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In sense of classical mechanics+special relativity what is lagrangian of a photon? Lagrangian of a relativistic massive particle is as follows: $$ L_{massive}= -mc\sqrt{c^2-v^2} $$ So is it a zero?

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    Concerning the Lagrangian of a massless scalar particle, see http://physics.stackexchange.com/q/107921/2451 – Qmechanic Oct 12 '15 at 14:47

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Classical electrodynamics has a lagrangian for the classical fields, see discussion here .

The photon is an elementary particle and does not have a classical existence.

Here is on page 5 the Lagrangian for a photon

photon langrangian

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