Lagrangian and Action question?

[Livethefire]

Ive been doing some research on the title concepts...
And would love it if someone could answer some questions because I can't seem to find the answer anywhere.

1) How was the lagrangian found? I know its kind of defined, and there are other lagrangians- but is there an idea behind it or was it trail and error?

2) Similarly, how was the action integral found? Or were they just trying to sum the lagrangian over the entire trajectory... for fun, and found that it needed to be minimum?

3)Finally a general question, are there problems that the lagrangian would overcomplicate? (ie Newtons laws are easier to use?) And why specifically do we see the hamiltonian more readily in Quantum physics, as opposed to the lagrangian?

Thank you for anyone who sheds light on these questions. Like I said, I can't seem to find a direct answer where I have looked.

Thanks.

 

[jambaugh]

I suggest you first look at the very nice wikipedia article on http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange" . He originally developed the Variational Calculus then applied it to Newtonian Mechanics.

As far as how the action is found, one may begin by considering the form of the Eulier-Lagrange equations to the specific equations of a the system in question. Then observe the form the Lagrangian must take to yield those E-L equations.

For general mechanical systems one can typically define separate kinetic energy, T and potential energy V. The Lagrangian which then gives the proper energy conservation is L=T-V.

In more speculative settings one makes use of http://en.wikipedia.org/wiki/Noether%27s_theorem" which she developed at the behest of Einstein. It states that any symmetry of the Lagrangian will manifest a corresponding conserved quantity. One then selects forms of Lagrangian's with the postulated and/or observed symmetries seen in nature. Given sufficient symmetry one has a pretty restricted class of lagrangians one might consider.

Finally as to your question as to why we use Hamiltonians more so than Lagrangians in QM, that in part is because particle velocities (and their correspondents in general systems) are not in and of themselves observables of QM systems. Rather it is the momentum which is.
The transition between x, dx/dt <---> to x,p is what is known as a Legendre transformation. When applied it transforms the Lagrangian to the Hamiltonian and the Euler-Lagrange equations to Hamilton's equations for the dynamics.

One is effectively introducing an extra set of independent variables while reducing the order of the dynamic equations. The fact that particles in QM do not follow classical paths as such and that velocities are not quantum observables, dictates a position,momentum (operator) representation which in turn best fits in the Hamiltonian scheme.

On the other hand, working in QFT one rather makes more use of the Lagrangian formulation than Hamilton's.