If the position of a material system as measured by a particular observer changes with respect to time, that system is said to be in motion with respect to the observer. Absolute motion, then, has no significance, and only relative motion may be defined; what one observer measures to be at rest, another observer in a different frame of reference may regard as being in motion. See Frame of reference, Relative motion
The time derivatives of the various coordinates used to specify the system may be used to prescribe the motion at any instant of time. How the motion develops in subsequent instants is then determined by the laws of motion. In classical dynamics it is supposed that in principle the motion and configuration of the system may be specified to an arbitrary precision, although in quantum mechanics it is recognized that the measurement of the one disturbs the other.
The most general theory of motion that has yet been developed is quantum field theory, which combines both quantum mechanics and relativity theory, as well as the experimentally observed fact that elementary particles can be created and annihilated. See Degree of freedom (mechanics), Dynamics, Hamilton's equations of motion, Harmonic motion, Kinematics, Kinetics (classical mechanics), Lagrange's equations, Newton's laws of motion, Oscillation, Periodic motion, Quantum field theory, Quantum mechanics, Rectilinear motion, Relativity, Rotational motion
in geometry, a transformation of space that preserves the properties of a figure (dimensions, shape, and so forth). The concept of a motion was formulated as an abstraction of actual motions of rigid bodies.
A motion of Euclidean space is a geometric transformation that preserves the distance between points. It is called proper or improper depending on whether it preserves or changes orientation. Apart from a translation, a motion is an orthogonal transformation.
A proper motion in a plane can be specified relative to a rectangular system of coordinates (x, y) by the formulas
x̄ = x cos φ - y sin φ + a
ȳ = y sin φ + y cos φ + b
that show that the set of all proper motions in a plane depends on the three parameters a, b, and φ. By assigning particular values to a and b we determine a translation of the plane by the vector (a, b), and by assigning a particular value to φ we determine a rotation of the plane about the origin by the angle φ. A proper motion can be represented as a translation or as a rotation about some point. An improper motion can be represented as a reflection in a line or as a product (the result of successive application) of a translation in some direction and a reflection in a line having the same direction.
A proper motion in space can be represented as a rotation about an axis, or a translation, or a screw motion (a rotation about an axis followed by a translation in the direction of this axis). An improper motion in space can be represented as a reflection in a plane, or a product of a reflection in a plane and a rotation about an axis perpendicular to this plane, or a product of a reflection in a plane and a translation determined by a vector parallel to this plane.
Apart from a translation, a motion in space can be represented analytically by a linear transformation with an orthogonal matrix having a determinant equal to 1 or -1 depending on whether the motion is proper or improper.
The concept of a motion is carried over into Riemannian spaces and spaces of affine connection. The concept of a motion plays an important role in the Riemannian spaces associated with the theory of relativity (the strong asymmetry of gravitational fields imposes restrictions on the motion of rigid bodies in such spaces).
Motion may be used as an undefined term in the axiomatic development of geometry. In this case, the axioms of a motion are substituted for the axioms of congruence. The congruence of segments, angles, and other figures is then defined in terms of motions (two figures are said to be congruent if there exists a motion that carries one figure into the other). The totality of motions forms a group.
E. G. POZNIAK
a mode of matter’s existence and its most important attribute. In its most general aspect motion is “change in general” (F. Engels, in K. Marx and F. Engels, Soch.,2nd ed., vol. 20, p. 563), any kind of interaction between material objects.
The concept of the universality of motion originated in remote antiquity among the thinkers of China, India, and Greece. The ancient Greek philosophers, including the Milesian school, Heraclitus, Democritus, and Epicurus, regarded the primal principles of all things—water, apeiron, air, fire, and atoms—as being in a state of constant motion and change. Aristotle believed that an “ignorance of motion of necessity means an ignorance of nature” (Physics,III 1, 200 b; Russian translation, Moscow, 1936). The conception of motion as a mode of matter’s existence was precisely formulated in the 18th century by the British philosopher J. Toland and subsequently by the French materialist P. Holbach. However, they conceived of motion itself as being merely mechanical displacement and interaction. Profound ideas concerning the interpretation of motion were expressed by the objective idealists G. W. Leibniz, G. Hegel, and others. Thus, Hegel transcends the conception of motion as merely mechanical displacement and formulates general laws of motion—the law of the transition from quantitative changes to qualitative ones, the law of the conflict of opposites, and the law of the negation of the negation.
A new and higher stage in the understanding of motion as a mode of matter’s being was attained with the creation of dialectical materialism by K. Marx and F. Engels. This doctrine concerning motion was further developed during the 20th century in the works of V. I. Lenin. Dialectical materialism has substantiated the link between matter and motion in a new way and has established the principle that matter in motion can be neither created nor destroyed. “Matter without motion is just as inconceivable as motion without matter. Motion, therefore, can no more be created or destroyed than matter itself (F. Engels, in K. Marx and F. Engels, Soch.,2nd ed., vol. 20, p. 59). The principles of the connection between matter and motion and of the inability of moving matter to be destroyed or created acquired a special significance in the light of the great discoveries in the natural sciences of the 19th and 20th centuries. Lenin put forward the principle of the unity of matter and motion in order to counter the attempts of the so-called energeticism school to reduce matter to energy. Lenin emphasized that matter is not something inert, something to which motion “is added,” nor is it the empty “subject,” of the predicate “to move.” Rather, matter is the foundation and universal vehicle of all states of motion and development. “Whether we say the world is moving matter or that the world is material motion makes no difference whatever” (Poln. sobr. soch.,5th ed., vol. 18, p. 286). Dialectical materialism holds that, along with its material quality, the principal characteristics of motion are its absoluteness and its contradictory quality. The motion of matter is absolute, whereas all rest is relative and represents one of the aspects of motion. It determines all the characteristics and manifestations of the surrounding world and the inner content of all objects and phenomena. The contradictory nature of motion consists in the unbroken unity of two opposing factors—changeability and stability, motion and rest. In fact, the concept of change makes sense only in connection with the idea of a relatively stable, continuously fixed state. This very change, however, is at the same time also a fixed state, which continues and maintains itself; that is, it also possesses stability. In this contradictory unity of changeability and stability the leading role is played by changeability, for everything new in the world first appears by means of it, whereas stability and rest merely fix what has been attained through this process.
The motion of matter is diverse in its manifestations and exists in various forms. Classification of the principal forms of motion involves making a distinction between inorganic matter and the biological and social spheres. Motion may occur in an ascending line, advancing from simple forms to more complex ones, from what is lower to what is higher; such motion is called development. Motion may also proceed along a descending line, toward simpler forms; that is, it may be regressive.
Motion occurs in space and time, which, as the theory of relativity has established, are merely relative “aspects” of a single form of matter’s existence, that is, space-time.
V. I. SVIDERSKII