Write down the solution to the oscillator equation for the case \(\omega_{0}>\gamma\) if the oscillator starts from \(x=0\) with velocity \(v\). Show that, as \(\omega_{0}\) is reduced to the critical value \(\gamma\), the solution tends to the corresponding solution for the critically damped oscillator.

Understanding the behavior of a damped harmonic oscillator is pivotal in physics, especially when dealing with systems that oscillate and gradually come to a rest due to resistive forces, like friction or air resistance. An underdamped oscillator solution is particularly important as it represents a system in which the damping is not strong enough to prevent oscillations.

In mathematical terms, when the natural frequency of the system \(\omega_{0}\) is greater than the damping coefficient \(\gamma\), the system is underdamped and the solution to the oscillator equation reflects this by showing a combination of decaying exponential behavior and sinusoidal oscillation. The general solution for such a system can be expressed as:
\[x(t) = A e^{-\gamma t} \cos(\omega t + \phi)\]
Here, \(A\) represents the amplitude of the oscillation, \(\omega\) is the damped natural frequency, and \(\phi\) is the phase constant. The damped natural frequency is calculated as \(\omega = \sqrt{\omega_{0}^{2} - \gamma^{2}}\).

The inclusion of the exponential term \(e^{-\gamma t}\) reflects how the amplitude of the oscillations diminishes over time due to the damping forces at play. Students often find visualizing the graph of this function helpful, as it clearly shows oscillations that progressively decrease in amplitude. In practical applications, this type of motion might be observed in systems like car suspensions or the strings of musical instruments.

Critical damping occurs at the precise point where the damping force is strong enough to prevent oscillations but not too strong to over-dampen the system, resulting in the quickest return to equilibrium without oscillating. This concept is integral in systems design, where engineers strive for rapid stabilization without overshoot.

Mathematically, a system is critically damped when the damping coefficient \(\gamma\) equals the natural frequency \(\omega_{0}\). At this point, the solution to the displacement \(x(t)\) of a critically damped oscillator is not oscillatory, unlike the underdamped case. Instead, it is a smooth decay back to equilibrium, typified by a rapid decrease in amplitude.

An interesting aspect of critical damping is that as \(\omega_{0}\) approaches \(\gamma\) in an underdamped oscillator, the sinusoidal nature of the solution diminishes and the characteristic equation of the system no longer has complex roots, leading to a solution involving real exponentials. The solution of a critically damped oscillator can be expressed as:
\[x(t) = (C_1 + C_2t)e^{-\gamma t}\]
With the correct initial conditions, the system will move back to the equilibrium position as rapidly as possible without overshooting, characteristic of critical damping.

The initial conditions of a harmonic oscillator define its state at the beginning of the motion and are crucial for determining its subsequent behavior. These are typically given in terms of initial displacement \(x(0)\) and initial velocity \(\dot{x}(0)\). In the context of the damped harmonic oscillator, applying these initial conditions allows us to find specific solutions to the equation of motion that satisfy the given starting state of the system.

For example, if an underdamped oscillator starts at rest from a certain displacement, the initial conditions would be \(x(0) = x_{0}\) and \(\dot{x}(0) = 0\). On the other hand, if it starts from the equilibrium position with an initial velocity, the initial conditions would be \(x(0) = 0\) and \(\dot{x}(0) = v\), as seen in the given exercise. By applying these specific initial conditions to the general solution, one can derive coefficients that tailor the generic solution to the particular scenario of the system's initial state.

It is the application of these conditions that make solving physics problems practical; they allow us to predict the motion of systems under various starting scenarios. Teaching students to correctly apply these initial conditions is therefore fundamental in fostering a deeper understanding of physical systems and their analysis.