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As you probably know, Newton thought that energy is linearly proportional to motion. The second law's original formulation reads: "Mutationem motus proportionalem esse vi motrici impressae" = "change of motion is proportional to the force impressed" It was Gottfried Leibniz, as early as ca. 1680, who first realized that KE is proportional to squared ...


You've already got some answers, but nobody mentioned Noether's Theorem yet. Noether's theorem maps a conserved quantity to each continuous symmetry. The relevant continuous symmetry needed to prove the conservation of energy is the one that leaves the laws of nature invariant, meaning the laws of physics don't change with time. Each continuous symmetry ...

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