File size: 2,405 Bytes
f18e7de | 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 | import OAI.MathematicalPhysics.Boltzmann.Branching.NormalReflectionGeometry
namespace OAI
/-! # Orthogonal changes of coordinates preserve the actual sphere area measure -/
noncomputable section
open MeasureTheory Set
open scoped Pointwise
namespace BoltzmannNonuniqueness
def sphereIsometry (R : Velocity ≃ₗᵢ[ℝ] Velocity) : Sphere ≃ᵐ Sphere where
toFun n := ⟨R n.1,by simpa only [Metric.mem_sphere,dist_zero_right,R.norm_map] using n.property⟩
invFun n := ⟨R.symm n.1,by simpa only [Metric.mem_sphere,dist_zero_right,R.symm.norm_map] using n.property⟩
left_inv n := by apply Subtype.ext; exact R.symm_apply_apply _
right_inv n := by apply Subtype.ext; exact R.apply_symm_apply _
measurable_toFun := by
exact (R.continuous.measurable.comp measurable_subtype_coe).subtype_mk
measurable_invFun := by
exact (R.symm.continuous.measurable.comp measurable_subtype_coe).subtype_mk
theorem sphereIsometry_cone (R : Velocity ≃ₗᵢ[ℝ] Velocity) (S : Set Sphere) :
(Ioo (0:ℝ) 1) • ((↑) '' ((sphereIsometry R) ⁻¹' S)) =
R ⁻¹' ((Ioo (0:ℝ) 1) • ((↑) '' S)) := by
ext x
constructor
· rintro ⟨r,hr,y,⟨n,hn,hny⟩,hxy⟩
subst y
change R x ∈ (Ioo (0:ℝ) 1) • ((↑) '' S)
refine ⟨r,hr,R n.1,⟨sphereIsometry R n,hn,rfl⟩,?_⟩
change r • R n.1 = R x
change r • n.1 = x at hxy
rw [← R.map_smul,hxy]
· rintro ⟨r,hr,y,⟨n,hn,hny⟩,hxy⟩
subst y
refine ⟨r,hr,R.symm n.1,⟨(sphereIsometry R).symm n,?_,rfl⟩,?_⟩
· change sphereIsometry R ((sphereIsometry R).symm n) ∈ S
simpa only [MeasurableEquiv.apply_symm_apply] using hn
· apply R.injective
rw [R.map_smul,R.apply_symm_apply]
exact hxy
theorem sphereIsometry_measurePreserving (R : Velocity ≃ₗᵢ[ℝ] Velocity) :
MeasurePreserving (sphereIsometry R) sphereArea sphereArea := by
refine ⟨(sphereIsometry R).measurable,?_⟩
apply Measure.ext
intro S hS
rw [Measure.map_apply (sphereIsometry R).measurable hS,sphereArea,
Measure.toSphere_apply' _ ((sphereIsometry R).measurable hS),Measure.toSphere_apply' _ hS,
sphereIsometry_cone]
congr 1
change volume (R.toHomeomorph ⁻¹' _) = volume _
rw [← R.toHomeomorph.measurableEmbedding.map_apply]
change (volume.map R) _ = volume _
rw [R.measurePreserving.map_eq]
end BoltzmannNonuniqueness
end
end OAI
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