Material point

Based on the possibility of localization of physical objects in time and space, in classical mechanics, the study of the laws of displacement begins with the simplest case. This is the motion of a material point. Schematic idea of an elementary particle analytical mechanics forms the prerequisites for the presentation of the basic laws of dynamics.

A material point is an object that has an infinitesimal size and finite mass. This idea completely corresponds to the notion of the discreteness of matter. Previously, physicists have tried to define it as a set of elementary particles in a state of displacement. In connection with this, the material point in its dynamics became precisely the instrument necessary for theoretical constructions.

The dynamics of the object under consideration is based on the inertial principle. According to him, a material point that is not under the influence of external forces maintains its state of rest (or displacement) over time. This provision is carried out quite strictly.

In accordance with the principle of inertia, the material point (free) moves uniformly and rectilinearly. Considering a particular case in which the velocity is zero, we can say that the object maintains a state of rest. In this connection, it can be assumed that the influence of a certain force on the subject under consideration reduces simply to a change in its velocity. The simplest hypothesis is the assumption that the change in velocity that a material point has is directly proportional to the indicator of the force acting on it. The proportionality coefficient decreases with increasing inertia.

The characteristic of a material point is natural, using the magnitude of the inertia-mass coefficient. In this case, the main law of the dynamics of an object can be formulated as follows: the reported acceleration at each instant of time is equal to the ratio of the force that acts on the object to its mass. The exposition of kinematics, therefore, precedes the presentation of dynamics. The mass, which in dynamics characterizes the material point, is introduced a posteriori (from experience), while the presence of a trajectory, position, acceleration, speed is allowed a priori.

In this connection, the equations of the dynamics of the object state that the product of the mass of the object under consideration is one of the components of its acceleration 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 determination of coordinates for a material point in accordance with time is carried out by means of 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 the assignment of coordinates, as well as their first derivatives in some initial time interval. In other words, given the known position of the material point and its velocity at a certain point, 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 under consideration is in absolute accordance with the principle of physical determinism. According to him, the upcoming state of the material world can be predicted completely in the presence of parameters that determine its position at a certain previous moment.

In connection with the fact that the size of a 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 the force takes place, which determines the movement to the next infinitesimal time interval.