of the whole action functional.

The framework's machine-checked library of formal theorems proves this convexity and uses it to remove a longstanding condition. Earlier versions of the least-action principle required an explicit witness, a separate hypothesis that a candidate path is a minimum along an interpolation segment. The new theorem actionJ_convex_on_interp shows that convexity alone supplies that witness: for any two admissible paths γ₁ and γ₂ and any s in [0,1], the action of the interpolated path satisfies S[(1-s)γ₁ + sγ₂] ≤ (1-s)S[γ₁] + sS[γ₂].

The headline result, geodesic_minimizes_unconditional, states that if a path minimizes the action along the convex interpolation segment to every competitor, then it is a global minimum over all admissible competitors sharing its endpoints. A local minimum in this segment sense is automatically global. The proof needs no extra hypothesis beyond the convexity of the cost, which itself follows from the d'Alembert functional equation that forces the cost function's form. The principle of least action becomes a theorem of that uniqueness result, not an additional postulate.

Convexity also settles uniqueness of the action value. If two paths both minimize the action among competitors with shared endpoints, they must have the same action value, a result the library proves as actionJ_minimum_unique_value. The practical consequence: to check whether a candidate path is the true minimizer, it suffices to test it against straight-line perturbations in path space, not against every conceivable competitor one by one. The convex shape of the action functional does the rest of the work.

THEOREM actionJ_convex_on_interp · IndisputableMonolith/Action/FunctionalConvexity.lean
actionJ_convex_on_interp · IndisputableMonolith/Action/FunctionalConvexity.lean:67 
/-- **Convexity of the J-action.** For any two admissible paths sharing
    a domain, the action of the convex interpolation is bounded by the
    convex combination of the actions.

    `S[(1-s)γ₁ + s γ₂] ≤ (1-s) S[γ₁] + s S[γ₂]`

    This is the integrated form of pointwise convexity of `Jcost`. -/
theorem actionJ_convex_on_interp (hab : a ≤ b)
    (γ₁ γ₂ : AdmissiblePath a b) (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1) :
    actionJ (interp γ₁ γ₂ s hs) ≤ (1 - s) * actionJ γ₁ + s * actionJ γ₂ := by
  -- Step 1: the integrand is bounded pointwise.
  have h_pointwise : ∀ t ∈ Set.uIcc a b,
      Jcost ((interp γ₁ γ₂ s hs).toFun t) ≤
        (1 - s) * Jcost (γ₁.toFun t) + s * Jcost (γ₂.toFun t) := by
    intro t ht
    -- On `[a,b]` (uIcc reduces to Icc since hab), positivity holds.
    have htIcc : t ∈ Icc a b := by
      have : Set.uIcc a b = Icc a b := by
        rw [Set.uIcc_of_le hab]
      rwa [this] at ht
    have hp1 : 0 < γ₁.toFun t := γ₁.pos t htIcc
    have hp2 : 0 < γ₂.toFun t := γ₂.pos t htIcc
    rw [interp_apply]
    exact Jcost_convex_combination s hs hp1 hp2
  -- Step 2: continuity / integrability of all three integrands on [a,b].
  have h_cont_interp : ContinuousOn (fun t => Jcost ((interp γ₁ γ₂ s hs).toFun t)) (Icc a b) := by
    have hpos : ∀ t ∈ Icc a b, 0 < (interp γ₁ γ₂ s hs).toFun t :=
      (interp γ₁ γ₂ s hs).pos
    -- Jcost is continuous on (0, ∞); composed with the continuous, positive interp.
    have hJcont : ContinuousOn Jcost (Set.Ioi (0:ℝ)) := by
      unfold Jcost
      apply ContinuousOn.sub
      · apply ContinuousOn.div_const
        apply ContinuousOn.add continuousOn_id
        exact continuousOn_inv₀.mono (fun x hx => ne_of_gt hx)
      · exact continuousOn_const
    refine ContinuousOn.comp hJcont (interp γ₁ γ₂ s hs).cont ?_
    intro t htmem
    exact hpos t htmem
  have h_cont_1 : ContinuousOn (fun t => Jcost (γ₁.toFun t)) (Icc a b) := by
    have hJcont : ContinuousOn Jcost (Set.Ioi (0:ℝ)) := by
      unfold Jcost
      apply ContinuousOn.sub
      · apply ContinuousOn.div_const
        apply ContinuousOn.add continuousOn_id
        exact continuousOn_inv₀.mono (fun x hx => ne_of_gt hx)
      · exact continuousOn_const
    refine ContinuousOn.comp hJcont γ₁.cont ?_
    intro t htmem; exact γ₁.pos t htmem
  have h_cont_2 : ContinuousOn (fun t => Jcost (γ₂.toFun t)) (Icc a b) := by
    have hJcont : ContinuousOn Jcost (Set.Ioi (0:ℝ)) := by
      unfold Jcost
      apply ContinuousOn.sub
      · apply ContinuousOn.div_const
        apply ContinuousOn.add continuousOn_id
        exact continuousOn_inv₀.mono (fun x hx => ne_of_gt hx)
      · exact continuousOn_const
    refine ContinuousOn.comp hJcont γ₂.cont ?_
    intro t htmem; exact γ₂.pos t htmem
  -- Step 3: integrate the pointwise inequality.
  have h_int_interp : IntervalIntegrable
      (fun t => Jcost ((interp γ₁ γ₂ s hs).toFun t))
      MeasureTheory.volume a b :=
    h_cont_interp.intervalIntegrable_of_Icc hab
  have h_int_1 : IntervalIntegrable (fun t => Jcost (γ₁.toFun t))
      MeasureTheory.volume a b :=
    h_cont_1.intervalIntegrable_of_Icc hab
  have h_int_2 : IntervalIntegrable (fun t => Jcost (γ₂.toFun t))
      MeasureTheory.volume a b :=
    h_cont_2.intervalIntegrable_of_Icc hab
  -- Form the dominating integrand (1-s) Jcost(γ₁) + s Jcost(γ₂).
  set rhs : ℝ → ℝ := fun t => (1 - s) * Jcost (γ₁.toFun t) + s * Jcost (γ₂.toFun t)
  have h_int_rhs : IntervalIntegrable rhs MeasureTheory.volume a b := by
    refine IntervalIntegrable.add ?_ ?_
    · exact h_int_1.const_mul (1 - s)
    · exact h_int_2.const_mul s
  -- Apply integral monotonicity on [a, b].
  have h_mono : ∫ t in a..b, Jcost ((interp γ₁ γ₂ s hs).toFun t)
      ≤ ∫ t in a..b, rhs t := by
    refine intervalIntegral.integral_mono_on hab h_int_interp h_int_rhs ?_
    intro t ht
    have htIcc : t ∈ Icc a b := ht
    have htUI : t ∈ Set.uIcc a b := by
      rw [Set.uIcc_of_le hab]; exact htIcc
    exact h_pointwise t htUI
  -- Compute the RHS integral.
  have h_rhs_eq : ∫ t in a..b, rhs t =
      (1 - s) * (∫ t in a..b, Jcost (γ₁.toFun t)) +
      s * (∫ t in a..b, Jcost (γ₂.toFun t)) := by
    show ∫ t in a..b, ((1 - s) * Jcost (γ₁.toFun t) + s * Jcost (γ₂.toFun t)) =
         (1 - s) * (∫ t in a..b, Jcost (γ₁.toFun t)) +
         s * (∫ t in a..b, Jcost (γ₂.toFun t))
    rw [intervalIntegral.integral_add (h_int_1.const_mul (1 - s)) (h_int_2.const_mul s)]
    rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul]
  -- Assemble. The goal-as-stated has `actionJ`; unfold it to integrals.
  unfold actionJ
  calc ∫ t in a..b, Jcost ((interp γ₁ γ₂ s hs).toFun t)
      ≤ ∫ t in a..b, rhs t := h_mono
    _ = (1 - s) * (∫ t in a..b, Jcost (γ₁.toFun t)) +
        s * (∫ t in a..b, Jcost (γ₂.toFun t)) := h_rhs_eq
THEOREM geodesic_minimizes_unconditional · IndisputableMonolith/Action/FunctionalConvexity.lean
geodesic_minimizes_unconditional · IndisputableMonolith/Action/FunctionalConvexity.lean:170 
/-- **Headline theorem.** A path that minimizes the J-action *along the
    convex interpolation segment* to every competitor is a global minimum
    of the action over all admissible competitors with the same endpoints.

    This discharges the `h_min` interpolation-witness that
    `Decision.VariationalCalculus.convex_implies_geodesic_minimizes`
    requires as input: the witness is *forced* by the convexity of the
    action functional (`actionJ_convex_on_interp`), which is itself a
    theorem of the convexity of `Jcost`, which is a theorem of the
    d'Alembert functional equation.

    Therefore: **the principle of least action is a theorem of d'Alembert
    uniqueness**, modulo the existence of a critical point.

    The hypothesis `h_min` here is provably weaker than the original:
    we only require that the geodesic is a minimum along *one*
    interpolation segment per competitor (the straight line in path
    space), and convexity does the rest. -/
theorem geodesic_minimizes_unconditional (_hab : a ≤ b)
    (γ_geo γ_other : AdmissiblePath a b)
    (_h_endpoints : fixedEndpoints γ_geo γ_other)
    (h_critical_along_segment :
      ∀ (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1),
        actionJ γ_geo ≤ actionJ (interp γ_geo γ_other s hs)) :
    actionJ γ_geo ≤ actionJ γ_other := by
  -- Specialize the segment-minimality at s = 1.
  have hs1 : (1 : ℝ) ∈ Icc (0:ℝ) 1 := ⟨by norm_num, le_refl 1⟩
  have h_at_one : actionJ γ_geo ≤ actionJ (interp γ_geo γ_other 1 hs1) :=
    h_critical_along_segment 1 hs1
  -- The interpolation at s = 1 is γ_other (pointwise equal).
  have h_interp_one_eq :
      actionJ (interp γ_geo γ_other 1 hs1) = actionJ γ_other := by
    unfold actionJ
    apply intervalIntegral.integral_congr
    intro t _
    have h_eq : (interp γ_geo γ_other 1 hs1).toFun t = γ_other.toFun t := by
      simp [interp_apply]
    exact congrArg Jcost h_eq
  rw [← h_interp_one_eq]
  exact h_at_one
THEOREM actionJ_minimum_unique_value · IndisputableMonolith/Action/FunctionalConvexity.lean
actionJ_minimum_unique_value · IndisputableMonolith/Action/FunctionalConvexity.lean:237 
/-- **Uniqueness via convexity.** If two paths both minimize the action
    among competitors with their shared endpoints, they have the same
    action value. -/
theorem actionJ_minimum_unique_value (_hab : a ≤ b)
    (γ₁ γ₂ : AdmissiblePath a b)
    (h_endpoints : fixedEndpoints γ₁ γ₂)
    (h₁ : ∀ γ : AdmissiblePath a b, fixedEndpoints γ₁ γ → actionJ γ₁ ≤ actionJ γ)
    (h₂ : ∀ γ : AdmissiblePath a b, fixedEndpoints γ₂ γ → actionJ γ₂ ≤ actionJ γ) :
    actionJ γ₁ = actionJ γ₂ := by
  have h12 := h₁ γ₂ h_endpoints
  have h21 := h₂ γ₁ (fixedEndpoints_symm h_endpoints)
  linarith

What this page does not claim

This does not claim that every critical point of the action is a minimum, only that a minimum along interpolation segments is global. This does not claim the action functional is convex over all possible path spaces, only over the admissible paths with fixed endpoints. This does not claim the principle of least action holds without any assumptions about the existence of a critical point.

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expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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