| 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 |
|
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| end OAI |
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