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?

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    $\begingroup$ You do know that Lagrange was able to formulate his mechanics without (and before) Hamliton's Principle, right? The work is a little more involved, of course and requires swallowing the principle of virtual work along the way. It's done in Goldstein. $\endgroup$ – dmckee Feb 8 '14 at 15:08
  • $\begingroup$ I really didn't know about that @dmckee. I'll take a look on Goldstein's book then. Thanks for pointing this out. $\endgroup$ – user1620696 Feb 8 '14 at 15:11
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    $\begingroup$ Have fun. I find Lagrange' approach to be pretty difficult, but it does avoid "How did he know to use that function?" which confused the heck out of me between college (where I got the "Least Action") version and grad school. $\endgroup$ – dmckee Feb 8 '14 at 15:15
  • $\begingroup$ Related: physics.stackexchange.com/q/78138/2451 and links therein. $\endgroup$ – Qmechanic May 12 '16 at 21:02
  • $\begingroup$ how is possible that Lagrange didn¡t discover that ? it is directly de4rvivated from Euler Langrange Equation for a conservative potential $\endgroup$ – Jose Javier Garcia Jul 9 at 17:39

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


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