# Invariance

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## invariance

[in′ver·ē·əns]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Invariance

in automatic control systems, the independence of some coordinate of a system from external actions applied to it. The independence of one of the controlled coordinates of the system from all external actions or the independence of all coordinates from one particular action is called polyinvariance. Often the conditions of invariance cannot be met precisely; in this case we speak of invariance with a accuracy up to a certain preas-signed magnitude. In order to realize the conditions of invariance the system must have at least two channels for propagation of the action between the point of application of the external action and the coordinate whose invariance must be secured (B. N. Petrov’s principle of two channels). The ideas of invariance are used in automatic control systems for aircraft and ships, in the control of chemical processes, and in the construction of servomechanism and, especially, combined systems in which the principles of deviation and disturbance control are used simultaneously.

### REFERENCES

Kukhtenko, A. I.*Problema invariantnosti v anomatike*. Kiev, 1963.

Petrov, B. N., and V. Iu. Rutkovskii. “Dvukhkratnaia invariantnost’sistem avtomaticheskogo upravleniia.”

*Dokl. AN SSSR*, 1965, vol. 161, no. 4

V. IU. RUTKOVSKII

## Invariance

invariability, independence of the physical conditions. Invariance is considered most often in a mathematical sense as the invariability of some quantity under certain transformations. For example, if we consider the motion of a material particle in two coordinate systems rotated at some angle to each other, the projections of the velocity of motion will change on transition from one frame of reference to the other, but the square of the velocity and consequently the kinetic energy will remain unchanged, that is, the kinetic energy is invariant with respect to spatial rotations of the frame of reference. The transformations of coordinates and time when switching from one inertial frame of reference to another (Lorentz transformations) are an important case of transformations. Quantities that do not change during such transformations are said to be Lorentz-invariant. An example of such an invariant is the so-called four-dimensional interval whose square is equal to *s*_{12}^{2} = (x_{1} − x_{2})^{2} + (y_{1} − y_{2})^{2} + (z_{1} − z_{2}) − c^{2}(t_{1} −_{2})^{2}, where *x*_{1}, *y*_{1}, *z*_{1} and *x*_{2}, *y*_{2}, *z*_{2} are the coordinates of two points in space at which some event transpires, *t*_{1} and *t*_{2} are the moments of time at which the event occurs, and c is the velocity of light. Another example: the intensities of an electric field E and a magnetic field H change during Lorentz transformations, but *E*^{2} − *H*^{2} and (EH) are Lorentz-invariant. In the general theory of relativity (the theory of gravitation), quantities that are invariant with respect to transformations to arbitrary curvilinear coordinates are considered.

The importance of the concept of invariance is determined by the fact that with its help it is possible to single out quantities independent of the choice of the frame of reference, that is, quantities that characterize the internal properties of the object being investigated.

Invariance is closely connected with the laws of conservation, which are of great importance. The equivalence of all points in space (the homogeneity of space), which is expressed mathematically in the form of the requirement for the invariance of some function defining the equations of motion (Lagrange) with respect to the translations of the origin of coordinates, leads to the law of conservation of momentum. Similarly, the equivalence of all directions in space (the isotropy of space) leads to the law of conservation of the moment of momentum, the equivalence of all moments of time leads to the law of conservation of energy, and so forth (the Noether theorem).

V. I. GRIGOR’EV