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My professor gave a lecture on the material, and his explanation struck me as a truism.

Essentially, he argued that the difference between the Lagrangian evaluated along the parameters describing the true path and the Lagrangian evaluated along parameters corresponding to a mild perturbation of the parameters by a function αη(x), where α is a scale factor, is zero.

Where exactly is the profundity in this statement? I understood it as "If we deviate the parameters away from the parameters that minimize the integral, and then take the limit as that deviation vanishes, the difference between the path described by these two sets of parameters is zero and the path must be the true path." Well of course this is true. What am I missing?

Alternatively are there any decent texts that outline this principle at an undergraduate level?