Least action principle for a free relativistic particle (Landau)

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The discussion revolves around the derivation of the variation of the action for a relativistic free particle as presented in Landau's "The Classical Theory of Fields." The user expresses confusion regarding the correct formulation of the variation of the invariant measure ds, specifically questioning why the expression involves a factor of dxi rather than a factor of 2 in the denominator. Clarification is sought on the calculation of the variation of the product of coordinates, δ(x_ix^i), and its implications for the derivation. The user is particularly interested in understanding how the terms cancel out to align with Landau's final expression. The thread highlights the complexities of covariant variation calculus in the context of relativistic physics.
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Reading the Landau's "The classical theory of fields" (chapter 2, section 9 ) I have some doubts in explaining the steps in derivig the formula for the variation of the action for the relativistic free particle http://books.google.it/books?id=QIx...age&q="to set up the expression for"&f=false". Given the invariant element of measure:

ds=\sqrt{dx_idx^i}

where x^i ( x_i ) are the four contravariant (covariant) coordinates which parametrize the world line of the free particle, I have to vary respect x^i, that is I make the variation \delta x^i. So my doubts are about the second step of the formula before the 9.10, that is why:

\delta(ds)=\frac{d x_i \delta d x^i}{ds}

is obtained, instead of (IMH and erroneous O):

\delta(ds)=\frac{d x_i \delta d x^i}{2 \cdot ds}

?

Can someone be so kind to show me the steps?
 
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What is \delta \left(x_{i}x^{i}\right) = ?
 
It should be:

\delta (x_i x^i) = \delta (c^2t^2-r^2) = 2 (c^2 t \delta x^0 - r \delta x^i)

but, sorry, I don't get the point... that is... should I calculate

\delta (dx_i dx^i)

?
 
The "d" in the brackets is not important. That 2 you have obtained in front cancels the one in the denominator, thus giving you the final expression from Landau's book.
 
thx for the moment. I hope to need no more help in covariant variation calculus... ;-)
 
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