I'm going over the Landau's(adsbygoogle = window.adsbygoogle || []).push({}); Mechanics, and can't get over two hurdles.

The first is the following(I'm not good with latex here, so please bear with me):

Landau asserts that the action S' between t1 and t2 of a Lagrangian L such that L is a function of (q+dq) and (q'+dq') minus the action S between t1 and t2 of a Lagrangian L such that L is a function of q and q', the gives a difference, that, expanded in powers of dq and dq' equals zero. Also, dq(t1)=dq(t2)=0.

OK, for this to be the case, S' must equal S' plus some residue R. R must then be formed of integrals of terms that are multiples of dq and dq' so that when integrated (definite) give 0.

Landau therefore expands S'-S 'in powers of' dq ad dq'. Here's my problem: How can you expand L in powers of dq and dq'? Talyor expansion allows for expansion in terms of q and q', and if I substitute q for q+dq I do in fact get the terms in dq and dq', but I also have annoying terms like 2q*dq and the like- I can see that q and q^2 terms will vanish due to S expanded.

So the question still stands: How can you expand L in powers of dq and dq'? is the above way satisfactory?

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# Landau's derivation of the Langangian expression

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