| import OAI.Geometry.CylinderCovering.Singular.StreamDefinitions |
|
|
| namespace OAI |
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|
| noncomputable section |
|
|
| namespace CylinderCovering.SingularStream |
| open RuledApproximation |
| open scoped ContDiff |
|
|
| lemma potential_eq (p b r : ℝ) : |
| potential p b r = (1 / 2 : ℝ) * (2 * r) ^ (p - 2) * (b ^ 2 - r ^ 2) := by |
| unfold potential |
| rw [show r / (1 / 2 : ℝ) = 2 * r by ring] |
|
|
| theorem potential_contDiffAt (p : ℝ) {z : Plane} (hz : 0 < z 1) : |
| ContDiffAt ℝ ∞ (fun w : Plane => potential p (w 0) (w 1)) z := by |
| have hs : ContDiffAt ℝ ∞ (fun w : Plane => w 1 / (1 / 2 : ℝ)) z := by |
| fun_prop |
| have hp := hs.rpow_const_of_ne (p := p - 2) (by positivity) |
| exact (contDiffAt_const.mul hp).mul (by fun_prop) |
|
|
| theorem potential_wall_pos (p r : ℝ) : potential p r r = 0 := by |
| simp [potential] |
|
|
| theorem potential_wall_neg (p r : ℝ) : potential p (-r) r = 0 := by |
| simp [potential] |
|
|
| theorem potential_homogeneous (p b : ℝ) {r t : ℝ} (hr : 0 < r) (ht : 0 < t) : |
| potential p (t * b) (t * r) = t ^ p * potential p b r := by |
| rw [potential_eq, potential_eq, |
| show 2 * (t * r) = t * (2 * r) by ring, |
| Real.mul_rpow ht.le (by positivity : (0 : ℝ) ≤ 2 * r)] |
| have hpow : t ^ p = t ^ (p - 2) * t ^ (2 : ℕ) := by |
| calc |
| t ^ p = t ^ ((p - 2) + 2) := by congr 1; ring |
| _ = t ^ (p - 2) * t ^ (2 : ℕ) := by rw [Real.rpow_add ht, Real.rpow_two] |
| rw [hpow] |
| ring |
|
|
| theorem potential_deriv_b (p b : ℝ) {r : ℝ} (_hr : 0 < r) : |
| deriv (fun x : ℝ => potential p x r) b = singularField p (vec2 b r) 1 := by |
| have hd := ((hasDerivAt_pow 2 b).sub_const (r ^ 2)).const_mul |
| ((1 / 2 : ℝ) * (2 * r) ^ (p - 2)) |
| have he : (fun x : ℝ => (1 / 2 : ℝ) * (2 * r) ^ (p - 2) * |
| (x ^ 2 - r ^ 2)) = (fun x => potential p x r) := by |
| funext x |
| exact (potential_eq p x r).symm |
| rw [he] at hd |
| rw [hd.deriv] |
| dsimp only [singularField, vec2] |
| simp only [Matrix.cons_val_zero, Matrix.cons_val_one] |
| norm_num only [Nat.cast_ofNat, pow_one, Nat.reduceSub] |
| rw [show r / (1 / 2 : ℝ) = 2 * r by ring] |
| ring |
|
|
| theorem potential_neg_deriv_r (p b : ℝ) {r : ℝ} (hr : 0 < r) : |
| -deriv (fun y : ℝ => potential p b y) r = singularField p (vec2 b r) 0 := by |
| have hpow := ((hasDerivAt_id r).const_mul 2).rpow_const |
| (p := p - 2) (Or.inl (by positivity : (2 : ℝ) * r ≠ 0)) |
| have hd := ((hpow.const_mul (1 / 2 : ℝ)).mul |
| ((hasDerivAt_pow 2 r).const_sub (b ^ 2))) |
| have he : (fun y : ℝ => (1 / 2 : ℝ) * (2 * y) ^ (p - 2) * |
| (b ^ 2 - y ^ 2)) = (fun y => potential p b y) := by |
| funext y |
| exact (potential_eq p b y).symm |
| change HasDerivAt (fun y : ℝ => (1 / 2 : ℝ) * (2 * y) ^ (p - 2) * |
| (b ^ 2 - y ^ 2)) _ r at hd |
| rw [he] at hd |
| rw [hd.deriv] |
| have hlow : (2 * r) ^ (p - 2 - 1) = (2 * r) ^ (p - 2) / (2 * r) := by |
| rw [Real.rpow_sub_one (by positivity : (2 : ℝ) * r ≠ 0)] |
| have hhigh : (2 * r) ^ (p - 1) = (2 * r) ^ (p - 2) * (2 * r) := by |
| rw [show p - 1 = (p - 2) + 1 by ring, |
| Real.rpow_add_one (by positivity : (2 : ℝ) * r ≠ 0)] |
| dsimp only [singularField, vec2] |
| simp only [Matrix.cons_val_zero, Matrix.cons_val_one] |
| simp only [id_eq, Nat.cast_ofNat, Nat.reduceSub, pow_one] |
| rw [show r / (1 / 2 : ℝ) = 2 * r by ring, hlow, hhigh] |
| field_simp [hr.ne'] |
| ring |
| |
| theorem field_hamiltonian (p b : ℝ) {r : ℝ} (hr : 0 < r) : |
| singularField p (vec2 b r) = |
| vec2 (-deriv (fun y : ℝ => potential p b y) r) |
| (deriv (fun x : ℝ => potential p x r) b) := by |
| rw [potential_neg_deriv_r p b hr, potential_deriv_b p b hr] |
| ext i |
| fin_cases i <;> simp [vec2] |
| |
| end CylinderCovering.SingularStream |
| |
| end |
| |
| end OAI |
| |
