Interpretation of abstract dynamics

As I understand the abstract approach to dynamics, the state of "a system" is defined by some (possibly vector valued) function f(x,t) where t is time and x is a vector that includes information about 1 or more particles - their position, momentum etc. It can also include information about various partial derivatives of f evaluated at t. The dynamics of the system is defined by the transformations T(dt) that transform f(x,t) to f(x, t + dt).

Can this interpretation be applied to represent fields, such as the electro-magnetic field, or a material medium with sound waves passing through it?

A common interpretation of abstract dynamics is that f(x,t) evaluated at x=x0,t=t0 represents the state of a possibly complex system (e.g. a container of gas) and transformations T(dt) show how that that state proceeds from this initial condition to its state at a later time. We don't think of f(x1,t0) and f(x2,t0) as being two things that both exist at t0. Instead, in a manner of speaking, we think of f(x1,t0) and f(x2,t0) as denoting the same thing set up in two different initial states at time t0.

This interpretation is different that thinking of f(x,t0) as a set of distinct locations in space (with associated properties of various sorts) that all exist simultaneously at time t0. For example we might think of f(x1,t0) and f(x2,t0) as representing the states of two different locations in a field at time t0.

Is this latter interpretation used?