# Relativistic PLA in 4D: Integrating with respect to a small change?

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?

Thanks!