| import OAI.Geometry.Arnold.Parameter |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators ContDiff |
| open scoped Topology NNReal |
| open Set Filter |
|
|
| namespace ArnoldCounterexample.Field |
| open ArnoldCounterexample.Quadric ArnoldCounterexample.ThreePoints ArnoldCounterexample.Factors |
| open scoped ContDiff Topology Matrix |
| open Set Filter |
|
|
| lemma casimir_unit (z : UnitQuadric) (hz : (z:CV) 0 = 0) : casimir z = 1 := by |
| have h := (sphereFactors_mem z hz).1 |
| simpa [casimir, sphereFactors, sqNorm, Fin.sum_univ_succ, add_assoc] using h |
|
|
| lemma pairing_scale (c : ℂ) (z : CV) : pairing (c • z) = Complex.normSq c ^ 2 * pairing z := by |
| simp only [pairing, left_scale, right_scale, Pi.smul_apply, smul_eq_mul, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
|
|
| lemma casimir_scale (c : ℂ) (z : CV) : casimir (c • z) = Complex.normSq c ^ 2 * casimir z := by |
| simp only [casimir, left_scale, Pi.smul_apply, smul_eq_mul, mul_pow, Finset.mul_sum] |
|
|
| lemma mixed_scale (c : ℂ) (z : CV) : mixedField (c • z) = Complex.normSq c • (c • mixedField z) := by |
| simp only [mixedField, left_scale, right_scale, Pi.smul_apply, smul_eq_mul, left_phase, right_phase, |
| Finset.smul_sum, smul_add, smul_comm c, smul_smul] |
|
|
| lemma self_scale (c : ℂ) (z : CV) : selfField (c • z) = Complex.normSq c • (c • selfField z) := by |
| simp only [selfField, left_scale, Pi.smul_apply, smul_eq_mul, left_phase, |
| Finset.smul_sum, smul_comm c, smul_smul] |
|
|
| lemma core_unit (z : UnitQuadric) (hz : (z:CV) 0 = 0) : |
| core z = leftOperator 0 z - rightOperator 2 z + mixedField z - pairing z • selfField z := by |
| simp only [core, casimir_unit z hz, radial_at_one.1, radial_at_one.2, one_smul] |
| module |
|
|
| lemma core_homogeneous (z : UnitQuadric) (hz : (z:CV) 0 = 0) (c : ℂ) |
| (hc : Complex.normSq c ∈ Ioo (1/2:ℝ) 2) : core (c • (z:CV)) = c • core z := by |
| let r := Complex.normSq c |
| have hr : r ≠ 0 := ne_of_gt (lt_trans (by norm_num) hc.1) |
| have he : 2 * (-(2*r^3)⁻¹) * (r^2 * pairing z) * r = -pairing z := by |
| field_simp |
| rw [core, casimir_scale, casimir_unit z hz, mul_one, |
| (radial_squared r hc).1, (radial_squared r hc).2, pairing_scale, mixed_scale, self_scale, |
| core_unit z hz] |
| simp only [left_phase, right_phase, smul_add, smul_sub, smul_smul, ← smul_comm c] |
| change c • leftOperator 0 z - c • rightOperator 2 z + c • ((r⁻¹*r) • mixedField z) + |
| c • ((2 * (-(2*r^3)⁻¹) * (r^2 * pairing z) * r) • selfField z) = _ |
| rw [inv_mul_cancel₀ hr, one_smul, he, neg_smul] |
| module |
|
|
| lemma left_bracket (i j : Fin 3) (z : CV) : symplectic (leftOperator i z) (leftOperator j z) = |
| 2 * (![![0,leftSphere z 2,-leftSphere z 1], ![-leftSphere z 2,0,leftSphere z 0], |
| ![leftSphere z 1,-leftSphere z 0,0]] i j) := by |
| fin_cases i <;> fin_cases j <;> |
| simp [symplectic_apply, omega, leftOperator, rotor_apply, Pi.single_apply, |
| leftSphere, moment_real, Fin.sum_univ_succ, Complex.mul_im] <;> ring |
|
|
| lemma right_bracket (i j : Fin 3) (z : CV) : symplectic (rightOperator i z) (rightOperator j z) = |
| -2 * (![![0,rightSphere z 2,-rightSphere z 1], ![-rightSphere z 2,0,rightSphere z 0], |
| ![rightSphere z 1,-rightSphere z 0,0]] i j) := by |
| fin_cases i <;> fin_cases j <;> |
| simp [symplectic_apply, omega, rightOperator, rotor_apply, Pi.single_apply, |
| rightSphere, moment_real, Fin.sum_univ_succ, Complex.mul_im] <;> ring |
|
|
| lemma mixed_bracket (i j : Fin 3) (z : CV) : symplectic (leftOperator i z) (rightOperator j z) = 0 := by |
| fin_cases i <;> fin_cases j <;> |
| simp [symplectic_apply, omega, leftOperator, rightOperator, rotor_apply, Pi.single_apply, |
| Fin.sum_univ_succ, Complex.mul_im] <;> ring |
|
|
| lemma left_core_derivative (z : UnitQuadric) (hz : (z:CV) 0 = 0) (i : Fin 3) : |
| symplectic (leftOperator i z) (core z) = 2 * ((east + rightSphere z) ⨯₃ leftSphere z) i := by |
| rw [core_unit z hz] |
| simp only [mixedField, selfField, map_sub, map_add, map_sum, map_smul, smul_eq_mul, |
| left_bracket, mixed_bracket, mul_zero, add_zero, sub_zero] |
| fin_cases i <;> simp [Fin.sum_univ_succ, crossProduct, east] <;> ring |
|
|
| lemma right_core_derivative (z : UnitQuadric) (hz : (z:CV) 0 = 0) (i : Fin 3) : |
| symplectic (rightOperator i z) (core z) = -2 * ((leftSphere z - pole) ⨯₃ rightSphere z) i := by |
| have hmix (i j : Fin 3) : symplectic (rightOperator i z) (leftOperator j z) = 0 := by |
| rw [symplectic_skew, mixed_bracket, neg_zero] |
| rw [core_unit z hz] |
| simp only [mixedField, selfField, map_sub, map_add, map_sum, map_smul, smul_eq_mul, |
| right_bracket, hmix, mul_zero, zero_add] |
| fin_cases i <;> simp [Fin.sum_univ_succ, crossProduct, pole] <;> ring |
|
|
| lemma cross_zero_multiplier (x a : R3) (hx : sqNorm x = 1) (ha : a ⨯₃ x = 0) : |
| a = (a ⬝ᵥ x) • x := by |
| have h := congrArg (fun y : R3 ↦ x ⨯₃ y) ha |
| rw [cross_cross_eq_smul_sub_smul', map_zero] at h |
| have hxx : x ⬝ᵥ x = 1 := by simpa [dotProduct, sqNorm, Fin.sum_univ_succ, pow_two, add_assoc] using hx |
| rw [hxx, one_smul] at h |
| exact sub_eq_zero.mp h |
|
|
| lemma core_vertical_critical (z : UnitQuadric) (hz : (z:CV) 0 = 0) (a : ℝ) |
| (ha : core z = a • (Complex.I • (z:CV))) : MultiplierCritical (sphereFactors z) := by |
| have hσ (v : CV) (hv : (bilinear (star (z:CV)) v).re = 0) : |
| symplectic v (Complex.I • (z:CV)) = 0 := by |
| rw [symplectic_apply] |
| have he : omega v (Complex.I • (z:CV)) = 2 * (bilinear (star (z:CV)) v).re := by |
| simp only [omega, bilinear, Pi.star_apply, Pi.smul_apply, smul_eq_mul, |
| Complex.im_sum, Complex.re_sum, Complex.mul_im, Complex.mul_re, |
| Complex.I_re, Complex.I_im, Complex.star_def, Complex.conj_re, Complex.conj_im] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| rw [he, hv, mul_zero] |
| have hi (i : Fin 3) := hσ (leftOperator i z) (left_skew i (z:CV)).1 |
| have hj (i : Fin 3) := hσ (rightOperator i z) (right_skew i (z:CV)).1 |
| have hL : (east + rightSphere z) ⨯₃ leftSphere z = 0 := by |
| ext i |
| have h := left_core_derivative z hz i |
| rw [ha, map_smul, hi, smul_zero] at h |
| simpa only [Pi.zero_apply] using (mul_eq_zero.mp h.symm).resolve_left (by norm_num) |
| have hR : (leftSphere z - pole) ⨯₃ rightSphere z = 0 := by |
| ext i |
| have h := right_core_derivative z hz i |
| rw [ha, map_smul, hj, smul_zero] at h |
| simpa only [Pi.zero_apply] using (mul_eq_zero.mp h.symm).resolve_left (by norm_num) |
| have hmem := sphereFactors_mem z hz |
| have eL := cross_zero_multiplier (leftSphere z) (east+rightSphere z) hmem.1 hL |
| have eR := cross_zero_multiplier (rightSphere z) (leftSphere z-pole) hmem.2 hR |
| refine ⟨hmem, (east+rightSphere z) ⬝ᵥ leftSphere z, |
| (leftSphere z-pole) ⬝ᵥ rightSphere z, ?_⟩ |
| have eL0 := congrFun eL 0; have eL1 := congrFun eL 1; have eL2 := congrFun eL 2 |
| have eR0 := congrFun eR 0; have eR1 := congrFun eR 1; have eR2 := congrFun eR 2 |
| simpa [sphereFactors, east, pole] using And.intro eL0 (And.intro eL1 (And.intro eL2 |
| (And.intro eR0 (And.intro eR1 eR2)))) |
|
|
| lemma core_complex_vertical_critical (z : UnitQuadric) (hz : (z:CV) 0 = 0) (c : ℂ) |
| (hc : core z = c • (z:CV)) : MultiplierCritical (sphereFactors z) := by |
| have hr := (core_skew (z:CV)).1 |
| rw [hc, bilinear_smul_right, bilinear_star_self, z.property.1] at hr |
| simp only [Complex.ofReal_one, mul_one] at hr |
| apply core_vertical_critical z hz c.im |
| rw [hc] |
| have he : c = (c.im : ℂ) * Complex.I := by |
| apply Complex.ext <;> simp [hr] |
| rw [he, mul_smul] |
| ext i |
| simp [Complex.real_smul] |
|
|
| end ArnoldCounterexample.Field |
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| end |
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| end OAI |
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