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Here you go ON A GENERAL METHOD IN DYNAMICS By William Rowan Hamilton https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/GenMeth.pdf

In Hamilton's "on a general method in dynamics", he starts with varying the function ##U## and writes the equation: $$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$ Then he defines ##T## to be: $$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$ Then by ##dT=dU##, he...