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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 | import OAI.Geometry.IsometricImmersion.Flows.FlowThirdVariation
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 timeFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ := deriv (Y p.2) p.1
def timeSecondFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
deriv (fun u => deriv (Y p.2) u) p.1
def mixedFlowDerivative (Y : ℝ → ℝ → ℝ) (p : ℝ × ℝ) : ℝ :=
deriv (fun u => timeFlowDerivative Y (p.1, u)) p.2
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
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
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
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]
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)
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
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
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
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)
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)
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
end ActualFlow
end SmoothLocal.Flow
end
end OAI
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