I have seen that Lagrange's equations are some times derived from Hamilton's principle. This makes me wonder what the historical development of these ideas was. Hamilton lived in the nineteenth century while Lagrange lived in the eighteenth century.(adsbygoogle = window.adsbygoogle || []).push({});

The principle that minimizes the integral of the Lagrangian over time is called "Hamilton's principle", so I would assume that Lagrange had not thought about this. But he did have his "Lagrangian" (even if he didn't call it by that name). I would assume that Lagrange's equations were developed by Lagrange, and Lagrange's equations use the Lagrangian.

So, if Lagrange was using the Lagrangian, how come he didn't think of integrating the Lagrangian over time?.

Some place I read that the "Lagrangian approach" uses the integral of the viz viva (mv^{2}) over time. This would be equivalent to the integral of p*ds (which is the Maupertuisian action).

So, this story sounds a little strange. Lagrange uses a principle of least action which uses mv^{2}as it's integrand instead of using the Lagrangian (which he invented). Then comes Hamilton who thinks up a new "least action principle" which uses the Lagrangian as argument.

Am I missing something here, or am I wrong about some of what I just said?

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# Hamilton's principle and Lagrangian mechanics

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