Dynamics is a mathematical model which aims to both describe and predict the motions of the various objects which we encounter in the world around us. The general principles of this theory were first enunciated by Sir Isaac Newton in a work entitled Philosophiae Naturalis Principia Mathematica (1687), which is commonly known as the Principa.
Up until the beginning of the 20th century, Newton's theory of motion was thought to constitute a complete description of all types of motion occurring in the Universe. We now know that this is not the case. The modern view is that Newton's theory is an approximation which is generally valid when describing the low speed (compared to the speed of light) motions of macroscopic objects. Newton's theory breaks down, and must be replaced by Einstein's theory of relativity, when objects start to move at speeds approaching the speed of light. Newton's theory also breaks down on the atomic scale, and must be replaced by quantum mechanics.
Newton's theory of motion is an axiomatic system. Like all axiomatic systems (e.g., Euclidean geometry), it starts from a set of terms which are undefined within the theory. In the present case, the fundamental terms are mass, position, time, and force. It is taken for granted that we understand what these terms mean, and, furthermore, that they correspond to measurable quantities which can be ascribed to, or associated with, objects in the world around us. In particular, it is assumed that the ideas of position in space, distance in space, and position as a function of time in space, are correctly described by the vector algebra and calculus. The next component of an axiomatic system is a set of axioms. These are a set of unproven propositions, involving the undefined terms, from which all other propositions in the system can be derived via logic and mathematical analysis. In the present case, the axioms are called Newton's laws of motion, and can only be justified via experimental observation.
Note, incidentally, that Newton's laws, in their primitive form, are only applicable to point objects. Newton's laws can be applied to extended object by treating them as collections of point objects. In the following, it is assumed that we know how to set up a Cartesian frame of reference, and also know how to measure the positions of point objects as functions of time within that frame. In addition, it is assumed that we have some basic familiarity with the laws of mechanics, and that we understand standard mathematics up to, and including, calculus.
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