Trajectory velocity at zero
ClassicalMechanics.DampedHarmonicOscillator.trajectory_velocity_at_zero
Plain-language statement
The selected trajectory has initial velocity IC.v₀.
Exact Lean statement
lemma trajectory_velocity_at_zero (IC : InitialConditions) :
∂ₜ (S.trajectory IC) 0 = IC.v₀Formal artifact
Lean source
lemma trajectory_velocity_at_zero (IC : InitialConditions) : ∂ₜ (S.trajectory IC) 0 = IC.v₀ := by rcases S.isUnderdamped_or_isCriticallyDamped_or_isOverdamped with hS | hS | hS · rw [S.trajectory_eq_of_underdamped IC hS, exp_decay_smul_velocity S.decayRate (S.underdampedBase IC) (by unfold underdampedBase; fun_prop), show ∂ₜ (S.underdampedBase IC) = _ from S.underdampedBase_velocity IC hS] simp [underdampedBase] · rw [S.trajectory_eq_of_criticallyDamped IC hS, exp_decay_smul_velocity S.decayRate (S.criticallyDampedBase IC) (by unfold criticallyDampedBase; fun_prop), S.criticallyDampedBase_velocity IC] simp [criticallyDampedBase] · rw [S.trajectory_eq_of_overdamped IC hS, exp_decay_smul_velocity S.decayRate (S.overdampedBase IC) (by unfold overdampedBase; fun_prop), show ∂ₜ (S.overdampedBase IC) = _ from S.overdampedBase_velocity IC hS] simp [overdampedBase]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean:379-396
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