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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Angular Momentum eq inertia Tensor mul Vec

RigidBody.angularMomentum_eq_inertiaTensor_mulVec

Plain-language statement

The angular momentum of a rigid body equals its inertia tensor applied to the angular velocity: L = I ω.

Exact Lean statement

theorem angularMomentum_eq_inertiaTensor_mulVec (R : RigidBody 3) (ω : Fin 3 → ℝ) :
    R.angularMomentum ω = R.inertiaTensor *ᵥ ω

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem angularMomentum_eq_inertiaTensor_mulVec (R : RigidBody 3) (ω : Fin 3  ) :    R.angularMomentum ω = R.inertiaTensor *ᵥ ω := by  funext i  simp only [angularMomentum, mulVec, dotProduct, inertiaTensor]  have hsmul :  j : Fin 3,      R.ρ fun x => (if i = j then 1 else 0) * ∑ k, (x k) ^ 2 - x i * x j,        ContDiff.contMDiff <| by fun_prop * ω j      = R.ρ (ω j • fun x => (if i = j then 1 else 0) * ∑ k, (x k) ^ 2 - x i * x j,        ContDiff.contMDiff <| by fun_prop) := by    intro j    rw [map_smul, smul_eq_mul, mul_comm]  rw [Finset.sum_congr rfl (fun j _ => hsmul j),  map_sum]  congr 1  ext x  simp only [ContMDiffMap.coeFn_mk]  rw [cross_cross_self_apply,  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]  fin_cases i <;> simp [Fin.sum_univ_three] <;> ring
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/ClassicalMechanics/RigidBody/AngularMomentum.lean:42-61

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Project-declaredLean 4.32.0

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adiabatic_relation_log

Plain-language statement

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physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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Project-declaredLean 4.32.0

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adiabatic_relation_UaUbVaVb

Plain-language statement

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physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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