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VILet’s generalize
Time to formulate the Principle of Least Action and derive Euler-Lagrange equations in covariant form
1) Covariant means that under Lorentz transformations we must obtain the same motion equations
2) In this generalized approach the generalized coordinates and velocities are field potential and its derivatives correspondingly
3) This variational form can be used for any field be it:
s=0 - scalar particle - meson - strong interactions
s=1 - vector - photon - electromagnetic
s=2 - tensor - graviton - gravitation
s stands for spin
4) I am going to use the generalized Stoke’s theorem for hypervolume and hypersurface
5) From classical mechanics (previous reel on this topic) to electrodynamics, instead of Lagrangian we will use the Lagrangian density
6) This action S is represented in a form of 4-dimensional integral over the invariant 4-volume. Now we set its variation to zero: δS=0
7) Using the product differentiation rule, we obtain two terms. But according to the Stoke’s theorem and taking into consideration that the variation of q vanishes at the boundaries δq|_Σ = 0
8) Yay! Now we obtain generalized or covariant form of Euler-Lagrange equations.
Almost forgot: for this Landau II or IV
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