| import Mathlib | |
| import OAI.Geometry.BallPacking.Forms.HorizontalCoupling | |
| namespace OAI | |
| noncomputable section | |
| namespace PackingSufficiencySupport.Hamiltonian | |
| section | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem manifoldExteriorOneForm_sub {α β : ManifoldOneForm E M} | |
| (hα : ∀ c,ContDiffOn ℝ ∞ (chartOneForm α c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (hβ : ∀ c,ContDiffOn ℝ ∞ (chartOneForm β c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (x : M) : manifoldExteriorOneForm (α-β) x= | |
| manifoldExteriorOneForm α x-manifoldExteriorOneForm β x := by | |
| have hx : extChartAt 𝓘(ℝ,E) x x∈(extChartAt 𝓘(ℝ,E) x).target := | |
| (extChartAt 𝓘(ℝ,E) x).map_source (mem_extChartAt_source x) | |
| have hn := (isOpen_extChartAt_target (I := 𝓘(ℝ,E)) x).mem_nhds hx | |
| have he : chartOneForm (α-β) x=chartOneForm α x-chartOneForm β x := | |
| funext (chartOneForm_sub α β x) | |
| simp only [manifoldExteriorOneForm,euclideanExteriorOneForm,he, | |
| fderiv_sub ((hα x).contDiffAt hn |>.differentiableAt (by simp)) | |
| ((hβ x).contDiffAt hn |>.differentiableAt (by simp))] | |
| apply ContinuousLinearMap.ext | |
| intro u | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| simp only [ContinuousLinearMap.bilinearComp_apply,sub_apply, | |
| ContinuousLinearMap.flip_apply] | |
| ring | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem manifoldExteriorOneForm_smul (a : ℝ) {α : ManifoldOneForm E M} | |
| (hα : ∀ c,ContDiffOn ℝ ∞ (chartOneForm α c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (x : M) : manifoldExteriorOneForm (a • α) x=a • manifoldExteriorOneForm α x := by | |
| have hx : extChartAt 𝓘(ℝ,E) x x∈(extChartAt 𝓘(ℝ,E) x).target := | |
| (extChartAt 𝓘(ℝ,E) x).map_source (mem_extChartAt_source x) | |
| have hn := (isOpen_extChartAt_target (I := 𝓘(ℝ,E)) x).mem_nhds hx | |
| have he : chartOneForm (a • α) x=a • chartOneForm α x := funext (chartOneForm_smul a α x) | |
| simp only [manifoldExteriorOneForm,he,euclideanExteriorOneForm_smul a | |
| ((hα x).contDiffAt hn |>.differentiableAt (by simp))] | |
| rfl | |
| theorem manifold_closed_exact_normalization {α β : ManifoldOneForm E M} | |
| (hα : ∀ c,ContDiffOn ℝ ∞ (chartOneForm α c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (hβ : ∀ c,ContDiffOn ℝ ∞ (chartOneForm β c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (hclosed : ∀ x,manifoldExteriorOneForm β x=0) | |
| {F : M → ℝ} (hF : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ F) (a : ℝ) (x : M) : | |
| manifoldExteriorOneForm (α-a • β-manifoldScalarDifferential F) x= | |
| manifoldExteriorOneForm α x := by | |
| have hb : ∀ c,ContDiffOn ℝ ∞ (chartOneForm (a • β) c) | |
| (extChartAt 𝓘(ℝ,E) c).target := by | |
| intro c | |
| simpa only [show chartOneForm (a • β) c=fun y => a • chartOneForm β c y | |
| from funext (chartOneForm_smul a β c)] using (hβ c).const_smul a | |
| have hab : ∀ c,ContDiffOn ℝ ∞ (chartOneForm (α-a • β) c) | |
| (extChartAt 𝓘(ℝ,E) c).target := by | |
| intro c | |
| exact ((hα c).sub (hb c)).congr (fun y _ => chartOneForm_sub α (a • β) c y) | |
| rw [manifoldExteriorOneForm_sub hab (manifoldScalarDifferential_smooth hF), | |
| manifoldScalarDifferential_closed hF,manifoldExteriorOneForm_sub hα hb, | |
| manifoldExteriorOneForm_smul a hβ,hclosed] | |
| apply ContinuousLinearMap.ext | |
| intro u | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| simp | |
| end | |
| section | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] | |
| {M : Type*} [TopologicalSpace M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] | |
| [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| theorem exists_smooth_manifold_cutoff {K U : Set M} | |
| (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) : | |
| ∃ χ : M → ℝ, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ χ ∧ HasCompactSupport χ ∧ | |
| tsupport χ ⊆ U ∧ (∀ᶠ x in 𝓝ˢ K, χ x=1) ∧ (∀ x, χ x ∈ Icc (0:ℝ) 1) := by | |
| let : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace E M | |
| obtain ⟨D,hD,hKD,hDU⟩ := exists_compact_between hK hU hKU | |
| obtain ⟨χ,hχ1,hχ0,hχ⟩ := exists_contMDiffMap_one_nhds_of_subset_interior | |
| (n := (⊤ : ℕ∞)) 𝓘(ℝ,E) hK.isClosed hKD | |
| have hs : tsupport (χ : M → ℝ) ⊆ D := | |
| closure_minimal (fun x hx => by by_contra hn; exact hx (hχ0 x hn)) hD.isClosed | |
| exact ⟨χ,χ.contMDiff,HasCompactSupport.of_support_subset_isCompact hD | |
| ((subset_tsupport χ).trans hs),hs.trans hDU,hχ1,hχ⟩ | |
| end | |
| section | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem manifold_local_potential_smooth {A : Set M} (hA : IsOpen A) | |
| {F ζ : M → ℝ} (hF : ContMDiffOn 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ F A) | |
| (hζ : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ ζ) (hs : tsupport ζ ⊆ A) : | |
| ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (fun x => ζ x*F x) := by | |
| intro x | |
| by_cases hx : x∈A | |
| · exact hζ.contMDiffAt.mul (hF.contMDiffAt (hA.mem_nhds hx)) | |
| · have ht : x∉tsupport ζ := fun ht => hx (hs ht) | |
| apply (contMDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq | |
| filter_upwards [isClosed_closure.isOpen_compl.mem_nhds ht] with y hy | |
| have hz : ζ y=0 := notMem_support.mp (fun hv => hy (subset_closure hv)) | |
| simp only [hz,zero_mul] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem scalar_product_tsupport_left (F ζ : M → ℝ) : | |
| tsupport (fun x => ζ x*F x) ⊆ tsupport ζ := by | |
| apply closure_minimal _ isClosed_closure | |
| intro x hx | |
| apply subset_closure | |
| intro hzero | |
| exact hx (by simp only [hzero,zero_mul]) | |
| variable [FiniteDimensional ℝ E] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] | |
| theorem exists_global_local_potential {K A : Set M} (hK : IsCompact K) | |
| (hA : IsOpen A) (hKA : K ⊆ A) {F : M → ℝ} | |
| (hF : ContMDiffOn 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ F A) : | |
| ∃ G : M → ℝ, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ G ∧ | |
| HasCompactSupport G ∧ tsupport G ⊆ A ∧ | |
| (∀ᶠ x in 𝓝ˢ K,G x=F x) ∧ | |
| HasCompactSupport (manifoldScalarDifferential (E := E) G) ∧ | |
| support (manifoldScalarDifferential (E := E) G) ⊆ A ∧ | |
| (∀ x,manifoldExteriorOneForm (manifoldScalarDifferential (E := E) G) x=0) ∧ | |
| (∀ x ∈ K,manifoldScalarDifferential (E := E) G x= | |
| manifoldScalarDifferential (E := E) F x) := by | |
| obtain ⟨ζ,hζ,hζc,hζA,hζ1,_⟩ := exists_smooth_manifold_cutoff (E := E) hK hA hKA | |
| let G : M → ℝ := fun x => ζ x*F x | |
| have hG : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ G := | |
| manifold_local_potential_smooth hA hF hζ hζA | |
| have hs : tsupport G ⊆ tsupport ζ := scalar_product_tsupport_left F ζ | |
| have hGc : HasCompactSupport G := | |
| HasCompactSupport.of_support_subset_isCompact hζc (subset_closure.trans hs) | |
| have hEq : ∀ᶠ x in 𝓝ˢ K,G x=F x := by | |
| filter_upwards [hζ1] with x hx | |
| simp only [G,hx,one_mul] | |
| have hd : support (manifoldScalarDifferential (E := E) G) ⊆ tsupport G := | |
| manifoldScalarDifferential_support G | |
| refine ⟨G,hG,hGc,hs.trans hζA,hEq, | |
| HasCompactSupport.of_support_subset_isCompact hGc hd,hd.trans (hs.trans hζA), | |
| manifoldScalarDifferential_closed hG,?_⟩ | |
| intro x hx | |
| have he : G =ᶠ[𝓝 x] F := hEq.filter_mono (nhds_le_nhdsSet hx) | |
| exact he.mfderiv_eq | |
| end | |
| section | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {P E : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] | |
| theorem manifoldFamily_local_potential_smoothOn {V : Set P} {A : Set M} | |
| (hV : IsOpen V) (hA : IsOpen A) {F : P × M → ℝ} {ζ : M → ℝ} | |
| (hF : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ F (V ×ˢ A)) | |
| (hζ : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ ζ) (hs : tsupport ζ ⊆ A) : | |
| ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ | |
| (fun q : P × M => ζ q.2*F q) (V ×ˢ univ) := by | |
| intro q hq | |
| by_cases hy : q.2∈A | |
| · exact ((hζ.comp contMDiff_snd).contMDiffAt.mul | |
| (hF.contMDiffAt ((hV.prod hA).mem_nhds ⟨hq.1,hy⟩))).contMDiffWithinAt | |
| · have ht : q.2∉tsupport ζ := fun ht => hy (hs ht) | |
| have he : (fun y : P × M => ζ y.2*F y) =ᶠ[𝓝 q] (fun _ => (0:ℝ)) := by | |
| have hn := (continuous_snd.continuousAt : ContinuousAt (Prod.snd : P × M → M) q) | |
| filter_upwards [hn (isClosed_closure.isOpen_compl.mem_nhds ht)] with y hy | |
| have hz : ζ y.2=0 := notMem_support.mp (fun hv => hy (subset_closure hv)) | |
| simp only [hz,zero_mul] | |
| exact ((contMDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq he).contMDiffWithinAt | |
| variable [FiniteDimensional ℝ E] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| [T2Space M] [NormalSpace M] [SigmaCompactSpace M] | |
| theorem exists_joint_global_local_potential {V : Set P} {K A : Set M} | |
| (hV : IsOpen V) (hK : IsCompact K) (hA : IsOpen A) (hKA : K ⊆ A) | |
| {F : P × M → ℝ} | |
| (hF : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ F (V ×ˢ A)) : | |
| ∃ (G : P × M → ℝ) (L : Set M), IsCompact L ∧ L ⊆ A ∧ | |
| ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ G (V ×ˢ univ) ∧ | |
| (∀ p,tsupport (fun y => G (p,y)) ⊆ L) ∧ | |
| (∀ᶠ y in 𝓝ˢ K,∀ p,G (p,y)=F (p,y)) ∧ | |
| (∀ p,support (manifoldScalarDifferential (E := E) (fun y => G (p,y))) ⊆ L) ∧ | |
| (∀ p ∈ V,∀ y,manifoldExteriorOneForm | |
| (manifoldScalarDifferential (E := E) (fun z => G (p,z))) y=0) ∧ | |
| (∀ p,∀ y ∈ K,manifoldScalarDifferential (E := E) (fun z => G (p,z)) y= | |
| manifoldScalarDifferential (E := E) (fun z => F (p,z)) y) := by | |
| obtain ⟨ζ,hζ,hζc,hζA,hζ1,_⟩ := exists_smooth_manifold_cutoff (E := E) hK hA hKA | |
| let G : P × M → ℝ := fun q => ζ q.2*F q | |
| have hG := manifoldFamily_local_potential_smoothOn hV hA hF hζ hζA | |
| have hs : ∀ p,tsupport (fun y => G (p,y)) ⊆ tsupport ζ := | |
| fun p => scalar_product_tsupport_left (fun y => F (p,y)) ζ | |
| have he : ∀ᶠ y in 𝓝ˢ K,∀ p,G (p,y)=F (p,y) := by | |
| filter_upwards [hζ1] with y hy p | |
| simp only [G,hy,one_mul] | |
| refine ⟨G,tsupport ζ,hζc,hζA,hG,hs,he,?_,?_,?_⟩ | |
| · intro p | |
| exact (manifoldScalarDifferential_support (fun y => G (p,y))).trans (hs p) | |
| · intro p hp y | |
| have hfib : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (fun z => G (p,z)) := by | |
| intro z | |
| exact (hG.contMDiffAt (((hV.prod isOpen_univ)).mem_nhds ⟨hp,mem_univ z⟩)).comp z | |
| (contMDiff_const.prodMk contMDiff_id).contMDiffAt | |
| exact manifoldScalarDifferential_closed hfib y | |
| · intro p y hy | |
| have he' : (fun z => G (p,z)) =ᶠ[𝓝 y] (fun z => F (p,z)) := by | |
| filter_upwards [he.filter_mono (nhds_le_nhdsSet hy)] with z hz | |
| exact hz p | |
| exact he'.mfderiv_eq | |
| end | |
| section | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {P E : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem manifoldFamily_scalar_fiber_smooth {V : Set P} (hV : IsOpen V) | |
| {G : P × M → ℝ} | |
| (hG : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ G (V ×ˢ univ)) | |
| {p : P} (hp : p∈V) : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (fun z => G (p,z)) := by | |
| intro z | |
| exact (hG.contMDiffAt ((hV.prod isOpen_univ).mem_nhds ⟨hp,mem_univ z⟩)).comp z | |
| (contMDiff_const.prodMk contMDiff_id).contMDiffAt | |
| theorem manifoldFamily_scalar_chart_smooth {V : Set P} (hV : IsOpen V) | |
| {G : P × M → ℝ} | |
| (hG : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ G (V ×ˢ univ)) | |
| (c : M) : ContDiffOn ℝ ∞ | |
| (fun q : P × E => G (q.1,(extChartAt 𝓘(ℝ,E) c).symm q.2)) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target) := by | |
| intro q hq | |
| have hi := (contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,E)) (n := ∞) c).contMDiffAt | |
| ((isOpen_extChartAt_target (I := 𝓘(ℝ,E)) c).mem_nhds hq.2) | |
| have hfst : ContMDiffAt 𝓘(ℝ,P × E) 𝓘(ℝ,P) ∞ (Prod.fst : P × E → P) q := | |
| contDiff_fst.contMDiff.contMDiffAt | |
| have hsnd : ContMDiffAt 𝓘(ℝ,P × E) 𝓘(ℝ,E) ∞ (Prod.snd : P × E → E) q := | |
| contDiff_snd.contMDiff.contMDiffAt | |
| have hinc := hfst.prodMk (hi.comp q hsnd) | |
| have hh := (hG.contMDiffAt ((hV.prod isOpen_univ).mem_nhds | |
| (show (q.1,(extChartAt 𝓘(ℝ,E) c).symm q.2)∈V ×ˢ univ from ⟨hq.1,mem_univ _⟩))).comp q hinc | |
| exact hh.contDiffAt.contDiffWithinAt | |
| omit [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem joint_scalar_fderiv_slice {F : P × E → ℝ} {q : P × E} | |
| (hF : DifferentiableAt ℝ F q) : | |
| fderiv ℝ (fun y => F (q.1,y)) q.2= | |
| (fderiv ℝ F q).comp ((0 : E →L[ℝ] P).prod (ContinuousLinearMap.id ℝ E)) := by | |
| have hi : HasFDerivAt (fun y : E => (q.1,y)) | |
| ((0 : E →L[ℝ] P).prod (ContinuousLinearMap.id ℝ E)) q.2 := | |
| (hasFDerivAt_const q.1 q.2).prodMk (hasFDerivAt_id q.2) | |
| exact (hF.hasFDerivAt.comp q.2 hi).fderiv | |
| theorem manifoldFamilyScalarDifferential_smooth {V : Set P} (hV : IsOpen V) | |
| {G : P × M → ℝ} | |
| (hG : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ G (V ×ˢ univ)) | |
| (c : M) : ContDiffOn ℝ ∞ | |
| (fun q : P × E => chartOneForm | |
| (manifoldScalarDifferential (E := E) (fun z => G (q.1,z))) c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target) := by | |
| let F : P × E → ℝ := fun q => G (q.1,(extChartAt 𝓘(ℝ,E) c).symm q.2) | |
| have hF : ContDiffOn ℝ ∞ F (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target) := | |
| manifoldFamily_scalar_chart_smooth hV hG c | |
| intro q hq | |
| have hn := (hV.prod (isOpen_extChartAt_target (I := 𝓘(ℝ,E)) c)).mem_nhds hq | |
| have hdf : ContDiffAt ℝ ∞ (fun z => fderiv ℝ F z) q := | |
| (hF.contDiffAt hn).fderiv_right (by simp) | |
| have hd := hdf.clm_comp | |
| (contDiffAt_const (c := ((0 : E →L[ℝ] P).prod (ContinuousLinearMap.id ℝ E)))) | |
| apply (hd.congr_of_eventuallyEq ?_).contDiffWithinAt | |
| filter_upwards [hn] with z hz | |
| rw [manifoldScalarDifferential_chart (manifoldFamily_scalar_fiber_smooth hV hG hz.1) hz.2] | |
| exact joint_scalar_fderiv_slice ((hF.contDiffAt | |
| ((hV.prod (isOpen_extChartAt_target (I := 𝓘(ℝ,E)) c)).mem_nhds hz)).differentiableAt (by simp)) | |
| def normalizedManifoldPrimitive (Γ : P → ManifoldOneForm E M) | |
| (β : ManifoldOneForm E M) (a : P → ℝ) (G : P × M → ℝ) (p : P) : | |
| ManifoldOneForm E M := Γ p-a p • β-manifoldScalarDifferential (fun y => G (p,y)) | |
| omit [NormedAddCommGroup P] [NormedSpace ℝ P] [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem normalizedManifoldPrimitive_chart | |
| (Γ : P → ManifoldOneForm E M) (β : ManifoldOneForm E M) | |
| (a : P → ℝ) (G : P × M → ℝ) (p : P) (c : M) (y : E) : | |
| chartOneForm (normalizedManifoldPrimitive Γ β a G p) c y= | |
| chartOneForm (Γ p) c y-a p • chartOneForm β c y- | |
| chartOneForm (manifoldScalarDifferential (fun z => G (p,z))) c y := by | |
| simp only [normalizedManifoldPrimitive,chartOneForm_sub,chartOneForm_smul] | |
| theorem normalizedManifoldPrimitive_smooth {V : Set P} (hV : IsOpen V) | |
| {Γ : P → ManifoldOneForm E M} {β : ManifoldOneForm E M} {a : P → ℝ} | |
| {G : P × M → ℝ} | |
| (hΓ : ∀ c,ContDiffOn ℝ ∞ (fun q : P × E => chartOneForm (Γ q.1) c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) | |
| (hβ : ∀ c,ContDiffOn ℝ ∞ (chartOneForm β c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (ha : ContDiffOn ℝ ∞ a V) | |
| (hG : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ G (V ×ˢ univ)) | |
| (c : M) : ContDiffOn ℝ ∞ | |
| (fun q : P × E => chartOneForm (normalizedManifoldPrimitive Γ β a G q.1) c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target) := by | |
| have hma : ContDiffOn ℝ ∞ (fun q : P × E => a q.1) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target) := | |
| ha.comp contDiff_fst.contDiffOn (fun _ hx => hx.1) | |
| have hmb : ContDiffOn ℝ ∞ (fun q : P × E => chartOneForm β c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target) := | |
| (hβ c).comp contDiff_snd.contDiffOn (fun _ hx => hx.2) | |
| exact (((hΓ c).sub (hma.smul hmb)).sub | |
| (manifoldFamilyScalarDifferential_smooth hV hG c)).congr | |
| (fun q _ => normalizedManifoldPrimitive_chart Γ β a G q.1 c q.2) | |
| variable [FiniteDimensional ℝ E] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] | |
| theorem exists_normalized_manifold_primitive {V : Set P} {K A : Set M} | |
| (hV : IsOpen V) (hK : IsCompact K) (hA : IsOpen A) (hKA : K ⊆ A) | |
| {Γ N : P → ManifoldOneForm E M} {β : ManifoldOneForm E M} | |
| {a : P → ℝ} {F : P × M → ℝ} | |
| (hΓ : ∀ c,ContDiffOn ℝ ∞ (fun q : P × E => chartOneForm (Γ q.1) c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) | |
| (hβ : ∀ c,ContDiffOn ℝ ∞ (chartOneForm β c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (hβclosed : ∀ x,manifoldExteriorOneForm β x=0) | |
| (hβcompact : HasCompactSupport β) (ha : ContDiffOn ℝ ∞ a V) | |
| (hF : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ F (V ×ˢ A)) | |
| (hFd : ∀ p ∈ V,∀ x ∈ A, | |
| manifoldScalarDifferential (E := E) (fun y => F (p,y)) x= | |
| Γ p x-N p x-a p • β x) : | |
| ∃ (Γ₁ : P → ManifoldOneForm E M) (L : Set M), IsCompact L ∧ L ⊆ A ∧ | |
| (∀ c,ContDiffOn ℝ ∞ (fun q : P × E => chartOneForm (Γ₁ q.1) c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) ∧ | |
| (∀ p,support (Γ₁ p-Γ p) ⊆ tsupport β ∪ L) ∧ | |
| (∀ p,HasCompactSupport (Γ₁ p-Γ p)) ∧ | |
| (∀ p ∈ V,∀ x,manifoldExteriorOneForm (Γ₁ p) x=manifoldExteriorOneForm (Γ p) x) ∧ | |
| (∀ p ∈ V,∀ x ∈ K,Γ₁ p x=N p x) := by | |
| obtain ⟨G,L,hLc,hLA,hG,_,_,hGs,_,hGd⟩ := | |
| exists_joint_global_local_potential hV hK hA hKA hF | |
| let Γ₁ := normalizedManifoldPrimitive Γ β a G | |
| have hs : ∀ p,support (Γ₁ p-Γ p) ⊆ tsupport β ∪ L := by | |
| intro p x hx | |
| by_contra hout | |
| have hβz : β x=0 := notMem_support.mp (fun hv => hout (Or.inl (subset_closure hv))) | |
| have hdz : manifoldScalarDifferential (E := E) (fun y => G (p,y)) x=0 := | |
| notMem_support.mp (fun hv => hout (Or.inr (hGs p hv))) | |
| apply hx | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| change Γ p x v-a p*β x v-manifoldScalarDifferential (E := E) (fun y => G (p,y)) x v- | |
| Γ p x v=0 | |
| rw [hβz,hdz] | |
| simp | |
| refine ⟨Γ₁,L,hLc,hLA,normalizedManifoldPrimitive_smooth hV hΓ hβ ha hG,hs, | |
| (fun p => HasCompactSupport.of_support_subset_isCompact (hβcompact.union hLc) (hs p)),?_,?_⟩ | |
| · intro p hp x | |
| have hΓp : ∀ c,ContDiffOn ℝ ∞ (chartOneForm (Γ p) c) | |
| (extChartAt 𝓘(ℝ,E) c).target := by | |
| intro c | |
| exact (hΓ c).comp ((contDiff_const.prodMk contDiff_id).contDiffOn) | |
| (fun y hy => ⟨hp,hy⟩) | |
| exact manifold_closed_exact_normalization hΓp hβ hβclosed | |
| (manifoldFamily_scalar_fiber_smooth hV hG hp) (a p) x | |
| · intro p hp x hx | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| change Γ p x v-a p*β x v-manifoldScalarDifferential (E := E) (fun y => G (p,y)) x v=N p x v | |
| rw [hGd p x hx,hFd p hp x (hKA hx)] | |
| simp only [sub_apply,smul_apply,smul_eq_mul] | |
| ring | |
| theorem exists_normalized_manifold_primitive_neighborhood {V : Set P} {K A : Set M} | |
| (hV : IsOpen V) (hK : IsCompact K) (hA : IsOpen A) (hKA : K ⊆ A) | |
| {Γ N : P → ManifoldOneForm E M} {β : ManifoldOneForm E M} | |
| {a : P → ℝ} {F : P × M → ℝ} | |
| (hΓ : ∀ c,ContDiffOn ℝ ∞ (fun q : P × E => chartOneForm (Γ q.1) c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) | |
| (hβ : ∀ c,ContDiffOn ℝ ∞ (chartOneForm β c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (hβclosed : ∀ x,manifoldExteriorOneForm β x=0) | |
| (hβcompact : HasCompactSupport β) (ha : ContDiffOn ℝ ∞ a V) | |
| (hF : ContMDiffOn ((𝓘(ℝ,P)).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ F (V ×ˢ A)) | |
| (hFd : ∀ p ∈ V,∀ x ∈ A, | |
| manifoldScalarDifferential (E := E) (fun y => F (p,y)) x= | |
| Γ p x-N p x-a p • β x) : | |
| ∃ (Γ₁ : P → ManifoldOneForm E M) (L W : Set M), | |
| IsCompact L ∧ L ⊆ A ∧ IsOpen W ∧ K ⊆ W ∧ W ⊆ A ∧ | |
| (∀ c,ContDiffOn ℝ ∞ (fun q : P × E => chartOneForm (Γ₁ q.1) c q.2) | |
| (V ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) ∧ | |
| (∀ p,support (Γ₁ p-Γ p) ⊆ tsupport β ∪ L) ∧ | |
| (∀ p,HasCompactSupport (Γ₁ p-Γ p)) ∧ | |
| (∀ p ∈ V,∀ x,manifoldExteriorOneForm (Γ₁ p) x=manifoldExteriorOneForm (Γ p) x) ∧ | |
| (∀ p ∈ V,∀ x ∈ W,Γ₁ p x=N p x) := by | |
| let : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace E M | |
| obtain ⟨D,hD,hKD,hDA⟩ := exists_compact_between hK hA hKA | |
| obtain ⟨Γ₁,L,hL,hLA,hs,hsp,hcp,hd,he⟩ := | |
| exists_normalized_manifold_primitive hV hD hA hDA hΓ hβ hβclosed hβcompact ha hF hFd | |
| exact ⟨Γ₁,L,interior D,hL,hLA,isOpen_interior,hKD,interior_subset.trans hDA, | |
| hs,hsp,hcp,hd,fun p hp x hx => he p hp x (interior_subset hx)⟩ | |
| end | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| {M N : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| [TopologicalSpace N] [ChartedSpace F N] [IsManifold 𝓘(ℝ,F) ∞ N] | |
| def handlePushforwardOneForm (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) | |
| (α : ManifoldOneForm E M) : ManifoldOneForm F N := | |
| extendManifoldOneForm e.target (manifoldPullbackOneForm (fun _ => α) e.symm 0) | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] [IsManifold 𝓘(ℝ,F) ∞ N] in | |
| theorem handlePushforwardOneForm_zero | |
| (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) | |
| {α : ManifoldOneForm E M} {K : Set M} | |
| (hz : ∀ x, x ∉ K → α x=0) {y : N} (hy : y ∉ e '' K) : | |
| handlePushforwardOneForm e α y=0 := by | |
| by_cases ht : y∈e.target | |
| · have hk : e.symm y∉K := by | |
| intro hk | |
| exact hy ⟨e.symm y,hk,e.right_inv ht⟩ | |
| rw [handlePushforwardOneForm,extendManifoldOneForm_inside ht] | |
| simp only [manifoldPullbackOneForm,hz _ hk,ContinuousLinearMap.zero_comp] | |
| · exact extendManifoldOneForm_outside ht | |
| theorem handlePushforwardOneForm_smooth [T2Space N] | |
| (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) | |
| {α : ManifoldOneForm E M} {K : Set M} (hK : IsCompact K) (hKs : K⊆e.source) | |
| (hα : ∀ c, ContDiffOn ℝ ∞ (chartOneForm α c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (hz : ∀ x, x∉K → α x=0) : | |
| SmoothOneFormFamily (fun _ : ℝ => handlePushforwardOneForm e α) := by | |
| have hKi : IsCompact (e '' K) := | |
| hK.image_of_continuousOn (e.contMDiffOn.continuousOn.mono hKs) | |
| change SmoothOneFormFamily (fun p : ℝ => extendManifoldOneForm e.target | |
| (manifoldPullbackOneForm (fun _ : ℝ => α) e.symm p)) | |
| apply extendManifoldOneForm_smooth e.open_target hKi.isClosed | |
| · rintro y ⟨x,hx,rfl⟩ | |
| exact e.map_source (hKs hx) | |
| · exact manifoldPullbackOneForm_smooth_on (α := fun _ : ℝ => α) | |
| e.open_target e.symm.contMDiffOn | |
| (fun c => (hα c).comp contDiffOn_snd (fun _ hp => hp.2)) | |
| · intro p y hyt hy | |
| have hk : e.symm y∉K := by | |
| intro hk | |
| exact hy ⟨e.symm y,hk,e.right_inv hyt⟩ | |
| simp only [manifoldPullbackOneForm,hz _ hk,ContinuousLinearMap.zero_comp] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] [IsManifold 𝓘(ℝ,F) ∞ N] in | |
| theorem handlePushforwardOneForm_tsupport [T2Space N] | |
| (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) | |
| {α : ManifoldOneForm E M} {K : Set M} (hK : IsCompact K) (hKs : K⊆e.source) | |
| (hz : ∀ x, x∉K → α x=0) : | |
| tsupport (handlePushforwardOneForm e α) ⊆ e '' K := by | |
| apply closure_minimal _ (hK.image_of_continuousOn | |
| (e.contMDiffOn.continuousOn.mono hKs)).isClosed | |
| intro y hy | |
| by_contra hn | |
| exact hy (handlePushforwardOneForm_zero e hz hn) | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] [IsManifold 𝓘(ℝ,F) ∞ N] in | |
| theorem handlePushforwardOneForm_compact [T2Space N] | |
| (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) | |
| {α : ManifoldOneForm E M} {K : Set M} (hK : IsCompact K) (hKs : K⊆e.source) | |
| (hz : ∀ x, x∉K → α x=0) : HasCompactSupport (handlePushforwardOneForm e α) := by | |
| apply HasCompactSupport.intro (hK.image_of_continuousOn | |
| (e.contMDiffOn.continuousOn.mono hKs)) | |
| exact fun _ hy => handlePushforwardOneForm_zero e hz hy | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem manifoldExteriorOneForm_zero (x : M) : | |
| manifoldExteriorOneForm (0 : ManifoldOneForm E M) x=0 := by | |
| unfold manifoldExteriorOneForm | |
| have hz : chartOneForm (0 : ManifoldOneForm E M) x = fun _ => 0 := | |
| funext fun y => chartOneForm_zero x y | |
| simp only [hz,euclideanExteriorOneForm,fderiv_const_apply, | |
| ContinuousLinearMap.flip_zero] | |
| ext u v | |
| change (0 : ℝ)-0=0 | |
| ring | |
| theorem handlePushforwardOneForm_closed [T2Space N] | |
| (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) | |
| {α : ManifoldOneForm E M} {K : Set M} (hK : IsCompact K) (hKs : K⊆e.source) | |
| (hα : ∀ c, ContDiffOn ℝ ∞ (chartOneForm α c) (extChartAt 𝓘(ℝ,E) c).target) | |
| (hz : ∀ x, x∉K → α x=0) (hd : ∀ x, manifoldExteriorOneForm α x=0) (y : N) : | |
| manifoldExteriorOneForm (handlePushforwardOneForm e α) y=0 := by | |
| by_cases hy : y∈e.target | |
| · rw [handlePushforwardOneForm,extendManifoldOneForm_exterior_inside e.open_target _ hy, | |
| manifold_pullback_exterior_on hα e.open_target e.symm.contMDiffOn hy] | |
| simp only [manifoldPullbackTwoForm,hd] | |
| rfl | |
| · have hn : y∉e '' K := by | |
| rintro ⟨x,hx,rfl⟩ | |
| exact hy (e.map_source (hKs hx)) | |
| have hKi := hK.image_of_continuousOn (e.contMDiffOn.continuousOn.mono hKs) | |
| have he : handlePushforwardOneForm e α =ᶠ[𝓝 y] 0 := by | |
| filter_upwards [hKi.isClosed.isOpen_compl.mem_nhds hn] with z hzK | |
| exact handlePushforwardOneForm_zero e hz hzK | |
| rw [manifoldExteriorOneForm_congr_of_eventuallyEq he,manifoldExteriorOneForm_zero] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] [IsManifold 𝓘(ℝ,F) ∞ N] in | |
| theorem handle_mfderiv_symm_comp | |
| (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) {x : M} (hx : x∈e.source) : | |
| (mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) e.symm (e x)).comp | |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,F) e x) = ContinuousLinearMap.id ℝ E := by | |
| have he : (e.symm ∘ e) =ᶠ[𝓝 x] id := by | |
| filter_upwards [e.open_source.mem_nhds hx] with y hy | |
| exact e.left_inv hy | |
| have hd := mfderiv_comp x (e.symm.mdifferentiableAt (by simp) (e.map_source hx)) | |
| (e.mdifferentiableAt (by simp) hx) | |
| rw [← hd,he.mfderiv_eq,mfderiv_id] | |
| rfl | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] [IsManifold 𝓘(ℝ,F) ∞ N] in | |
| theorem handlePushforwardOneForm_pullback | |
| (e : PartialDiffeomorph 𝓘(ℝ,E) 𝓘(ℝ,F) M N ∞) | |
| (α : ManifoldOneForm E M) {x : M} (hx : x∈e.source) : | |
| manifoldPullbackOneForm (fun _ => handlePushforwardOneForm e α) e 0 x=α x := by | |
| have hd := handle_mfderiv_symm_comp e hx | |
| change (extendManifoldOneForm e.target | |
| (manifoldPullbackOneForm (fun _ => α) e.symm 0) (e x)).comp _ = _ | |
| rw [extendManifoldOneForm_inside (e.map_source hx)] | |
| ext v | |
| change α (e.symm (e x)) | |
| ((manifoldMapDifferential (E := E) (F := F) e.symm (e x)) | |
| ((manifoldMapDifferential (E := F) (F := E) e x) v)) = α x v | |
| have hαx : α (e.symm (e x))=α x := congrArg α (e.left_inv hx) | |
| rw [hαx] | |
| exact congrArg (α x) (congrArg (fun D : E →L[ℝ] E => D v) hd) | |
| end PackingSufficiencySupport.Hamiltonian | |
| end | |
| end OAI | |