Although the expressions of Lagrangian for quarks and leptons in the standard-model are very sophisticated, the mixture of indices for Lorentz space and inner space sometimes makes difficult to develop equations. To get better perspective, I tried to rewrite the Lagrangian with the differential form in the following site;

 

http://hecoaustralia.fortunecity.com/lagrangian/gauge4.htm

I suppose it depends on what you are used to. I find the traditional form much more intuitive.

What does Lagrangian mean in terms of modern and theoretical physics?

The link does not work.

 

So I don't know what you have done, but it is extremely useful to write the Lagrangian (density) as differential form on the total space of a jet bundle. This differential form is known as the Poincare-Cartan n-form.

 

You can then form all the Euler-Lagrange equations in terms of geometry, in fact in terms of the Lie derivative of the Poincare-Cartan form.

 

What does Lagrangian mean in terms of modern and theoretical physics?

 

The Lagrangian is a function of the fields and their derivatives and is assumed to contain all the physical information about the system both classically and quantum mechanically. The hard part is extracting this information!

What does Lagrangian mean in terms of modern and theoretical physics?

 

To expand on ajb's answer, as an example, the Lagrangian for QED is:

 

[math] {\cal L} = -\frac{1}{4}F^{\mu \nu} F_{\mu \nu} + \bar \psi \left( i \gamma^\mu \partial_\mu +m \right) \psi [/math]

 

In principle, everything you need for QED is in that formula (once you define the notation) - it completely specifies the theory (which is quite neat I think).

 

Edit: the link works fine for me.

Edit: the link works fine for me.

 

Sometimes it works, sometimes it doesn't.

To expand on ajb's answer, as an example, the Lagrangian for QED is:

 

[math] {\cal L} = -\frac{1}{4}F^{\mu \nu} F_{\mu \nu} + \bar \psi \left( i \gamma^\mu \partial_\mu +m \right) \psi [/math]

 

In principle, everything you need for QED is in that formula (once you define the notation) - it completely specifies the theory (which is quite neat I think).

 

Edit: the link works fine for me.

 

QWT, too:eyebrow:

QED = Quantum Electrodynamics, in case that was your question.