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Lagrangian density

| nosplit | ↑ parent "Lagrangian (field theory)" | words: 319
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When we particles particles, the action is obtained by integrating the Lagrangian over time: $$Action=∫t0​t​L(x(t),∂t∂x(t)​,t)dt (68)$$
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In the case of fields however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.
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Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.
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E.g. for a 2-dimensional field : $$Action=∫t0​t​L(f(x,y,t),∂x∂f(x,y,t)​,∂y∂f(x,y,t)​,∂t∂f(x,y,t)​,x,y,t)dxdydt (69)$$
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Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with : $$Action=∫t0​t​L(f(x,t),∇f(x,t),x,t)dxdt (70)$$
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And in the context of special relativity, people condense that even further by adding to the spacetime Four-vector as well, so you don't even need to write that separate pesky .
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The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.
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