| import OAI.Geometry.IsometricImmersion.Flows.FlowThirdVariation |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory Function |
| open scoped ContDiff Topology Interval |
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| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
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| def timeFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ := deriv (Y p.2) p.1 |
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| def timeSecondFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ := |
| deriv (fun u => deriv (Y p.2) u) p.1 |
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| def mixedFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ := |
| deriv (fun u => timeFlowDerivative Y (p.1, u)) p.2 |
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|
| section ActualFlow |
| variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} |
| variable (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| variable (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| variable (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) |
| variable (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) |
| variable (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) |
| include hq hU hSU hY hrange hstart hode |
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| theorem cap_flow_time_hasDerivAt {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (Y s) (-q (coordinatePoint t (Y s t))) t := by |
| have hsmooth := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode |
| have htime : DifferentiableAt ℝ (Y s) t := |
| ((hsmooth.contDiffAt ((pairRectangle_isOpen 2 (-2) 2).mem_nhds ⟨ht, hs⟩)).differentiableAt |
| (by simp)).comp t (differentiableAt_id.prodMk (differentiableAt_const s)) |
| have hderiv := (hode s ⟨hs.1.le, hs.2.le⟩ t ⟨ht.1.le, ht.2.le⟩).hasDerivAt |
| (Icc_mem_nhds ht.1 ht.2) |
| simpa only [hderiv.deriv] using htime.hasDerivAt |
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| theorem timeFlowDerivative_eq {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| timeFlowDerivative Y (t, s) = -q (coordinatePoint t (Y s t)) := |
| (cap_flow_time_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv |
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| theorem flowPullback_hasDerivAt_t {C : Coord → ℝ} (hC : ContDiffOn ℝ ∞ C U) |
| {s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun u => C (coordinatePoint u (Y s u))) |
| (coordPartial 0 C (coordinatePoint t (Y s t)) - |
| q (coordinatePoint t (Y s t)) * coordPartial 1 C (coordinatePoint t (Y s t))) t := by |
| have hT : HasDerivAt (fun u => coordinatePoint u (Y s u)) |
| (Pi.single 0 (1 : ℝ) - q (coordinatePoint t (Y s t)) • Pi.single 1 (1 : ℝ) : Coord) t := by |
| convert |
| ((hasDerivAt_id t).smul_const (Pi.single 0 (1 : ℝ) : Coord)).add |
| ((cap_flow_time_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).smul_const |
| (Pi.single 1 (1 : ℝ) : Coord)) using 1 <;> |
| first | rfl | simp only [coordinatePoint, one_smul, neg_smul, sub_eq_add_neg] |
| have hp := hSU (flowCoordinateMap_mem_modelSquare (p := (t, s)) hq hU hSU hY hrange hstart hode ⟨ht, hs⟩) |
| have hC' := ((hC.contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)).hasFDerivAt |
| convert hC'.comp_hasDerivAt t hT using 1 <;> |
| first | rfl | simp only [map_sub, map_smul, smul_eq_mul, coordPartial, flowCoordinateMap] |
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| theorem cap_flow_mixed_hasDerivAt {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun u => timeFlowDerivative Y (t, u)) |
| (-coordPartial 1 q (coordinatePoint t (Y s t)) * initialFlowDerivative Y (t, s)) s := by |
| have hd := (flowPullback_hasDerivAt_y hq hU hSU hY hrange hstart hode hq hs ht).neg |
| have he : (fun u => timeFlowDerivative Y (t, u)) =ᶠ[𝓝 s] |
| (fun u => -q (coordinatePoint t (Y u t))) := by |
| filter_upwards [isOpen_Ioo.mem_nhds hs] with u hu |
| exact timeFlowDerivative_eq hq hU hSU hY hrange hstart hode hu ht |
| convert hd.congr_of_eventuallyEq he using 1 |
| first | rfl | (dsimp [initialFlowDerivative]; ring) |
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| theorem mixedFlowDerivative_eq {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| mixedFlowDerivative Y (t, s) = |
| -coordPartial 1 q (coordinatePoint t (Y s t)) * initialFlowDerivative Y (t, s) := |
| (cap_flow_mixed_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv |
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| theorem cap_flow_time_second_hasDerivAt {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun u => timeFlowDerivative Y (u, s)) |
| (-coordPartial 0 q (coordinatePoint t (Y s t)) + |
| coordPartial 1 q (coordinatePoint t (Y s t)) * q (coordinatePoint t (Y s t))) t := by |
| have hd := (flowPullback_hasDerivAt_t hq hU hSU hY hrange hstart hode hq hs ht).neg |
| have he : (fun u => timeFlowDerivative Y (u, s)) =ᶠ[𝓝 t] |
| (fun u => -q (coordinatePoint u (Y s u))) := by |
| filter_upwards [isOpen_Ioo.mem_nhds ht] with u hu |
| exact timeFlowDerivative_eq hq hU hSU hY hrange hstart hode hs hu |
| convert hd.congr_of_eventuallyEq he using 1 |
| first | rfl | ring |
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| theorem timeSecondFlowDerivative_eq {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| timeSecondFlowDerivative Y (t, s) = |
| -coordPartial 0 q (coordinatePoint t (Y s t)) + |
| coordPartial 1 q (coordinatePoint t (Y s t)) * q (coordinatePoint t (Y s t)) := |
| (cap_flow_time_second_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).deriv |
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| theorem cap_flow_mixed_initial_hasDerivAt {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun u => mixedFlowDerivative Y (t, u)) |
| (-coordPartial 1 (coordPartial 1 q) (coordinatePoint t (Y s t)) * |
| (initialFlowDerivative Y (t, s))^2 - |
| coordPartial 1 q (coordinatePoint t (Y s t)) * initialSecondFlowDerivative Y (t, s)) s := by |
| have hdq := flowPullback_hasDerivAt_y hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 1) hs ht |
| have hdV : HasDerivAt (fun u => initialFlowDerivative Y (t, u)) |
| (initialSecondFlowDerivative Y (t, s)) s := |
| (cap_flow_initial_second_hasDerivAt hq hU hSU hY hrange hstart hode hs ht).differentiableAt.hasDerivAt |
| have hd := hdq.neg.mul hdV |
| have he : (fun u => mixedFlowDerivative Y (t, u)) =ᶠ[𝓝 s] |
| (fun u => -coordPartial 1 q (coordinatePoint t (Y u t)) * initialFlowDerivative Y (t, u)) := by |
| filter_upwards [isOpen_Ioo.mem_nhds hs] with u hu |
| exact mixedFlowDerivative_eq hq hU hSU hY hrange hstart hode hu ht |
| convert hd.congr_of_eventuallyEq he using 1 |
| first | rfl | (dsimp [initialFlowDerivative]; ring) |
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| theorem cap_flow_time_second_initial_hasDerivAt {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun u => timeSecondFlowDerivative Y (t, u)) |
| ((-coordPartial 1 (coordPartial 0 q) (coordinatePoint t (Y s t)) + |
| coordPartial 1 (coordPartial 1 q) (coordinatePoint t (Y s t)) * q (coordinatePoint t (Y s t)) + |
| (coordPartial 1 q (coordinatePoint t (Y s t)))^2) * initialFlowDerivative Y (t, s)) s := by |
| have hd0 := flowPullback_hasDerivAt_y hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 0) hs ht |
| have hd1 := flowPullback_hasDerivAt_y hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 1) hs ht |
| have hdq := flowPullback_hasDerivAt_y hq hU hSU hY hrange hstart hode hq hs ht |
| have hd := hd0.neg.add (hd1.mul hdq) |
| have he : (fun u => timeSecondFlowDerivative Y (t, u)) =ᶠ[𝓝 s] |
| (fun u => -coordPartial 0 q (coordinatePoint t (Y u t)) + |
| coordPartial 1 q (coordinatePoint t (Y u t)) * q (coordinatePoint t (Y u t))) := by |
| filter_upwards [isOpen_Ioo.mem_nhds hs] with u hu |
| exact timeSecondFlowDerivative_eq hq hU hSU hY hrange hstart hode hu ht |
| convert hd.congr_of_eventuallyEq he using 1 |
| first | rfl | (dsimp [initialFlowDerivative]; ring) |
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| theorem cap_flow_time_third_hasDerivAt {s t : ℝ} |
| (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) : |
| HasDerivAt (fun u => timeSecondFlowDerivative Y (u, s)) |
| (-coordPartial 0 (coordPartial 0 q) (coordinatePoint t (Y s t)) + |
| q (coordinatePoint t (Y s t)) * coordPartial 1 (coordPartial 0 q) (coordinatePoint t (Y s t)) + |
| (coordPartial 0 (coordPartial 1 q) (coordinatePoint t (Y s t)) - |
| q (coordinatePoint t (Y s t)) * coordPartial 1 (coordPartial 1 q) (coordinatePoint t (Y s t))) * |
| q (coordinatePoint t (Y s t)) + |
| coordPartial 1 q (coordinatePoint t (Y s t)) * |
| (coordPartial 0 q (coordinatePoint t (Y s t)) - |
| q (coordinatePoint t (Y s t)) * coordPartial 1 q (coordinatePoint t (Y s t)))) t := by |
| have hd0 := flowPullback_hasDerivAt_t hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 0) hs ht |
| have hd1 := flowPullback_hasDerivAt_t hq hU hSU hY hrange hstart hode (partial_contDiffOn hq hU 1) hs ht |
| have hdq := flowPullback_hasDerivAt_t hq hU hSU hY hrange hstart hode hq hs ht |
| have hd := hd0.neg.add (hd1.mul hdq) |
| have he : (fun u => timeSecondFlowDerivative Y (u, s)) =ᶠ[𝓝 t] |
| (fun u => -coordPartial 0 q (coordinatePoint u (Y s u)) + |
| coordPartial 1 q (coordinatePoint u (Y s u)) * q (coordinatePoint u (Y s u))) := by |
| filter_upwards [isOpen_Ioo.mem_nhds ht] with u hu |
| exact timeSecondFlowDerivative_eq hq hU hSU hY hrange hstart hode hs hu |
| convert hd.congr_of_eventuallyEq he using 1 |
| first | rfl | ring |
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| end ActualFlow |
| end SmoothLocal.Flow |
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| end |
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| end OAI |
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