A Note on the Time-Dependent Harmonic Oscillator

Method of comparison equations and generalized Ermakov’s equation

“…(68) and (69) cannot be represented as a system of equations of motion corresponding to some Lagrangian or Hamiltonian as in Ref. [19]. Perhaps, this system could be described in terms of non-Hamiltonian dissipative dynamics, but this question requires a further study.…”

“…Again, it is interesting to look at the generalized Ermakov-Pinney equation as an equation describing a two-dimensional physical system in the spirit of the reference [19]. In our case one has…”

“…(64) and (65) and sometimes allows one to find the solution of one of these equations provided the solution of the other one is known (just as in the case of the standard Ermakov system). However, for the case of an arbitrary couple of functions f and g a simple physical or geometrical interpretation of the Ermakov system analogous to that given in [19] is not known.…”

“…(1) and (2) and the solution of one of them (2) is well known and described. Using the previous convenient analogy [19], one can say that in this case one has two time-dependent oscillators, evolving with two different time-parameters. One can try to find the solution of Eq.…”

“…(10). In terms of the two-dimensional oscillator analogy [19] this means that we exclude from the equation for the radial coordinate the dependence on the angle coordinate σ by using its cyclicity, i.e. the conservation of the angular moment.…”