The material pointThe notion of a material point is widely used in mechanics. When the motion of a body is considered, the body is usually reduced to a point endowed with mass. This procedure can be soundly justified when applied to stars and their motion as observed from earth: the distance to the stars is so huge compared to their dimension, that the latter can be neglected. The reduction is not so obvious, however, when the material point replaces a ball, a hammer or a car. Is there any justification at all for that?
Any motion of a rigid body (well, here comes another idealization, since there is no absolutely rigid body in real life) can be decomposed into two simple kinds of motion: translation of the center of mass and rotation around this center. When rotation is beyond our interest or there is none, the body is replaced by a single point to which the entire mass of the object is assigned. The next step consists in writing down the corresponding motion equations for the point. Knowing its position, the position of any real point of the rigid body can be retrieved. The center of mass may, though need not, lie within the body. (For example, the center of mass of a hoop is none of its points.)
Well then, can we say that the material point is just a convenient way to describe real motion, even if it does not exist in nature? That is true, but not the whole true. According to our present knowledge, the most elementary objects, like electrones and quarks, are considered to be structureless point-like elements with mass. (Another elementary object, the photon, has no mass at all and in many cases behaves like a wave, so it has no well-determined position in space.) Obviously, what we got here are two different ideas of a material point. On the one hand, it is a fictitious object used to describe motion, on the other - it is the most elementary real component of matter. So, do they really exist? In guise of an answer let's observe that quite recently the proton and the neutron were also considered to be point-like objects. However, in the progress of high energy physics we have learned that they do have internal structure: they are composed of three point-like (?) quarks. Therefore we can still ask whether real material points exist or not and the only possible answer is: we don't know.