CONSERVATION-LAWS OF RHEO-LINEAR DYNAMIC-SYSTEMS WITH ONE-DEGREE AND 2-DEGREES-OF-FREEDOM

This paper exhibits a new method for finding quadratic conservation laws of rheo-linear dynamical systems with one- and two-degrees-of-freedom which possess Hamiltonian structure. The method depends upon the possibility of finding any solution (exact, approximate or numerical) of a system of non-linear ordinary differential equations subject to arbitrary initial conditions. For the case of a single-degree-of-freedom rheo-linear oscillator whose circular frequency and resisting-force coefficient are arbitrary functions of time, the conservation law and the auxiliary equation are identical with the results obtained by Lewis and many other authors. For the case of two-degrees-of-freedom, rheo-linear dynamical systems, the conservation laws and the method of their derivation, as far as the author is aware, have not been published previously.