# Conservation Law

(redirected from*Law of the Conservation of Momentum*)

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## conservation law

[‚kän·sər′vā·shən ‚lȯ]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Conservation Law

a physical law stating that the numerical values of some physical quantity do not vary with time in any process or in a certain class of processes. A complete description of a physical system is possible only within the framework of dynamic laws that define in detail the evolution of a system with time. In many cases, however, the dynamic law for a given system is unknown or too complicated. In such a situation conservation laws permit some conclusions to be drawn as to the character of the system’s behavior. The most important conservation laws are the laws of conservation of energy, momentum, angular momentum, and electric charge. These laws are valid for any isolated systems. In addition to universal conservation laws, there exist conservation laws that hold only for limited classes of systems and phenomena.

The idea of conservation originally appeared as a purely philosophical conjecture on the existence of something unchanging and stable in a perpetually changing world. The ancient materialist philosophers Anaxagoras, Empedocles, Democritus, Epicurus, and Lucretius arrived at the concept of matter as the indestructible and uncreatable basis of all that exists. On the other hand, the observation of continual changes in nature brought Thales, Anaximander, Anaximenes, Heraclitus of Ephesus, Leu-cippus, and Democritus to the conclusion that the most important property of matter is that matter is always in motion. With the development of the mathematical formulation of mechanics, two laws appeared on this foundation: the law of conservation of mass, set forth by M. V. Lomonosov and A. Lavoisier, and the law of conservation of mechanical energy, advanced by G. von Leibniz. J. R. von Mayer, J. Joule, and H. von Helmholtz subsequently discovered experimentally the law of conservation of energy in nonmechanical phenomena. Thus, by the mid-19th century the laws of conservation of mass and energy, which were interpreted as the conservation of matter and motion, had taken shape.

In the early 20th century, however, the development of the special theory of relativity brought a fundamental reconsideration of these conservation laws (*see*RELATIVITY, THEORY OF). The special theory of relativity replaced classical, Newtonian mechanics in the description of motion at high speeds comparable to the speed of light. Mass, as determined from the inertial properties of a body, was found to depend on the body’s speed. Consequently, mass characterizes not only the quantity of matter but also its motion. On the other hand, the concept of energy also underwent a change: according to Einstein’s famous equation *E* = *mc*^{2}, the total energy *E* is proportional to the mass *m;* here, *c* is the speed of light. Thus, the law of conservation of energy in the special theory of relativity united the laws of conservation of mass and energy that had existed in classical mechanics. When the laws of conservation of mass and energy are considered separately, they are not fulfilled—that is, the quantity of matter cannot be characterized without taking into account its motion.

The evolution of the law of conservation of energy shows that since conservation laws are drawn from experience, they require experimental verification and refinement from time to time. One cannot be sure that a given law or the specific statement of a law will remain valid forever, regardless of the increase in human experience. The law of conservation of energy is also interesting in that physics and philosophy are very closely interwoven in it. As the law was refined, it was gradually transformed from a vague and abstract philosophical statement into an exact quantitative formula. On the other hand, some conservation laws appeared directly in a quantitative form. Such laws include the laws of conservation of momentum, angular momentum, and electric charge and numerous conservation laws in the theory of elementary particles. Conservation laws are an essential part of modern physics.

An important role is played by conservation laws in quantum theory, particularly in the theory of elementary particles. For example, conservation laws determine selection rules, according to which elementary-particle reactions that would violate a conservation law cannot occur in nature. In addition to conservation laws that also hold in the physics of macroscopic bodies (conservation of energy, momentum, angular momentum, and electric charge), many specific conservation laws have appeared in elementary particle theory that permit explanation of experimentally observed selection rules. Examples are the laws of conservation of baryon number and lepton number; these laws are exact—that is, they hold in all types of interactions and in all processes. In addition to exact conservation laws, approximate conservation laws, which are satisfied in some processes and violated in others, also exist in the theory of elementary particles. Such approximate conservation laws have meaning if the class of processes and phenomena in which they are satisfied can be indicated precisely. Examples of approximate conservation laws are the laws of conservation of strangeness (or of hypercharge), iso-topic spin (*see*ISOTOPIC INVARIANCE), and parity. These laws are strictly satisfied in strong-interaction processes, which have a characteristic time of 10^{–23}—10^{–24} sec, but are violated in weak-interaction processes, whose characteristic time is approximately 10”^{10} sec. Electromagnetic interactions violate the law of conservation of isotopic spin. Thus, investigations of elementary particles have shown once again the necessity of verifying existing conservation laws in every domain of phenomena.

Conservation laws are closely related to the symmetry properties of physical systems. Here, symmetry is understood as the invariance of physical laws with respect to certain transformations of the quantities involved in the formulation of these laws. For a given system, the existence of a symmetry means that a conserved physical quantity exists (*see*NOETHERS THEOREM). Thus, if a system’s symmetry properties are known, then conservation laws can be found for it, and conversely.

As stated above, the laws of conservation of the mechanical quantities energy, momentum, and angular momentum are universal. The reason for this circumstance is that the corresponding symmetries may be regarded as symmetries of space-time (the universe), in which material bodies move. Thus, the conservation of energy follows from the homogeneity of time—that is, from the invariance of physical laws under a change in the origin of the time coordinate (translations of time). The conservation of momentum and the conservation of angular momentum follow, respectively, from the homogeneity of space (invariance under translations of space) and from the isotropy of space (invariance under rotations of space). Therefore, a verification of mechanical conservation laws constitutes a verification of the corresponding fundamental properties of space-time. It was long believed that, in addition to the symmetries listed above, space-time has reflection symmetry—that is, it is invariant under space inversion. Space parity should then be conserved. In 1957, however, the nonconservation of parity was experimentally detected in weak interactions. Once again beliefs regarding the underlying properties of the geometry of the universe had to be reexamined.

The development of the theory of gravitation will apparently necessitate a further reexamination of views on the symmetry of space-time and on fundamental conservation laws, particularly the laws of conservation of energy and momentum.

M. B. MENSKII