Trajectory equation Of Motion of overdamped
ClassicalMechanics.DampedHarmonicOscillator.trajectory_equationOfMotion_of_overdamped
Plain-language statement
In the overdamped regime, the selected trajectory satisfies the damped equation of motion.
Exact Lean statement
lemma trajectory_equationOfMotion_of_overdamped (IC : InitialConditions)
(hS : S.IsOverdamped) :
S.EquationOfMotion (S.trajectory IC)Formal artifact
Lean source
lemma trajectory_equationOfMotion_of_overdamped (IC : InitialConditions) (hS : S.IsOverdamped) : S.EquationOfMotion (S.trajectory IC) := by rw [S.trajectory_eq_of_overdamped IC hS] have hγ : S.γ = 2 * S.m * S.decayRate := S.gamma_eq_two_mul_m_mul_decayRate have hk : S.k = S.m * (S.decayRate^2 - S.angularFrequency^2) := by rw [S.k_eq_m_mul_ω_sq, S.angularFrequency_sq_of_overdamped hS] ring refine S.exp_decay_smul_equationOfMotion S.decayRate (S.angularFrequency^2) (S.overdampedBase IC) (by unfold overdampedBase fun_prop) ?_ (S.overdampedBase_acceleration IC hS) hγ hk rw [show ∂ₜ (S.overdampedBase IC) = _ from S.overdampedBase_velocity IC hS] fun_prop- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean:446-461
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