>>13799249You got a space Y like Y=R^3 and time X like X=R and are interested in curves c:X->R, i.e. a family of positions c(t).
You could fix a position y0 in Y write down a predicate like c(3)=y0.
With the _kinetic energy_ T, he just tacitly assumes that it's only dependent on the values and behaviour of the curve, such as
T(t) = m · v(t) / 2
where
v(t) := c'(t)
and not, for example, something weird like
T(t, x) = c(t) / x
The sensible heuristic is that object with position given by a curve c will not have a kinetic that for some reason varies throughout space.
Like the function that tells you how many hair you currently have on your head is not a spatial field that at all times also can be evaluated on the top of the Eiffel tower. Your hair concern you, not physical space. Similar with the kinetic energy of an object.
