📐 **Koenig's Theorem for Kinetic Energy**

The total kinetic energy is:
 'T = 1/2Σᵢ mᵢ |**r**̇ᵢ|² = 1/2Σᵢ mᵢ |**R**̇ + **r**̇ᵢ\'|²'
 Expanding:
 'T = 1/2M|**R**̇|² + **R**̇ • Σᵢ mᵢ**r**̇ᵢ\' + 1/2Σᵢ mᵢ |**r**̇ᵢ\'|²'
 The middle term vanishes since 'Σᵢ mᵢ**r**̇ᵢ\' = 0'.
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 The total kinetic energy separates into center-of-mass and relative contributions:
 'T = 1/2M|**R**̇|² + 1/2Σᵢ mᵢ |**r**̇ᵢ\'|²'
 The first term is the kinetic energy of the center of mass; the second is the internal kinetic
 energy (motion relative to the center of mass).

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