Equation Of Motion iff hamilton Eq Op eq zero
ClassicalMechanics.HarmonicOscillator.equationOfMotion_iff_hamiltonEqOp_eq_zero
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_iff_hamiltonEqOp_eq_zero (xₜ : Time → EuclideanSpace ℝ (Fin 1))
(hx : ContDiff ℝ ∞ xₜ) : S.EquationOfMotion xₜ ↔
hamiltonEqOp S (fun t => S.toCanonicalMomentum t (xₜ t) (∂ₜ xₜ t)) xₜ = 0Formal artifact
Lean source
lemma equationOfMotion_iff_hamiltonEqOp_eq_zero (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : ContDiff ℝ ∞ xₜ) : S.EquationOfMotion xₜ ↔ hamiltonEqOp S (fun t => S.toCanonicalMomentum t (xₜ t) (∂ₜ xₜ t)) xₜ = 0 := by rw [hamiltonEqOp, hamiltonEqOp_eq_zero_iff_hamiltons_equations] simp [toCanonicalMomentum_eq, gradient_hamiltonian_momentum_eq, gradient_hamiltonian_position_eq] rw [equationOfMotion_iff_newtons_2nd_law _ _ hx] simp [Time.deriv_smul _ S.m (deriv_differentiable_of_contDiff xₜ hx), force_eq_linear]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean:738-744
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