| import OAI.Geometry.Arnold.CompactField |
|
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| namespace OAI |
|
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| universe uE |
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| noncomputable section |
| open scoped BigOperators ContDiff |
|
|
| open scoped ContDiff Topology NNReal |
| open Set Filter |
|
|
| namespace ArnoldCounterexample.Midpoint |
| variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| omit [CompleteSpace E] in |
| lemma contraction (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (s : ℝ) (z : E) (hs : ‖s‖ * C < 1) : |
| ContractingWith (‖s‖₊ * C) (fun u ↦ z + s • X u) := by |
| refine ⟨?_, ?_⟩ |
| · exact_mod_cast hs |
| · apply LipschitzWith.of_dist_le_mul |
| intro x y |
| rw [dist_eq_norm, add_sub_add_left_eq_sub, ← smul_sub, norm_smul, ← dist_eq_norm] |
| simpa only [NNReal.coe_mul, coe_nnnorm, mul_assoc] using |
| mul_le_mul_of_nonneg_left (hX.dist_le_mul x y) (norm_nonneg s) |
|
|
| /-- The inverse of id-sX, for a smooth uniformly small parameter s. -/ |
| def resolvent (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (p : ℝ × E) : E := |
| (contraction X hX (τ p.1) p.2 (hτ p.1)).fixedPoint (fun u ↦ p.2 + τ p.1 • X u) |
|
|
| lemma resolvent_eq (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (p : ℝ × E) : |
| resolvent X hX τ hτ p = p.2 + τ p.1 • X (resolvent X hX τ hτ p) := by |
| exact ((contraction X hX (τ p.1) p.2 (hτ p.1)).fixedPoint_isFixedPt).symm |
|
|
| lemma resolvent_unique (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (p : ℝ × E) (u : E) |
| (hu : u = p.2 + τ p.1 • X u) : u = resolvent X hX τ hτ p := by |
| exact (contraction X hX (τ p.1) p.2 (hτ p.1)).fixedPoint_unique hu.symm |
|
|
| omit [CompleteSpace E] in |
| lemma implicit_partial (X : E → E) (hX : ContDiff ℝ ∞ X) |
| (τ : ℝ → ℝ) (hτ : ContDiff ℝ ∞ τ) (p : ℝ × E) (u : E) : |
| fderiv ℝ (fun q : (ℝ × E) × E ↦ q.2 - q.1.2 - τ q.1.1 • X q.2) (p,u) ∘L |
| ContinuousLinearMap.inr ℝ (ℝ × E) E = |
| ContinuousLinearMap.id ℝ E - τ p.1 • fderiv ℝ X u := by |
| have hg : ContDiff ℝ ∞ (fun q : (ℝ × E) × E ↦ q.2 - q.1.2 - τ q.1.1 • X q.2) := by fun_prop |
| have hu : HasFDerivAt (fun v ↦ v - p.2 - τ p.1 • X v) |
| (ContinuousLinearMap.id ℝ E - τ p.1 • fderiv ℝ X u) u := |
| ((hasFDerivAt_id u).sub_const p.2).sub ((hX.differentiable (by simp) u).hasFDerivAt.const_smul _) |
| have hc := (hg.differentiable (by simp) (p,u)).hasFDerivAt.comp u |
| ((hasFDerivAt_const p u).prodMk (hasFDerivAt_id (𝕜 := ℝ) u)) |
| convert hc.unique hu using 1; congr 1 |
|
|
| lemma resolvent_smooth (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) |
| (hτs : ContDiff ℝ ∞ τ) : ContDiff ℝ ∞ (resolvent X hX τ hτ) := by |
| rw [contDiff_iff_contDiffAt] |
| intro p |
| let u := resolvent X hX τ hτ p |
| let g : (ℝ × E) × E → E := fun q ↦ q.2 - q.1.2 - τ q.1.1 • X q.2 |
| have hg : ContDiffAt ℝ ∞ g (p,u) := by dsimp [g]; fun_prop |
| have hi : (fderiv ℝ g (p,u) ∘L ContinuousLinearMap.inr ℝ (ℝ × E) E).IsInvertible := by |
| rw [show g = (fun q : (ℝ × E) × E ↦ q.2 - q.1.2 - τ q.1.1 • X q.2) from rfl, |
| implicit_partial X hXs τ hτs] |
| have hb : ‖τ p.1 • fderiv ℝ X u‖ < 1 := by |
| rw [norm_smul] |
| exact lt_of_le_of_lt (mul_le_mul_of_nonneg_left |
| (norm_fderiv_le_of_lipschitz ℝ hX) (norm_nonneg _)) (hτ p.1) |
| obtain ⟨A,hA⟩ := isUnit_one_sub_of_norm_lt_one hb |
| exact ⟨ContinuousLinearEquiv.ofUnit A, hA⟩ |
| apply (hg.contDiffAt_implicitFunction (by simp) hi).congr_of_eventuallyEq |
| filter_upwards [hg.eventually_apply_implicitFunction (by simp) hi] with q hq |
| have hbase : g (p,u) = 0 := by |
| dsimp [g] |
| have he := resolvent_eq X hX τ hτ p |
| change u = _ at he |
| calc |
| u - p.2 - τ p.1 • X u = u - (p.2 + τ p.1 • X u) := by abel |
| _ = 0 := sub_eq_zero.mpr he |
| rw [hbase] at hq |
| have hu : hg.implicitFunction (by simp) hi q = q.2 + τ q.1 • X (hg.implicitFunction (by simp) hi q) := by |
| dsimp [g] at hq |
| exact (sub_eq_iff_eq_add.mp (sub_eq_zero.mp hq)).trans (add_comm _ _) |
| exact (resolvent_unique X hX τ hτ q _ hu).symm |
|
|
| lemma neg_small {C : ℝ≥0} {τ : ℝ → ℝ} (hτ : ∀ t, ‖τ t‖ * C < 1) : |
| ∀ t, ‖-τ t‖ * C < 1 := by simpa only [norm_neg] using hτ |
|
|
| /-- The implicit midpoint step. The actual step size is 2τ. -/ |
| def step (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (p : ℝ × E) : E := |
| (2 : ℝ) • resolvent X hX τ hτ p - p.2 |
|
|
| lemma step_center (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (p : ℝ × E) : |
| step X hX τ hτ p = resolvent X hX τ hτ p + τ p.1 • X (resolvent X hX τ hτ p) := by |
| have he : resolvent X hX τ hτ p - p.2 = τ p.1 • X (resolvent X hX τ hτ p) := by |
| exact sub_eq_iff_eq_add.mpr ((resolvent_eq X hX τ hτ p).trans (add_comm _ _)) |
| calc |
| step X hX τ hτ p = resolvent X hX τ hτ p + (resolvent X hX τ hτ p - p.2) := by |
| dsimp [step]; module |
| _ = _ := by rw [he] |
|
|
| lemma resolvent_step (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (p : ℝ × E) : |
| resolvent X hX (fun t ↦ -τ t) (neg_small hτ) (p.1, step X hX τ hτ p) = |
| resolvent X hX τ hτ p := by |
| symm |
| apply resolvent_unique |
| simp only [step_center, neg_smul] |
| abel |
|
|
| lemma step_inverse (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (p : ℝ × E) : |
| step X hX (fun t ↦ -τ t) (neg_small hτ) (p.1, step X hX τ hτ p) = p.2 := by |
| rw [step, resolvent_step] |
| dsimp [step] |
| module |
|
|
| lemma step_smooth (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) |
| (hτs : ContDiff ℝ ∞ τ) : ContDiff ℝ ∞ (step X hX τ hτ) := |
| ((resolvent_smooth X hX hXs τ hτ hτs).const_smul (2 : ℝ)).sub contDiff_snd |
|
|
| lemma resolvent_zero (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (t : ℝ) (ht : τ t = 0) (z : E) : |
| resolvent X hX τ hτ (t,z) = z := by |
| simpa only [ht, zero_smul, add_zero] using resolvent_eq X hX τ hτ (t,z) |
|
|
| lemma step_zero (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (t : ℝ) (ht : τ t = 0) (z : E) : |
| step X hX τ hτ (t,z) = z := by |
| rw [step_center, resolvent_zero _ _ _ _ _ ht] |
| simp [ht] |
|
|
| lemma resolvent_symmetry (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (S : E →ₗ[ℝ] E) |
| (hS : ∀ z, X (S z) = S (X z)) (p : ℝ × E) : |
| resolvent X hX τ hτ (p.1, S p.2) = S (resolvent X hX τ hτ p) := by |
| symm |
| apply resolvent_unique |
| calc |
| S (resolvent X hX τ hτ p) = S (p.2 + τ p.1 • X (resolvent X hX τ hτ p)) := |
| congrArg S (resolvent_eq X hX τ hτ p) |
| _ = _ := by simp only [map_add, map_smul, hS] |
|
|
| lemma step_symmetry (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (S : E →ₗ[ℝ] E) |
| (hS : ∀ z, X (S z) = S (X z)) (p : ℝ × E) : |
| step X hX τ hτ (p.1, S p.2) = S (step X hX τ hτ p) := by |
| simp [step, resolvent_symmetry X hX τ hτ S hS p] |
|
|
| lemma resolvent_fderiv_equation (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) |
| (hτs : ContDiff ℝ ∞ τ) (p : ℝ × E) (a : ℝ) (v : E) : |
| fderiv ℝ (resolvent X hX τ hτ) p (a,v) = v + |
| (a * deriv τ p.1) • X (resolvent X hX τ hτ p) + |
| τ p.1 • fderiv ℝ X (resolvent X hX τ hτ p) |
| (fderiv ℝ (resolvent X hX τ hτ) p (a,v)) := by |
| let R := resolvent X hX τ hτ |
| have hR := ((resolvent_smooth X hX hXs τ hτ hτs).differentiable (by simp) p).hasFDerivAt |
| have hτR := ((hτs.differentiable (by simp) p.1).hasDerivAt).comp_hasFDerivAt p (hasFDerivAt_fst (𝕜 := ℝ) (p := p)) |
| have hXR := (hXs.differentiable (by simp) (R p)).hasFDerivAt.comp p hR |
| have hd := hasFDerivAt_snd.add (hτR.smul hXR) |
| have heq : (fun q : ℝ × E ↦ q.2 + τ q.1 • X (R q)) = R := by |
| funext q |
| exact (resolvent_eq X hX τ hτ q).symm |
| change HasFDerivAt (fun q : ℝ × E ↦ q.2 + τ q.1 • X (R q)) _ p at hd |
| rw [heq] at hd |
| have he := congrArg (fun L : (ℝ × E) →L[ℝ] E ↦ L (a,v)) (hR.unique hd) |
| change fderiv ℝ (resolvent X hX τ hτ) p (a,v) = |
| v + (τ p.1 • fderiv ℝ X (resolvent X hX τ hτ p) |
| (fderiv ℝ (resolvent X hX τ hτ) p (a,v)) + |
| (deriv τ p.1 * a) • X (resolvent X hX τ hτ p)) at he |
| simpa only [add_assoc, add_comm, add_left_comm, mul_comm] using he |
|
|
| omit [CompleteSpace E] in |
| lemma hamiltonian_linearization (σ : E →L[ℝ] E →L[ℝ] ℝ) |
| (hσ : ∀ a b, σ a b = -σ b a) (X : E → E) (hX : ContDiff ℝ ∞ X) |
| (K : E → ℝ) (hK : ContDiff ℝ ∞ K) (u : E) |
| (hHK : ∀ᶠ w in 𝓝 u, σ (X w) = fderiv ℝ K w) (a b : E) : |
| σ (fderiv ℝ X u a) b = -σ a (fderiv ℝ X u b) := by |
| have hd := σ.hasFDerivAt.comp u ((hX.differentiable (by simp) u).hasFDerivAt) |
| have hdK := hd.congr_of_eventuallyEq (show (fun w ↦ σ (X w)) =ᶠ[𝓝 u] fderiv ℝ K from hHK).symm |
| have hs := (hK.contDiffAt (x := u)).isSymmSndFDerivAt (by |
| simp only [minSmoothness_of_isRCLikeNormedField] |
| change ((2 : ℕ∞) : ℕ∞ω) ≤ ((⊤ : ℕ∞) : ℕ∞ω) |
| exact WithTop.coe_le_coe.mpr le_top) a b |
| rw [hdK.fderiv] at hs |
| simpa only [ContinuousLinearMap.comp_apply, hσ _ a] using hs |
|
|
| end ArnoldCounterexample.Midpoint |
|
|
| namespace ArnoldCounterexample.Midpoint |
| variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| def hamiltonian (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) (K : E → ℝ) (p : ℝ × E) : ℝ := |
| (2 * deriv τ p.1) * K (resolvent X hX (fun t ↦ -τ t) (neg_small hτ) p) |
|
|
| lemma hamiltonian_smooth (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) |
| (hτs : ContDiff ℝ ∞ τ) (K : E → ℝ) (hK : ContDiff ℝ ∞ K) : |
| ContDiff ℝ ∞ (hamiltonian X hX τ hτ K) := by |
| have hd : ContDiff ℝ ∞ (deriv τ) := hτs.deriv' |
| exact (contDiff_const.mul (hd.comp contDiff_fst)).mul |
| (hK.comp (resolvent_smooth X hX hXs _ (neg_small hτ) hτs.neg)) |
|
|
| lemma step_time_derivative (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) |
| (hτs : ContDiff ℝ ∞ τ) (p : ℝ × E) : |
| deriv (fun t ↦ step X hX τ hτ (t,p.2)) p.1 = |
| (2 : ℝ) • fderiv ℝ (resolvent X hX τ hτ) p (1,0) := by |
| have hR := ((resolvent_smooth X hX hXs τ hτ hτs).differentiable (by simp) p).hasFDerivAt |
| have ht : HasDerivAt (fun t : ℝ ↦ (t,p.2)) (1,0) p.1 := |
| (hasDerivAt_id p.1).prodMk (hasDerivAt_const p.1 p.2) |
| have hd := ((hR.comp_hasDerivAt p.1 ht).const_smul (2 : ℝ)).sub_const p.2 |
| exact hd.deriv |
|
|
| lemma step_hamiltonian_equation (σ : E →L[ℝ] E →L[ℝ] ℝ) |
| (hσ : ∀ a b, σ a b = -σ b a) (X : E → E) {C : ℝ≥0} (hX : LipschitzWith C X) |
| (hXs : ContDiff ℝ ∞ X) (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖ * C < 1) |
| (hτs : ContDiff ℝ ∞ τ) (K : E → ℝ) (hK : ContDiff ℝ ∞ K) (p : ℝ × E) |
| (hHK : ∀ᶠ w in 𝓝 (resolvent X hX τ hτ p), σ (X w) = fderiv ℝ K w) (v : E) : |
| σ (deriv (fun t ↦ step X hX τ hτ (t,p.2)) p.1) v = |
| fderiv ℝ (fun w ↦ hamiltonian X hX τ hτ K (p.1,w)) (step X hX τ hτ p) v := by |
| let u := resolvent X hX τ hτ p |
| let y := step X hX τ hτ p |
| let A := fderiv ℝ (resolvent X hX τ hτ) p (1,0) |
| let R := resolvent X hX (fun t ↦ -τ t) (neg_small hτ) |
| let B := fderiv ℝ R (p.1,y) (0,v) |
| have hRu : R (p.1,y) = u := resolvent_step X hX τ hτ p |
| have hA : A = deriv τ p.1 • X u + τ p.1 • fderiv ℝ X u A := by |
| simpa only [zero_add, one_mul] using resolvent_fderiv_equation X hX hXs τ hτ hτs p 1 0 |
| have hB : B = v - τ p.1 • fderiv ℝ X u B := by |
| have h := resolvent_fderiv_equation X hX hXs (fun t ↦ -τ t) (neg_small hτ) hτs.neg (p.1,y) 0 v |
| change B = v + (0 * deriv (fun t ↦ -τ t) p.1) • X (R (p.1,y)) + |
| (-τ p.1) • fderiv ℝ X (R (p.1,y)) B at h |
| simpa only [zero_mul, zero_smul, add_zero, hRu, neg_smul, sub_eq_add_neg] using h |
| have hlin := hamiltonian_linearization σ hσ X hXs K hK u hHK A B |
| have eeA : σ A B = deriv τ p.1 * σ (X u) B + τ p.1 * σ (fderiv ℝ X u A) B := by |
| conv_lhs => rw [hA] |
| simp only [map_add, map_smul, add_apply, smul_apply, smul_eq_mul] |
| have eeB : σ A B = σ A v - τ p.1 * σ A (fderiv ℝ X u B) := by |
| conv_lhs => rw [hB] |
| simp only [map_sub, map_smul, smul_eq_mul] |
| have ee : σ A v = deriv τ p.1 * σ (X u) B := by rw [hlin] at eeA; nlinarith |
| have hKX : fderiv ℝ K u = σ (X u) := (Filter.EventuallyEq.eq_of_nhds hHK).symm |
| have hRd := ((resolvent_smooth X hX hXs _ (neg_small hτ) hτs.neg).differentiable (by simp) (p.1,y)).hasFDerivAt |
| have hinc : HasFDerivAt (fun w : E ↦ (p.1,w)) (ContinuousLinearMap.inr ℝ ℝ E) y := |
| (hasFDerivAt_const p.1 y).prodMk (hasFDerivAt_id (𝕜 := ℝ) y) |
| have hKd := (hK.differentiable (by simp) (R (p.1,y))).hasFDerivAt |
| have hd := ((hKd.comp y (hRd.comp y hinc)).const_mul (2 * deriv τ p.1)) |
| change HasFDerivAt (fun w ↦ hamiltonian X hX τ hτ K (p.1,w)) _ y at hd |
| rw [hd.fderiv, step_time_derivative X hX hXs τ hτ hτs p] |
| change σ ((2 : ℝ) • A) v = _ |
| simp only [map_smul, smul_apply, smul_eq_mul, ContinuousLinearMap.comp_apply] |
| rw [hRu, hKX] |
| change 2 * σ A v = (2 * deriv τ p.1) * σ (X u) B |
| rw [ee] |
| ring |
|
|
| end ArnoldCounterexample.Midpoint |
|
|
| end |
|
|
| end OAI |
|
|
