Rotational Kinetic Energy eq integral
RigidBody.rotationalKineticEnergy_eq_integral
Plain-language statement
The rotational kinetic energy equals the mass integral of the local rotational speed squared: T = ½ ∫ |ω × r|² dm.
Exact Lean statement
theorem rotationalKineticEnergy_eq_integral (R : RigidBody 3) (ω : Fin 3 → ℝ) :
R.rotationalKineticEnergy ω
= (1 / (2 : ℝ)) * R.ρ ⟨fun x => (ω ⨯₃ (x : Fin 3 → ℝ)) ⬝ᵥ (ω ⨯₃ (x : Fin 3 → ℝ)),
ContDiff.contMDiff <| (contDiff_cross_dotProduct_cross ω).comp
(contDiff_pi.mpr fun i => Space.eval_contDiff i)⟩Formal artifact
Lean source
theorem rotationalKineticEnergy_eq_integral (R : RigidBody 3) (ω : Fin 3 → ℝ) : R.rotationalKineticEnergy ω = (1 / (2 : ℝ)) * R.ρ ⟨fun x => (ω ⨯₃ (x : Fin 3 → ℝ)) ⬝ᵥ (ω ⨯₃ (x : Fin 3 → ℝ)), ContDiff.contMDiff <| (contDiff_cross_dotProduct_cross ω).comp (contDiff_pi.mpr fun i => Space.eval_contDiff i)⟩ := by rw [rotationalKineticEnergy_eq_angularMomentum] congr 1 simp_rw [dotProduct, angularMomentum, ← smul_eq_mul, ← map_smul, ← map_sum] congr 1 ext x rw [← ContMDiffMap.coeFnAddMonoidHom_apply, map_sum, Finset.sum_apply] simp only [ContMDiffMap.coeFnAddMonoidHom_apply, ContMDiffMap.coe_smul, Pi.smul_apply, ContMDiffMap.coeFn_mk, smul_eq_mul] exact dotProduct_cross_cross_self (x : Fin 3 → ℝ) ω- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/RigidBody/KineticEnergy.lean:59-72
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