How Hamilton's Principle was found?

Question

Hamilton's principle states that the actual path a particle follows from points $p_1$ and $p_2$ in the configuration space between times $t_1$ and $t_2$ is such that the integral

$$S = \int_{t_1}^{t_2}L(q(t),\dot{q}(t),t)dt$$

is stationary. And then we have that $L = T - U$ is the lagrangian. Now, how this was found? I mean, how could someone find that picking the quantity $T-U$, considering the integral and extremizing it would give us the actual path on the configuration space?

I know that it works, and the books show this very well. But historically how physicists found that this would give the path? How they found the quantity $L$ and thought on studying it's integral?

Answer 1

Hamilton was guided by a hunch that since a minimum principle worked for optics, then perhaps a similar principle worked for mechanics. From his principle of least action he was then able to derive the Euler-Lagrange equations from his paper:

W.R. Hamilton, "On a General Method in Dynamics.", Philosophical Transaction of the Royal Society, 1834