![]() ![]() The complementary variables and Ermakov formalisms are then compared. In the vein of Ermakov’s formalism, the amplitude and phase nonlinear differential equations are derived for a time-dependent damped system. ![]() The energy ripples superimposed in the exponential decay are described by a dissipative modulation term. To this end, the kinetic energy is split into conservative and dissipative terms. ![]() This periodic energy transfer is compared with the usual oscillator energy of the damped system. An invariant \(Q_\ge 0\) is a measure of the back and forth energy exchange. The damping coefficient depends linearly on the velocity, but is allowed to have an arbitrary time dependence. This approach is extended here to systems with a dissipative force. The energy of a mechanical system as well as other invariants can be obtained using a complementary variable formulation. ![]()
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