Classical Mechanics is formulated for several particles. You can study systems constituted by a huge amount of particles, but they are all independent, there is no notion os "closeness". If we want our theory to be "local" we need to introduce fields.
Example.
Think in a particle $p_1$ oscillating around an equilibrium point as an entity to which we measure its deviation, that change with time. Let's call this measure $\phi_1(t)$. This would be a (0+1)-dimensional field.
If we had lots of particles $p_1,\ldots,p_N$ we would have lots of fields of this kind: $\phi_1(t),\ldots,\phi_N(t)$. But if we consider infinite (continuum) particles we have a $(1+1)$-dimensional field
$$ \phi(x,t) $$representing a string. See also this. $\blacksquare$
To my knowledge (20/4/22) there are different approaches to Classical Field Theory:
In classical physics, the primary reason for introducing the concept of the field is to construct laws of Nature that are local. The old laws of Coulomb and Newton involve “action at a distance.” This means that the force felt by an electron (or planet) changes immediately if a distant proton (or star) moves. This situation is philosophically unsatisfactory. More importantly, it is also experimentally incorrect. The field theories of Maxwell and Einstein remedy the situation, with all interactions mediated locally by the field. The requirement of locality remains a strong motivation for studying field theories in the quantum world.
Extracted from here.
See also this in page 17, although I don't understand yet. Also in this video.
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Author of the notes: Antonio J. Pan-Collantes
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