Trajectory velocity
ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_velocity
Project documentation
For zero initial conditions, the trajectory is zero. -/ @[simp] lemma trajectory_zero : trajectory S 0 = fun _ => 0 := by simp [trajectory_eq] /-! ### B.3. Smoothness of the trajectories The trajectories for any initial conditions are smooth functions of time. -/ @[fun_prop] lemma trajectory_contDiff (S : HarmonicOscillator) (IC : InitialConditions) {n :...
Exact Lean statement
lemma trajectory_velocity (IC : InitialConditions) : ∂ₜ (IC.trajectory S) =
fun t : Time => - S.ω • sin (S.ω * t.val) • IC.x₀ + cos (S.ω * t.val) • IC.v₀Formal artifact
Lean source
lemma trajectory_velocity (IC : InitialConditions) : ∂ₜ (IC.trajectory S) = fun t : Time => - S.ω • sin (S.ω * t.val) • IC.x₀ + cos (S.ω * t.val) • IC.v₀ := by funext t rw [trajectory_eq, Time.deriv, fderiv_fun_add (by fun_prop) (by fun_prop)] rw [fderiv_smul_const (by fun_prop), fderiv_smul_const (by fun_prop)] have h1 : (fderiv ℝ (fun t => sin (S.ω * t.val) / S.ω) t) = (1/ S.ω) • (fderiv ℝ (fun t => sin (S.ω * t.val)) t) := by rw [← fderiv_mul_const] congr funext t field_simp fun_prop simp [h1] rw [fderiv_cos (by fun_prop), fderiv_sin (by fun_prop), fderiv_fun_mul (by fun_prop) (by fun_prop)] simp only [fderiv_fun_const, Pi.zero_apply, smul_zero, add_zero, neg_smul, _root_.neg_apply, FunLike.coe_smul, Pi.smul_apply, fderiv_val, smul_eq_mul, mul_one] field_simp [S.ω_ne_zero] module- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean:415-434
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