The Two-Link Arm, by Hand

The viewer can (1) write both link twists of a planar 2R arm, including the adjoint term (sin q₂, cos q₂)q̇₁, (2) expand each link's ½𝒱ᵀ𝔗𝒱, (3) read M off T by collecting q̇₁², q̇₁q̇₂, q̇₂² terms, (4) interpret M₁₁ (stretched vs folded), why q₁ is absent, M₁₂ as coupling.