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Who, besides Hamilton, was instrumental in discovering at least part of the so-called Hamilton principle?

$$\delta \int_{a}^{b}L(q,\dot q ,t )dt$$

where $ L=T-V$, the Lagrangian.

What exactly, was Lagrange's role, especially in the part that bears his name?

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Although called Hamilton's Principle, there were a number of mathematical physicists of the day who contributed to the overall work. Contributions by mathematicians such as Euler, Lagrange, and similar or related work by Maupertius should not be ignored. Also, powerful add on work was done by Jacobi and Poisson. Plus, the original work with Calculus of Variations began (I think) with Johann Bernoulli's solution to the Brachistochrone Problem.

Also, I think that the pure Hamilton's Principle should be written as the stationary action: $$ \frac{\delta S}{\delta q_i(t)} = 0 $$ Where the action $S$ is written as you have in your post with the time integral of the Lagrangian.

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  • $\begingroup$ Maupertuis's is a different principle. This one was discovered by Hamilton. This is not "calculus of variations". This is physics (mechanics). $\endgroup$ – Alexandre Eremenko Apr 16 '17 at 21:19
  • $\begingroup$ Maupertius' Principle defines the action functional that is used and leads to the principle by Hamilton. Sure, they are slightly different but different parts of the same kinds of statements. Yes, I know it is physics. But, "mathematical" physics. I also disagree with your comment that this is not calculus of variations -- calculus of variations is at the heart of solving the functional problem for stationary action. $\endgroup$ – K7PEH Apr 17 '17 at 2:12
  • $\begingroup$ but did not Lagrange discover the calculus of variations ? the equations of variations are the same than classical mechanics $\endgroup$ – Jose Javier Garcia Apr 20 '17 at 19:35
  • $\begingroup$ @JoseJavierGarcia -- The first sentence under the History section of the Wikipedia page for Calculus of Variations says -- "The calculus of variations may be said to begin with the brachistochrone curve problem raised by Johann Bernoulli (1696)". $\endgroup$ – K7PEH Apr 21 '17 at 20:55

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