Classical mechanics

Classical mechanics ( классическая механика, or "Newtonian mechanics") is Science, branch of physics and mathematics based on the concepts of coordinate, time and material points (point mass, материальная точка); the coordinates are elements of the 3-dimensional Euclidean space, and time is universal real parameter. The basic elements of the classical mechanics are listed below.

Basic concepts

All the world is supposed to be made of material points. These material points are merged in the Euclidean space. By default, the Catresian orthogonal system of coordinates is assumed (id est, the Decart coordinates). Each material point coordinates, which form the 3-vector. Components of such a vector are real numbers. This vector is smooth function an additional coordinate that is interpreted as time. TIme is also real number. The coordinates of each point are smooth functions of time. For simplicity, assume that coordinate also mean the 3-vector function dependent on time. This function is interpreted as trajectory. The time is often denoted with letter \(t\).

Several material points may form a physical body, or physical system. The coordinate of the physical system is assumed to be combination of coordinates of all its material points.

Often, the coordinates are denoted with letter \(\vec x= \{ x[1], x[2], x[2]\}\). If the body counts with several material points, say, \(N\) points, then the material points are numbered and denoted as

\(\vec x_n=\vec x_n(t)\), \(n=1.. N\).

The component of each vector is not required for formulation of the basic axioms of classical mechanics, but if necessary, they can be indicated with an additional subscript:

\(\vec x_n=\{ x_n[1], x_n[2], x_n[3] \} = \{ x_{n,1}, x_{n,2}, x_{n,3}\}\)

However, in the last case, it is important to keep the order of subscripts to avoid confusions.

Each material point is attributed its universal property, called mass. The material points may have different masses; in this case, they are numbered and denoted with \(m_n\), \(n=1.. N\). In such a way, that material point of mass \(m_n\) has coordinates \(\vec x_n=\vec x_n(t)\). Mass of each material point is constant and cannot vary with time.

In some cases, the detailed state of the body is not important for the consideration; then the body is characterised with its mean coordinate called also center of mass; the coordinate of center of mass of a body of \(N\) material points is defined with

\(\displaystyle X(t)= \sum_{n=1}^{N} m_n~ x_n(t)\)

At the scientific slang, the coordinate of center of mass of a body can be called simply "coordinate of a body", although it actually consists of 3 components of vector \(\vec X\) and, in principle, may depend on time.

The time derivative of coordinate is called velocity and denoted with letter \(\vec v\);

\(\vec v_n(t)=\vec x_n'(t)\), where prime means derivative.

All velocities are differentiable functions, and the time derivative of velocity is called acceleration:

\(\vec w_n(t)=\vec v_n'(t)=\vec x_n''(t)\)

The interaction between material points is characterized with forces. The properties of masses, coordinates and forces are assumed to satisfy the Laws of Newton. Historically, the 4 Laws of Newton are postulated.

Keywords

Robert Hooke, Isaac Newton, First law of Newton, Second law of Newton, Third law of Newton, Conservation laws