Calculus of Variation: "Help Me Understand a Step!

[Gavroy]

hey

I do not understand a step here!

The integral is:

$$\delta S(x,t)=-mc \int_a^b u_i d \delta x^i =0$$

and now they say one should do integration by parts, but I do not know how this should work here?

Where are my two functions?As far as I see there is only the four-velocity and I do not how it depends on the differential so I wanted to ask you whether one of you could explain this step to me.

The result should be:

$$\delta S(x,t)=-mc u_i \delta x^i |^b_a + mc\int_a^b \delta x^i \frac{du_i}{ds} ds$$

I have not a clue what happened there! Can you help me?

 

[fzero]

It's probably clearer if you write the original integral in longhand:

$$\int_a^b u_i \frac{d (\delta x^i)}{ds} ds$$

The two functions are $$u_i$$ and $$\delta x^i$$.