Definition of Lagrangian Density?

[LarryS]

I understand the definitions of both the classical and relativistic (SR) Lagrangians. But I cannot find a precise mathematical definition of Lagrangian Density. Please assist. Thanks in advance.

 

[The_Duck]

In classical or quantum field theory you tend to write the Lagrangian of your system as an integral of some expression over all space. The integrand is the Lagrangian density. In the same way, total mass is the integral of mass density over space. But maybe you wanted something deeper?

 

[arkajad]

Density means that $\mathcal{L}\times volume$ must be invariant under coordinate transformations. The volume form is not invariant (it involves the Jacobian), so Lagrangian density L must also be not a scalar, to compensate for non-invariance of the volume form. Action - the integral should be invariant.

 

[LarryS]

In classical or quantum field theory you tend to write the Lagrangian of your system as an integral of some expression over all space. The integrand is the Lagrangian density. In the same way, total mass is the integral of mass density over space. But maybe you wanted something deeper?

It almost sounds as if the Lagrangian Density (LD) is defined as the solution of an integral equation, i.e. any function (integrand) of space and time whose integral over all space and time equals the Action, is a Lagrangian Density. However, I would like to find an mathematical example of a LD for a specific physical system.

 

[The_Duck]

http://en.wikipedia.org/wiki/Lagrangian#Lagrangians_and_Lagrangian_densities_in_field_theory".

 

[arkajad]

http://en.wikipedia.org/wiki/Lagrangian#Lagrangians_and_Lagrangian_densities_in_field_theory".

Not the best example, because it deals with the Minkowski space, where Lorentz transformation are all of $|det|=1$. Better check these http://www.math.princeton.edu/~seri/homepage/courses/GR2009.pdf" , Section 2, Classical field theory,