The Magical Lagrangian

[mezarashi]

1. It is really neat how the Lagrangian Equations of Motion can be derived from Hamilton's Variational Principle, and solve so many classical systems much easier than from a Newtonian approach. Does anybody have any leads to how Lagrange came up with his "Lagrangian", specifically the equation:

 

[math]L = T - U[/math]

where T is the kinetic energy and U is the potential

 

The equation seems somewhat arbitrary and made up yet it works!

 

2. The second question is, who really came up with this formalism? The Lagrange equations are derived from Hamilton's principle (or can they somehow be derived from each other?), but then again, Hamilton's principle has the Lagrangian in it, so... who came first and who really put it together.

 

[math]J = \int_a^b L dt[/math]

[Dave]

In regards to the second question, are you looking for the actual derivation from the principle, or a more historical explanation? I can't help with the latter, but I've studied the former in a course at the end of last year. However, I don't want to post something that you already know

[mezarashi]

In regards to the second question, are you looking for the actual derivation from the principle, or a more historical explanation? I can't help with the latter, but I've studied the former in a course at the end of last year. However, I don't want to post something that you already know

 

Well hmmm, maybe I'm looking for a bit of both ^^. I've seen Lagrange's equations being derived from Hamilton's and D'Alembert's principles. I guess I'm just confused on who got who's idea from who, because they seem all interlated, I'm not sure which is the more fundamental or atleast original of ideas.

[Dave]

From a mathematical standpoint, the entire thing comes from the calculus of variations; basically, you try to minimize or maximize integrals in the same way that you might work out the first derivative to find critical points on a function. Sounds useless and somewhat whimsicle, but it allows you to work out things like the shapes of hanging ropes or shortest route from A to B over rough terrain.

 

The general method was developed by Euler and Lagrange (hence they're called the Euler-Lagrange equations). However, I'm not totally boned up as to who did what when. It would seem logical, for me at least, that Hamilton and Lagrange had some sort of correspondence and came up with a method of implementing Hamilton's Principle using actions etc. After this it might have been generalized with some assistance from Euler. This is only me guessing though, so who knows

[Dave]

From reading the stuff in Wikipedia under Euler-Lagrange Equations:

 

The principle of least action was first formulated by Maupertuis [1] in 1746 and further developed (from 1748 onwards) by the mathematicians Euler, Lagrange, and Hamilton.

 

This would seem to confirm (in part) what I posted above about the various correspondence between those three guys.