I'm reading Landau and Lifshitz, 'The Classical Theory of Fields', and I'm a bit confused with part of the principle of least action derivation for a free particle.

They say that the variation in the action $S$ between events $a$ and $b$ is

$\delta S = - m c \delta \int_a^b ds$,

which I am on board with. They then use that $ds = \sqrt{dx_i dx^i}$ and some other intermediate steps to arrive at

$\delta S = - m c \int_a^b u_i d \delta x^i$.

This is really a sticking point in the derivation for me. It looks like they have gone from integrating a scalar with respect to interval $s$, to integrating a four-vector $u_i$ with respect to some $\textit{deformation}$ $\delta x^i$, which itself is a four-vector. This raises two points of confusion for me:

  1. What does it mean to integrate with respect to a four-vector?

  2. What does it mean to integrate with respect to a change in coordinate, rather than a coordinate itself? If we just forget that these are four-vectors, what does it mean if I were to write $\int_a^b f(x) d \delta x$, for instance?

I can't really make any sense of this. In fact, I think I may just be grossly misunderstanding the notation, but if so, I have no idea what the notation should mean. Can anyone shed any light on this?



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