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59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Flows.FlowBounds
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
namespace OAI
noncomputable section
open Set Filter MeasureTheory Function
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
def triangularLinearEquiv (a b : ℝ) (hb : b ≠ 0) : (ℝ × ℝ) ≃L[ℝ] (ℝ × ℝ) where
toFun v := (v.1, a * v.1 + b * v.2)
invFun w := (w.1, (w.2 - a * w.1) / b)
left_inv v := by
apply Prod.ext
· rfl
· dsimp
field_simp [hb]
ring
right_inv w := by
apply Prod.ext
· rfl
· dsimp
field_simp [hb]
ring
map_add' v w := by
apply Prod.ext <;> simp only [Prod.fst_add, Prod.snd_add]
ring
map_smul' c v := by
apply Prod.ext <;> simp only [Prod.smul_fst, Prod.smul_snd, smul_eq_mul, RingHom.id_apply]
ring
continuous_toFun := by fun_prop
continuous_invFun := by fun_prop
variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
theorem triangularFlow_hasFDerivAt_equiv
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
Y s t ∈ Icc (-3 : ℝ) 3)
(hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
{p : ℝ × ℝ} (hp : p ∈ pairRectangle 2 (-2) 2) :
∃ A : (ℝ × ℝ) ≃L[ℝ] (ℝ × ℝ), HasFDerivAt (triangularFlow Y) (A : (ℝ × ℝ) →L[ℝ] (ℝ × ℝ)) p := by
let F (v : ℝ × ℝ) := Y v.2 v.1
let a := -q (coordinatePoint p.1 (Y p.2 p.1))
let b := Real.exp (-(∫ r in 0..p.1, coordPartial 1 q (coordinatePoint r (Y p.2 r))))
have hF : ContDiffOn ℝ ∞ F (pairRectangle 2 (-2) 2) :=
cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hd := ((hF.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds hp)).differentiableAt
(by simp)).hasFDerivAt
have hdx : HasDerivAt (fun t => F (t, p.2)) a p.1 :=
(hode p.2 ⟨hp.2.1.le, hp.2.2.le⟩ p.1 ⟨hp.1.1.le, hp.1.2.le⟩).hasDerivAt
(Icc_mem_nhds hp.1.1 hp.1.2)
have hdy : HasDerivAt (fun s => F (p.1, s)) b p.2 :=
cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hp.2 hp.1
have hdx' : HasDerivAt (fun t => F (t, p.2)) (fderiv ℝ F p (1, 0)) p.1 := by
convert hd.comp_hasDerivAt p.1 ((hasDerivAt_id p.1).prodMk
(hasDerivAt_const p.1 p.2)) using 1
rfl
have hdy' : HasDerivAt (fun s => F (p.1, s)) (fderiv ℝ F p (0, 1)) p.2 := by
convert hd.comp_hasDerivAt p.2 ((hasDerivAt_const p.2 p.1).prodMk
(hasDerivAt_id p.2)) using 1
rfl
have hx : fderiv ℝ F p (1, 0) = a := hdx'.unique hdx
have hy : fderiv ℝ F p (0, 1) = b := hdy'.unique hdy
have hdeq : fderiv ℝ F p = jointDifferential a b := by
apply ContinuousLinearMap.ext
intro v
have hv : v = v.1 • ((1, 0) : ℝ × ℝ) + v.2 • ((0, 1) : ℝ × ℝ) := by
ext <;> simp
calc
fderiv ℝ F p v = v.1 * a + v.2 * b := by
conv_lhs => rw [hv]
simp only [map_add, map_smul, smul_eq_mul, hx, hy]
_ = jointDifferential a b v := by simp [jointDifferential]
have hFD : HasFDerivAt F (jointDifferential a b) p := hd.congr_fderiv hdeq
have hb : b ≠ 0 := (Real.exp_pos _).ne'
refine ⟨triangularLinearEquiv a b hb, ?_⟩
have hTF := (hasFDerivAt_fst (𝕜 := ℝ) (p := p)).prodMk hFD
convert hTF using 1 <;> first
| rfl
| (ext <;> simp [triangularLinearEquiv, jointDifferential])
theorem exists_triangularFlow_smooth_chart
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
Y s t ∈ Icc (-3 : ℝ) 3)
(hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
{p : ℝ × ℝ} (hp : p ∈ pairRectangle 2 (-2) 2) :
∃ e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ),
p ∈ e.source ∧ e.source ⊆ pairRectangle 2 (-2) 2 ∧
(e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y ∧
ContDiffOn ℝ ∞ e e.source ∧ ContDiffOn ℝ ∞ e.symm e.target := by
let D := pairRectangle 2 (-2) 2
have hD : IsOpen D := pairRectangle_isOpen 2 (-2) 2
have hTF : ContDiffOn ℝ ∞ (triangularFlow Y) D :=
contDiffOn_fst.prodMk (cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode)
have hTFp := hTF.contDiffAt (hD.mem_nhds hp)
obtain ⟨A, hA⟩ := triangularFlow_hasFDerivAt_equiv hq hU hSU hY hrange hstart hode hp
let e0 := hTFp.toOpenPartialHomeomorph (triangularFlow Y) hA (by simp)
let e := e0.restrOpen D hD
have heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y := rfl
have hsub : e.source ⊆ D := fun _ hx => hx.2
refine ⟨e, ⟨hTFp.mem_toOpenPartialHomeomorph_source hA (by simp), hp⟩,
hsub, heq, ?_, ?_⟩
· rw [heq]
exact hTF.mono hsub
· intro y hy
have hx : e.symm y ∈ D := hsub (e.map_target hy)
obtain ⟨L, hL⟩ := triangularFlow_hasFDerivAt_equiv hq hU hSU hY hrange hstart hode hx
have hLe : HasFDerivAt e (L : (ℝ × ℝ) →L[ℝ] (ℝ × ℝ)) (e.symm y) := by
rw [heq]
exact hL
have hfe : ContDiffAt ℝ ∞ e (e.symm y) := by
rw [heq]
exact hTF.contDiffAt (hD.mem_nhds hx)
exact (e.contDiffAt_symm hy hLe hfe).contDiffWithinAt
end SmoothLocal.Flow
end
end OAI
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