# Hamilton's principle and Lagrangian mechanics

1. Aug 18, 2004

### alexepascual

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.
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 (mv2) 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 mv2 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?

Last edited: Aug 18, 2004
2. Aug 18, 2004

### pervect

Staff Emeritus
I would *guess* that Lagrange got as far as Lagrange's equations, without realizing that his equations minimized a variational integral. But I don't know the actual history. One approach that leads to Lagrange's equations is the "virtual work" approach.

3. Aug 18, 2004

Staff Emeritus
History of the Lagrangian:
Maupertuis proposed the forerunner of the principle of least action.
Daniel Bernoulli brought it to the attention of Euler.
Euler, fascinated with the mathematical problem, created the Calculus of Variations and derived the Euler equations form the minimized (actually stationized) action.
Lagrange used Euler's result, formed the general Lagrangean representation of the action and used this, plus his virtual work idea, to create the science of analytical mechanics in his Mechanique Analitique.

Hamilton's problem with Lagrange's method was that the Euler equations in it came out as second order differential equations. Hamilton sought for an approach that would yield first degree equations, and found it.

4. Aug 23, 2004

### alexepascual

Thanks for your responses guys and sorry for the delay. I had forgotten to subscrive to this thread.
I still have some questions.
If you translate Maupertui's action S=integral{p ds} into energy, you get an integral of 2T over time, not the Lagrangian. So it looks like Maupertui's action is quite different conceptually form the Action you get from the Lagrangian.
From what I remember, the action obtained by integrating the Lagrangian is known as Hamiltonian Action, which would give the hint that Lagrange was not the one who came up with it.
My questions are:
(1) Did Lagrange at any point integrate the Lagrangian over time and talk about minimizing that or was Hamilton the first one to do it?
(2) Was Lagrange who discovered the concept of virtual work or was it D'alembert (or someone before him)?
Thanks again,
Alex