# How Hamilton's Principle was found?

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?

• 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. – dmckee Feb 8 '14 at 15:08
• I really didn't know about that @dmckee. I'll take a look on Goldstein's book then. Thanks for pointing this out. – user1620696 Feb 8 '14 at 15:11
• 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. – dmckee Feb 8 '14 at 15:15
• Related: physics.stackexchange.com/q/78138/2451 and links therein. – Qmechanic May 12 '16 at 21:02
• how is possible that Lagrange didn¡t discover that ? it is directly de4rvivated from Euler Langrange Equation for a conservative potential – Jose Javier Garcia Jul 9 at 17:39