Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
One motivation for the development of the Lagrangian formalism on fields, and more generally, for classical field theory, is to provide a clean mathematical foundation for quantum field theory, which is infamously beset by formal difficulties that make it unacceptable as a mathematical theory. The Lagrangians presented here are identical to their quantum equivalents, but, in treating the fields as classical fields, instead of being quantized, one can provide definitions and obtain solutions with properties compatible with the conventional formal approach to the mathematics of partial differential equations. This enables the formulation of solutions on spaces with well-characterized properties, such as Sobolev spaces. It enables various theorems to be provided, ranging from proofs of existence to the uniform convergence of formal series to the general settings of potential theory. In addition, insight and clarity is obtained by generalizations to Riemannian manifolds and fiber bundles, allowing the geometric structure to be clearly discerned and disentangled from the corresponding equations of motion. A clearer view of the geometric structure has in turn allowed highly abstract theorems from geometry to be used to gain insight, ranging from the Chern–Gauss–Bonnet theorem and the Riemann–Roch theorem to the Atiyah–Singer index theorem and Chern–Simons theory.
Hi all,
As a blind follower of QFT from the sidelines (the joys of the woefully inadequate teaching of theory to exp. particle physics students...), I have just realized that I've never actually gone further than deriving the Dirac equation, and then just used the Dirac Lagrangian density as...
Hello,
I took an Electrodynamics course this semester, where we derived Maxwell's equations from the field's Lagrangian density.
As a motivation, we "looked" for a scalar (in the relativistic sense) having something to do with EM fields - and had we found one we would have declared it a...
we know the lagrangian l=ke-pe right
in case of fields is called "lagrangian density"
let particle with mass "m" and position "x"
it kientic energy= 1/2(mv^2)
so lagrangian =1/2(mv^2)-v(x) , v(x)=potential energy
in case the field lagrangian density
how can i determine the lagragian...
Why does the Lagrangian density for the EM field in a dielectric medium take the form d^3 \bf x \left[ \epsilon \bf E^2 - \bf B^2 \right]? I can see that the expression for Lagrangian density has units of energy per unit volume as you would expect but that's about it. Much appreciated.
Here's the problem. For a neutral vector field V_{\mu} we have the Lagrangian density
\mathcal{L} = -\frac{1}{2}(\partial_{\mu}V_{\nu})(\partial^{\mu}V^{\nu})+\frac{1}{2}(\partial_{\mu}V^{\mu})(\partial_{\nu}V^{\nu})+\frac{1}{2}m^2V_{\mu}V^{\mu}
We are then going to use the Euler-Lagrange...