A classical particle or, as it is also referred to, a material point, is an abstraction that does not have a genuine counterpart in nature other than as an idealisation that can be mapped on certain physical systems in some circumstances.
Planets, for example, are so far away from each other and from the Sun that their movements can be described fairly accurately by treating them as material points. But a planet-moon system can no longer be analysed in these terms, because the bodies are sufficiently close to ``see'' each other's angular dimension and that has a profound effect on the dynamics of a planet-moon system.
All physical bodies, which we encounter in our every-day life have some physical extent, and thus the description of their motion and their interaction with other bodies becomes a very complex issue.
The use of the word ``particles'' as applied to Quantum particles is a misnomer, that tends to evoke quite inappropriate Classical Physics connotations. Quantum particles are very strange objects. Although they tend to interact with macroscopic apparatuses in a point-like manner, which is why the term ``particle'' has been slapped onto them, more sophisticated experiments reveal their extended, highly non-local nature. In one of those experiments a quantum particle has been demonstrated to ``stretch'' over a distance of 15 km and seemingly pass some sort of a communication (not readily accessible to us though) within that stretch at an infinite speed. Quantum Mechanics, as it has been formulated so far, suggests that a quantum particle may even ``stretch'' like that over the entire universe, which really forces us to revise the very notions of space and time, notions that are intrinsically classical and therefore essentially incompatible with the realm of Quantum Mechanics. Yet we continue to cling to those notions even in Quantum Field Theory, because we have nothing better to replace them with.
The clash between the concepts of a classical space-time continuum and Quantum reality is ultimately responsible for most mathematical difficulties in Quantum Field Theory. But to discuss those interesting issues in more depth would take us too far from the topic of this chapter, which is numerical simulation of a material point.