How does colpitts and hartley oscillators start oscillation?
While it's true that if you have an LC circuit, pass current through the L, and then cut it off, the circuit will oscillate for a while before the oscillations die away. However, that's irrelevant to how Colpitts or Hartley oscillators start up.

An oscillator of this type has greater than unity gain at a specific frequency round the loop that contains the active device, with the LC circuit acting as a filter. As there's more than unity gain, any small disturbance, either the switch-on transient or circuit noise, will be amplified round the loop and will build up oscillations. Eventually when the oscillations are big enough, some mechanism will reduce the loop gain to unity, and the level will stabilise.

In a simulator, the situation is slightly different. Unless it's been modelled (and we rarely do) there will be no circuit noise. In addition, the initial DC bias solve step may complete so accurately that there's no 'switch-on transient'. So although the oscillator has greater than unity loop gain, there's nothing to amplifier, and the oscillator never starts. In this case, it's necessary to inject a disturbance manually, either with a pulse generator, or more economically by setting the initial conditions on one or more reactive components.