Let $\mathscr L (\phi(\mathbf x), \partial \phi(\mathbf x))$ denote the Lagrangian density of field $\phi(\mathbf x)$. Then then actual value of the field $\phi(\mathbf x)$ can be computed from the principle of least action. In case of motion of particles, I know that principle of least action comes from Newton's second laws. But why does the principle of least action also hold for classical fields like EM field and gravitational field? Is there any deep reason why it holds for both EM and gravitational field?


marked as duplicate by Brandon Enright, Qmechanic Apr 11 '14 at 10:22

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/9/2451 , physics.stackexchange.com/q/3500/2451 , physics.stackexchange.com/q/15899/2451 and links therein. $\endgroup$ – Qmechanic Apr 10 '14 at 12:28
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    $\begingroup$ @Qmechanic The question I'm asking here is very different from those two questions. I'm not asking why motion of a particle follows the principle of least action, which comes from the Newton's laws. My question is for classical field theory like EM and gravitational field. One obvious answer is Newton's gravity law and Maxwell eqns imply the the principle of least action. But the fact that both of these field theories satisfy the least action principle, does that mean there is something deeper going around? Is it the particular of these laws which frorce them to satisfy the least action. $\endgroup$ – user774025 Apr 11 '14 at 2:22