Solid Sphere center Of Mass
RigidBody.solidSphere_centerOfMass
Plain-language statement
The center of mass of a solid sphere located at the origin is 0.
Exact Lean statement
lemma solidSphere_centerOfMass {d : ℕ} (m R : ℝ≥0) : (solidSphere d m R).centerOfMass = 0Formal artifact
Lean source
lemma solidSphere_centerOfMass {d : ℕ} (m R : ℝ≥0) : (solidSphere d m R).centerOfMass = 0 := by ext i simp only [centerOfMass, solidSphere, one_div, LinearMap.coe_mk, AddHom.coe_mk, ContMDiffMap.coeFn_mk, smul_eq_mul, Space.zero_apply, mul_eq_zero, inv_eq_zero, div_eq_zero_iff, coe_eq_zero] right right suffices ∫ x in Metric.closedBall (0 : Space d) R, x i ∂MeasureSpace.volume = -∫ x in Metric.closedBall (0 : Space d) R, x i ∂MeasureSpace.volume by linarith rw [← integral_neg] simp only [← integral_indicator measurableSet_closedBall, Set.indicator, Metric.mem_closedBall] rw [← integral_neg_eq_self] norm_num- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/RigidBody/SolidSphere.lean:61-73
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