| import OAI.Geometry.Arnold.MidpointGeometry | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators ContDiff | |
| open scoped Topology NNReal | |
| open Set Filter | |
| namespace ArnoldCounterexample.Construction | |
| open ArnoldCounterexample.Quadric ArnoldCounterexample.Field | |
| open scoped ContDiff Topology NNReal | |
| open Set Filter | |
| def constant : ℝ≥0 := vectorField_lipschitz.choose | |
| lemma lipschitz : LipschitzWith constant vectorField := vectorField_lipschitz.choose_spec | |
| def epsilon : ℝ := 1 / (100 * ((constant:ℝ)+1)) | |
| lemma epsilon_pos : 0 < epsilon := by unfold epsilon; positivity | |
| def parameter (t : ℝ) : ℝ := epsilon * Real.sin (Real.pi * t / 2) | |
| lemma parameter_smooth : ContDiff ℝ ∞ parameter := by unfold parameter; fun_prop | |
| lemma parameter_zero : parameter 0 = 0 := by simp [parameter] | |
| lemma parameter_one : parameter 1 = epsilon := by simp [parameter] | |
| lemma parameter_small : ∀ t, ‖parameter t‖ * constant < 1/10 := by | |
| intro t | |
| have hc : 0 ≤ (constant:ℝ) := constant.coe_nonneg | |
| have hden : 0 < 100 * ((constant:ℝ)+1) := by positivity | |
| have he : epsilon * (constant:ℝ) < 1/100 := by | |
| unfold epsilon | |
| rw [one_div_mul_eq_div, div_lt_iff₀ hden] | |
| linarith | |
| have hb : ‖parameter t‖ ≤ epsilon := by | |
| rw [parameter, norm_mul, Real.norm_eq_abs, abs_of_pos epsilon_pos, Real.norm_eq_abs] | |
| exact (mul_le_mul_of_nonneg_left (Real.abs_sin_le_one _) epsilon_pos.le).trans_eq (mul_one _) | |
| exact lt_trans (lt_of_le_of_lt (mul_le_mul_of_nonneg_right hb hc) he) (by norm_num) | |
| lemma parameter_contract : ∀ t, ‖parameter t‖ * constant < 1 := | |
| fun t ↦ lt_trans (parameter_small t) (by norm_num) | |
| def center : ℝ × CV → CV := Midpoint.resolvent vectorField lipschitz parameter parameter_contract | |
| def forwardMid : ℝ × CV → CV := Midpoint.step vectorField lipschitz parameter parameter_contract | |
| def backwardMid : ℝ × CV → CV := | |
| Midpoint.step vectorField lipschitz (fun t ↦ -parameter t) (Midpoint.neg_small parameter_contract) | |
| def midHamiltonian : ℝ × CV → ℝ := | |
| Midpoint.hamiltonian vectorField lipschitz parameter parameter_contract potential | |
| lemma center_smooth : ContDiff ℝ ∞ center := | |
| Midpoint.resolvent_smooth _ lipschitz vectorField_smooth _ _ parameter_smooth | |
| lemma forwardMid_smooth : ContDiff ℝ ∞ forwardMid := | |
| Midpoint.step_smooth _ lipschitz vectorField_smooth _ _ parameter_smooth | |
| lemma backwardMid_smooth : ContDiff ℝ ∞ backwardMid := | |
| Midpoint.step_smooth _ lipschitz vectorField_smooth _ _ parameter_smooth.neg | |
| lemma midHamiltonian_smooth : ContDiff ℝ ∞ midHamiltonian := | |
| Midpoint.hamiltonian_smooth _ lipschitz vectorField_smooth _ _ parameter_smooth _ potential_smooth | |
| lemma forwardMid_preserves (t : ℝ) (z : UnitQuadric) : IsUnitQuadric (forwardMid (t,z)) := | |
| midpoint_preserves lipschitz parameter parameter_contract t z | |
| lemma backwardMid_preserves (t : ℝ) (z : UnitQuadric) : IsUnitQuadric (backwardMid (t,z)) := | |
| midpoint_preserves lipschitz _ (Midpoint.neg_small parameter_contract) t z | |
| lemma center_phase_general (τ : ℝ → ℝ) (hτ : ∀ t, ‖τ t‖*constant < 1) | |
| (p : ℝ × CV) (c : ℂ) (hc : Complex.normSq c = 1) : | |
| Midpoint.resolvent vectorField lipschitz τ hτ (p.1,c•p.2) = | |
| c • Midpoint.resolvent vectorField lipschitz τ hτ p := by | |
| exact Midpoint.resolvent_symmetry _ lipschitz τ hτ (c • (LinearMap.id : CV →ₗ[ℝ] CV)) | |
| (fun z ↦ vectorField_phase c hc z) p | |
| lemma center_phase (p : ℝ × CV) (c : ℂ) (hc : Complex.normSq c = 1) : | |
| center (p.1,c•p.2) = c • center p := center_phase_general _ _ p c hc | |
| lemma forwardMid_phase (p : ℝ × CV) (c : ℂ) (hc : Complex.normSq c = 1) : | |
| forwardMid (p.1,c•p.2) = c • forwardMid p := by | |
| simp only [forwardMid, Midpoint.step, center_phase_general _ _ p c hc, smul_sub, smul_comm c] | |
| lemma backwardMid_phase (p : ℝ × CV) (c : ℂ) (hc : Complex.normSq c = 1) : | |
| backwardMid (p.1,c•p.2) = c • backwardMid p := by | |
| simp only [backwardMid, Midpoint.step, center_phase_general _ _ p c hc, smul_sub, smul_comm c] | |
| lemma midHamiltonian_phase (p : ℝ × CV) (c : ℂ) (hc : Complex.normSq c = 1) : | |
| midHamiltonian (p.1,c•p.2) = midHamiltonian p := by | |
| simp only [midHamiltonian, Midpoint.hamiltonian, center_phase_general _ _ p c hc, potential_phase c hc] | |
| lemma backward_forward (p : ℝ × CV) : backwardMid (p.1,forwardMid p) = p.2 := | |
| Midpoint.step_inverse _ lipschitz parameter parameter_contract p | |
| lemma forward_backward (p : ℝ × CV) : forwardMid (p.1,backwardMid p) = p.2 := by | |
| simpa only [neg_neg, forwardMid, backwardMid] using Midpoint.step_inverse _ lipschitz (fun t ↦ -parameter t) | |
| (Midpoint.neg_small parameter_contract) p | |
| lemma forwardMid_start (z : CV) : forwardMid (0,z) = z := | |
| Midpoint.step_zero _ lipschitz _ _ 0 parameter_zero z | |
| lemma center_bound (t : ℝ) (z : UnitQuadric) : normSquared (center (t,z)) ∈ Ioo (1/2:ℝ) 2 := | |
| ⟨center_norm_gt_half lipschitz _ _ parameter_small t z, | |
| lt_of_le_of_lt (center_norm_le_one lipschitz _ _ t z) (by norm_num)⟩ | |
| lemma mid_equation (t : ℝ) (z : UnitQuadric) (v : CV) : | |
| omega (deriv (fun s ↦ forwardMid (s,z)) t) v = | |
| fderiv ℝ (fun w ↦ midHamiltonian (t,w)) (forwardMid (t,z)) v := by | |
| simpa only [forwardMid, midHamiltonian, symplectic_apply] using Midpoint.step_hamiltonian_equation symplectic symplectic_skew _ lipschitz vectorField_smooth | |
| parameter parameter_contract parameter_smooth potential potential_smooth (t,z) | |
| (vectorField_hamiltonian (lt_trans (center_bound t z).2 (by norm_num))) v | |
| lemma center_identity (p : ℝ × CV) : center p = p.2 + parameter p.1 • vectorField (center p) := | |
| Midpoint.resolvent_eq _ lipschitz _ _ p | |
| lemma forwardMid_center (p : ℝ × CV) : forwardMid p = center p + parameter p.1 • vectorField (center p) := | |
| Midpoint.step_center _ lipschitz _ _ p | |
| lemma center_average (p : ℝ × CV) : (2:ℝ) • center p = p.2 + forwardMid p := by | |
| dsimp [forwardMid, Midpoint.step, center] | |
| module | |
| lemma center_zero_coordinate (p : ℝ × CV) : center p 0 = p.2 0 := by | |
| have h := congrFun (center_identity p) 0 | |
| simpa only [Pi.add_apply, Pi.smul_apply, vectorField_zero_coordinate, smul_zero, add_zero] using h | |
| lemma forwardMid_zero_coordinate (p : ℝ × CV) : forwardMid p 0 = p.2 0 := by | |
| rw [forwardMid_center] | |
| simp only [Pi.add_apply, Pi.smul_apply, vectorField_zero_coordinate, smul_zero, add_zero, | |
| center_zero_coordinate] | |
| end ArnoldCounterexample.Construction | |
| end | |
| end OAI | |