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vincelost September 15th, 2017 05:33 AM

Deducing Lagrangian of a system of free particles
 
Dear All,
In Landau - Vol.1 (Mechanics) - Chapter 1 - Paragraph 4
there is the passages to obtain Lagrangian of a system of free particles.
From the chapter before it was deduced that:

L = L(v^2), that is Lagrangian is only function of v^2

Than consider a new reference system with in which velocity of particle is:

v' = v + epsilon
so v'^2 = v^2 + 2v*eps + eps^2

so L(v'^2) = L(v^2 + 2v*eps + eps^2)

Then he said: "Expanding this expression in power of epsilon and neglecting terms above the first order, we obtain:"

L(v'^2) = L(v^2) + [partial_d (L) / partial_d (v^2)] * 2 * v * epsilon

Please could you kindly illustrate the passages to reach this result?

Thank you

Vincenzo


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