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Construction of Exact Invariants for Time Dependent Classical Dynamical Systems

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In the present work, we survey various methodsused for the construction of exact invariants fordynamical systems involving an explicit time dependence.More stress is placed on two-dimensional (2D) than one-dimensional (1D) systems. While bothharmonic and anharmonic time-dependent (TD) systems arediscussed in the 1D case, the construction of invariantsis carried out for several interesting central and noncentral systems in 2D. The method ofcomplexification of two space dimensions is described indetail. The TD coupled oscillator problem, which in analternative form suggests the generalization of Ermakov systems, is analyzed in greater detail. Theavailable methods in the 2D case provide only the firstinvariant, and that for a few TD systems. These methodsas such are still inadequate as far as the construction of the second invariant is concerned. The roleand scope of some of the derived invariants in thecontext of various physical problems are highlighted.The possibility of extension of some of these methods to 3D TD systems is also discussed.

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Kaushal, R.S. Construction of Exact Invariants for Time Dependent Classical Dynamical Systems. International Journal of Theoretical Physics 37, 1793–1856 (1998). https://doi.org/10.1023/A:1026605011434

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