Equation Of Motion tfae
ClassicalMechanics.HarmonicOscillator.equationOfMotion_tfae
Project documentation
The operator on the momentum-position phase-space whose vanishing is equivalent to the hamilton's equations between the momentum and position. -/ noncomputable def hamiltonEqOp (p : Time → EuclideanSpace ℝ (Fin 1)) (q : Time → EuclideanSpace ℝ (Fin 1)) := ClassicalMechanics.hamiltonEqOp (hamiltonian S) p q /-! ### G.5. Equation of motion if and only if Ha...
Exact Lean statement
lemma equationOfMotion_tfae (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : ContDiff ℝ ∞ xₜ) :
List.TFAE [S.EquationOfMotion xₜ,
(∀ t, S.m • ∂ₜ (∂ₜ xₜ) t = force S (xₜ t)),
hamiltonEqOp S (fun t => S.toCanonicalMomentum t (xₜ t) (∂ₜ xₜ t)) xₜ = 0,
(δ (q':=xₜ), ∫ t, lagrangian S t (q' t) (fderiv ℝ q' t 1)) = 0,
(δ (pq':= fun t => (S.toCanonicalMomentum t (xₜ t) (∂ₜ xₜ t), xₜ t)),
∫ t, ⟪(pq' t).1, ∂ₜ (Prod.snd ∘ pq') t⟫_ℝ - S.hamiltonian t (pq' t).1 (pq' t).2) = 0]Formal artifact
Lean source
lemma equationOfMotion_tfae (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : ContDiff ℝ ∞ xₜ) : List.TFAE [S.EquationOfMotion xₜ, (∀ t, S.m • ∂ₜ (∂ₜ xₜ) t = force S (xₜ t)), hamiltonEqOp S (fun t => S.toCanonicalMomentum t (xₜ t) (∂ₜ xₜ t)) xₜ = 0, (δ (q':=xₜ), ∫ t, lagrangian S t (q' t) (fderiv ℝ q' t 1)) = 0, (δ (pq':= fun t => (S.toCanonicalMomentum t (xₜ t) (∂ₜ xₜ t), xₜ t)), ∫ t, ⟪(pq' t).1, ∂ₜ (Prod.snd ∘ pq') t⟫_ℝ - S.hamiltonian t (pq' t).1 (pq' t).2) = 0] := by rw [← equationOfMotion_iff_hamiltonEqOp_eq_zero, ← equationOfMotion_iff_newtons_2nd_law] rw [hamiltons_equations_varGradient, euler_lagrange_varGradient] simp only [List.tfae_cons_self] rw [← gradLagrangian_eq_eulerLagrangeOp, ← equationOfMotion_iff_gradLagrangian_zero] simp only [List.tfae_cons_self] erw [← equationOfMotion_iff_hamiltonEqOp_eq_zero] simp only [List.tfae_cons_self, List.tfae_singleton] repeat fun_prop simp [toCanonicalMomentum_eq] repeat fun_prop- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean:759-775
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