Inertial system

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inertial frame

A reference system (frame) in classical mechanics and in special relativity theory in which Newton's first law is applicable. The notion of an inertial system is an abstraction, but for a wide class of physical phenomena (a description of strong gravity fields is outside their scope) there exist reference systems which are very close to inertial ones. In the case when no global inertial system exists (for example, in general relativity theory), it is possible to introduce at each point a reference system that is approximately inertial in a small neighbourhood of this point. In the case of general relativity theory, such reference systems are called local Galilean systems (Galilean frames). The existence of a local Galilean system implies that the tangent space at a given point approximates curved space-time.

Each reference system that moves without acceleration relative to an inertial system is also an inertial system. Various inertial systems in classical mechanics are related by the transformations of the inhomogeneous group of Galilean transformations (cf. Galilean transformation); in special relativity theory by transformations of the Poincaré group (see Lorentz transformation). The laws of classical mechanics and the laws of special relativity theory are invariant with respect to the inhomogeneous Galilean group and the Poincaré group, respectively (see Relativity principle). Various conservations laws (of energy, momentum, angular momentum) which hold only in inertial systems are a corollary of the principles of relativity. In special relativity theory an inertial system is usually defined as a Galilean coordinate system, and in classical mechanics as a Cartesian coordinate system.


[1] V.A. [V.A. Fok] Fock, "The theory of space, time and gravitation" , Macmillan (1964) (Translated from Russian)
[2] C. Møller, "The theory of relativity" , Clarendon Press (1972)



[a1] S. Weinberg, "Gravitation and cosmology" , Wiley (1972) pp. Chapt. 3
[a2] D.F. Lawden, "An introduction to calculus and relativity" , Methuen (1962)
[a3] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
How to Cite This Entry:
Inertial system. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article