How did Christiaan Huygens come across the fact that the quantity $mv^2$ is conserved during motion? I would like to know his thought process and how he derived or proved this result.
In "Tractatus de motu corporum ex percussione" Christiaan Huygens showed energy conservation during the elastic impact. His investigations were triggered by a question of the Royal Society. Also Wren and Wallis gave answers. Wallis treated the inelastic impact, Huygens and Wren the elastic impact.
Huygens considered the impact in two different inertial systems, one of them being the centre-of-mass system. There he found that for both cases the vis viva was conserved. He wrote the equation already in modern form: $m_1v_1^2 + m_2v_2^2 =m_1v_1'^2 + m_2v_2'^2 $
Source: Norbert Pucker: Physikalische Grundlagen der Energietechnik, Springer (2013)
F. Chareix’s La découverte des lois du choc par Christiaan Huygens (2003) discusses this in great detail, on the basis of Huygens manuscripts (1652–). Generally his arguments were of relativistic nature, i.e. he drew conclusions from the possibility of viewing the collision “from the ground”, then “from the boat”, etc. But while Chareix sketches (pp. 18-19) the simple proof he may have had along such lines, Huygens does not seem to have written that: instead, he states conservation of vis viva as an unproved “Axiom” (2, pp. 18, 52), “Proposition” (XI, p. 55), or “Rule” (VI, p. 57).