Wikipedia:Reference desk/Archives/Mathematics/2012 November 12

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November 12

Lagrangian density

Can someone explain to me Lagrangian density used to derived field equations like how ordinary lagrangian derives the trajectory of a particle? For example I don't understand what L is a function defined on for the density version, I know for ordinary Lagrangian L is defined on R^2n for n dimensional space, but I don't understand the field version. And also what's the motivation behind it? Ordinary Lagrangian wants the stationary point of the action due to a trajectory, so what about Lagrangian density? Money is tight (talk) 08:44, 12 November 2012 (UTC)

Perhaps the attached image will help. ~AH1 ^(discuss!) 04:06, 13 November 2012 (UTC)

This is a bit beyond me, but try looking at [1] - see Section 3 (p22) 'Lagrangian Mechanics' which leads into 3.2 'Principles of Lagrangian Field Theory' Christopherlumb (talk) 17:19, 14 November 2012 (UTC)

Lagrangian density is a function which integrated over spatial coordinates gives you the Lagrangian, as per the usual definition of density. Since you are dealing with fields, and not particles, in field theory, there is a contribution from each volume element to the total Lagrangian.