Energy exchange in the dissipative time-dependent harmonic oscillator: Physical interpretation of the Ermakov invariant
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The energy of a mechanical system as well as other invariants can be obtained using a complementary variable formulation. This approach is extended here to systems with a dissipative force. The damping coefficient depends linearly on the velocity, but is allowed to have an arbitrary time dependence. An invariant Q$_{00}$ is obtained in terms of linearly independent solutions. A semipositive definite version of this quantity is the Ermakov invariant. This scenario including damping, allow us to give a physical meaning to a closely related quantity $\varepsilon^{ex}_ω$ = $\frac{1}{2}$mq00, which is the energy exchanged between the kinetic and potential energies per unit frequency, or $\varepsilon^{ex}_t$ = $\frac{1} {2}$,κq00 which is the energy exchange per period. The q00 exchange energy is positive under light damping. Under critical or heavy damping, when no oscillations occur,q00 is either zero or negative. Thus,q00 $\geq$ 0 is a measure of the back and forth energy exchange. This periodic energy transfer is compared with the usual oscillator energy of the damped system.To this end, the kinetic energy is split into conservative and dissipative terms. The energy ripples superimposed in the exponential decay are described by a dissipative modulation term. In the vein of Ermakov’s formalism,the amplitude and phase nonlinear differential equations are derived for a time-dependent damped system. The complementary variables and Ermakov formalisms are then compared.
M FERNÁNDEZ GUASTI^{1} ^{}
Volume 97, 2023
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