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720fa25 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | 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
end
end OAI
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