Based on the possibility of localizing physical objects in time and space, in classical mechanics, the study of the laws of movement begins with the simplest case. This case is the movement of a material point. Analytical mechanics creates the prerequisites for the presentation of the basic laws of dynamics with the conceptual idea of an elementary particle .
A material point is an object that has an infinitesimal size and finite mass. This idea fully corresponds to the concept of discreteness of matter. Previously, physicists tried to define it as a set of elementary particles in a state of displacement. In this regard, the material point in its dynamics has become just that tool necessary for theoretical constructions.
The dynamics of the object in question proceeds from the inertial principle. According to him, a material point that is not under the influence of external forces maintains its state of rest (or movement) over time. This provision is implemented quite strictly.
In accordance with the principle of inertia, the material point (free) moves uniformly and rectilinearly. Considering a special case in which the velocity is zero, we can say that the object maintains a state of rest. In this regard, we can assume that the influence of a certain force on the subject in question is reduced simply to a change in its speed. The simplest hypothesis is the assumption that the change in speed that a material point has is directly proportional to the rate of force acting on it. In this case, the proportionality coefficient decreases with increasing inertia.
It is natural to characterize a material point using the value of the coefficient of inertia — mass. In this case, the main law of the object’s dynamics can be formulated as follows: the reported acceleration at each moment of time is equal to the ratio of the force that acts on the object to its mass. The presentation of kinematics, therefore, precedes the presentation of dynamics. The mass, which characterizes the material point in dynamics, is introduced a posteriori (from experience), while the presence of a trajectory, position, acceleration, speed is allowed a priori.
In this regard, the equations of dynamics of the object state that the product of the mass of the object in question for any of its acceleration components is equal to the corresponding component of the force acting on the object. Assuming that the force is a known function of time and coordinates, the coordinates for a material point are determined in accordance with time using three ordinary differential equations of the second order in time.
In accordance with the well-known theorem from the course of mathematical analysis, the solution of this system of equations is uniquely determined by setting the coordinates, as well as their first derivatives, in any initial time interval. In other words, given the known position of the material point and its speed at a certain moment, it is possible to accurately determine the nature of its movement in all future periods.
As a result, it becomes clear that the classical dynamics of the object in question is in absolute accordance with the principle of physical determinism. According to him, the upcoming state (position) of the material world can be completely predicted if there are parameters that determine its position at a certain previous moment.
Due to the fact that the size of the material point is infinitesimal, its trajectory will be a line occupying only a one-dimensional continuum in three-dimensional space . In each section of the trajectory, a certain value of force takes place, which determines the movement to the next infinitesimal period of time.