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(in the classical sense), a mechanical system with a finite number of degrees of freedom—for example, a system with a finite number of material particles or solid bodies that is moving in accordance with the laws of classical dynamics. The state of such a system is generally characterized by its configuration and by the speed of change of the configuration, and the law of motion indicates the rate at which the state of the system changes.
In the simplest cases, the state may be characterized by means of the values wl … , wm, which can assume random (real) values, in which case different states correspond to two different sets of values wl, … , wm and wl’, … , w’m, and vice versa. The proximity of all values wi to w’l signifies the proximity of corresponding states of the system. The law of motion, then, is recorded as a system of ordinary differential equations:
(1) Wi = fi(Wl, … , Wm) i = 1, … , m
Regarding the values w1, … , wm as coordinates of a point w in an m-dimensional space, the corresponding condition of the dynamic system may be conceptualized geometrically by means of the point w. This point is called the phase point (sometimes also the point of representation or presentation), and the space is called the phase space of the system (the adjective “phase” is used because the states of a system were formerly often called its phases). The change in the system with time is depicted as movement of the phase point along a certain line (the so-called phase trajectory; frequently called simply the trajectory) in phase space. A vector field that juxtaposes to each point w a vector f(W) that originates from it, with components
(f1(Wl, … , Wm), … , fm(Wl, … , Wm))
is defined within the phase space. The differential equations (1), which by means of the designations introduced above may be written in abbreviated form as
(2) W = f(W)
signify that at each moment of time the vector velocity of the movement of the phase point is equal to the vector f(w), which originates at that point w of the phase space at which the moving phase point is located at a given moment. This is the kinematic interpretation of a system of differential equations (1).
For example, the state of a particle without inner degrees of freedom (a mass point) that is moving in a potential field with the potential U(x1, x2, x3) is characterized by its position x = (x1, x2, x3) and velocity x; the momentum p = mx, where m is the mass of the particle, may be used instead of velocity. The law of motion of the particle may be written as
Formulas (3) are an abbreviated notation of a system of six ordinary differential equations of the first order. Here six-dimensional euclidian space serves as the phase space. The six components of the vector of phase speed are the components of ordinary velocity and force, and the projection of the phase trajectory onto the space of the positions of the particle (parallel to the space of the momenta) is a particle trajectory in the ordinary sense of the word.
The term “dynamic system” is also applied in a broader sense, signifying a random physical system (for example, an automatic control system or a radio-engineering system) described by differential equations of form (1) or (2), or simply a system of differential equations of such a form, irrespective of the origin of the system.
REFERENCESNemytskii, V. V., and V. V. Stepanov. Kachestvennaia teoriia differentsial’nykh uravenii, 2nd ed. Moscow-Leningrad, 1949.
Coddington, E. A., and N. Levinson. Teoriia obyknovennykh differentsial’nykh uravnenii. Moscow, 1958. Chapters 13–17. (Translated from English.)
Halmos, P. R. Lektsii po ergodicheskoi teorii. Moscow, 1959. (Translated from English.)
Lefschetz, S. Geometricheskaia teoriia differentsial’nykh uravnenii. Moscow, 1961. (Translated from English.)
D. V. ANOSOV