CONSTRUCTION OF EXACT INVARIANTS FOR TIME-DEPENDENT CLASSICAL DYNAMICAL-SYSTEMS

In the present work, we survey various methods used for the constructi on of exact invariants for dynamical systems involving an explicit tim e dependence. More stress is placed on two-dimensional (2D) than one-d imensional (1D) systems. While both harmonic and anharmonic time-depen dent (TD) systems are discussed in the 1D case, the construction of in variants is carried out for several interesting central and noncentral systems in 2D. The method of complexification of two space dimensions is described in detail. The TD coupled oscillator problem, which in a n alternative form suggests the generalization of Ermakov systems, is analyzed in greater detail. The available methods in the 2D case provi de only the first invariant, and that for a few TD systems. These meth ods as such are still inadequate as far as the construction of the sec ond invariant is concerned. The role and scope of some of the derived invariants in the context of various physical problems are highlighted . The possibility of extension of some of these methods to 3D TD syste ms is also discussed.