Kinetic Energy deriv
ClassicalMechanics.HarmonicOscillator.kineticEnergy_deriv
Project documentation
The energy of the harmonic oscillator is the kinetic energy plus the potential energy. -/ noncomputable def energy (xₜ : Time → EuclideanSpace ℝ (Fin 1)) : Time → ℝ := fun t => kineticEnergy S xₜ t + potentialEnergy S (xₜ t) /-! ### C.2. Simple equalities for the energies -/ lemma kineticEnergy_eq (xₜ : Time → EuclideanSpace ℝ (Fin 1)) : kineticEnergy S x...
Exact Lean statement
lemma kineticEnergy_deriv (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : ContDiff ℝ ∞ xₜ) :
∂ₜ (kineticEnergy S xₜ) = fun t => ⟪∂ₜ xₜ t, S.m • ∂ₜ (∂ₜ xₜ) t⟫_ℝFormal artifact
Lean source
lemma kineticEnergy_deriv (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : ContDiff ℝ ∞ xₜ) : ∂ₜ (kineticEnergy S xₜ) = fun t => ⟪∂ₜ xₜ t, S.m • ∂ₜ (∂ₜ xₜ) t⟫_ℝ := by funext t unfold kineticEnergy have hd : DifferentiableAt ℝ (∂ₜ xₜ) t := (deriv_differentiable_of_contDiff xₜ hx).differentiableAt rw [Time.deriv_eq, fderiv_const_mul (by fun_prop), _root_.smul_apply, fderiv_inner_apply (𝕜 := ℝ) hd hd, ← Time.deriv_eq] simp [inner_smul_right, real_inner_comm] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean:250-259
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