The ODE for a driven harmonic oscillator is given by
$$
\ddot{x}+2\gamma \dot{x}+\omega_0^2 x = \frac{F}{m}\cos(\omega_dt)
$$
By assuming balance of forces, i.e. energy conservation, one can solve for x and eventually arrive at
$$
\sigma=\arctan \left(\frac{\gamma \omega_d}{\omega_0^2-\omega_d^2}\right)
$$
which describes the phase relationship between the oscillator and the driving force. I understand that this phase shift gives rise to the Lorentzian line shape of the amplitude of the oscillator, since at resonance the driving force is in phase with the velocity of the oscillator, resulting in maximum power. At non-resonant frequencies the driving force is in phase only some of the times during a period. However, intuitively I'm unable to see why this phase shift is a necessary condition for (or how it relates to) energy conservation.
Best Answer
Balance of forces is not equivalent to energy conservation, and the driven oscillator is a good example of this.
Newton's second law, which is what I assume you used to find this equation of motion, does not follow from the conservation of energy. It just states that the acceleration of a body is determined by the forces acting on it according to
\begin{equation} \mathbf{a}(t) = \frac{1}{m} \mathbf{F}(t). \end{equation}
Energy, on the other hand, is a quantity related to time translation symmetry via Noether's theorem. Newton's second law does not require or follow from energy conservation.
For the driven oscillator, both the damping and the driving force break energy conservation. The damping term is a simple way to model the loss of energy of your oscillator to the environment, by heat dissipation for instance. The oscillatory driving force is constantly injecting energy into the oscillator. Since there exist these external systems which exchange energy with the oscillator, but whose dynamics are not considered in the force law, energy is not conserved.
If you decided to include the detailed description of the environment and whatever it is that is responsible for the cosine force, you would then find that energy is conserved as a whole, but can still flow from one subsystem to another. This, however, would make the equations of motion impossibly complicated and for most purposes this level of detail is not even necessary in order to understand the relevant physics, so it is usually better to just give up energy conservation and model how your system loses or gains energy phenomenologically, which is how you get the damped driven oscillator.