A particle of mass \(m\) is attached to the origin by a light elastic string of natural length \(l\), so that it is able to move freely along the \(x\)-axis if its distance from the origin is less than \(l\), but otherwise moves in a potential \(V(x)=\frac{1}{2} k(|x|-l)^{2}\) for \(|x|>l\). If the particle always moves in a straight line, sketch the potential and the phase-plane trajectories for different values of the energy \(E\). Show that \(E=\left(\sqrt{I \Omega+\beta^{2}}-\beta\right)^{2}\), where \(I\) is the action, \(\beta=\sqrt{2 k} l / \pi, \Omega=\sqrt{k / m}\). Explain briefly (without detailed calculation) how the angle variable \(\phi\) conjugate to the action may be found in the form \(\phi(x)\) and how \(x\) may be found as a function of the time \(t\). What happens in the (separate) limits of small and large energies \(E\) ?

In the realm of classical mechanics, potential energy is a concept that refers to the energy stored within a system due to the position or arrangement of different parts. Specifically, consider a particle of mass m attached to an elastic string as described in the exercise.

When the string is within its natural length l, the particle is in a state of equilibrium, and no elastic potential energy is stored in the string. This translates to the potential energy V(x) being zero, which we express mathematically as: \[V(x) = 0, \quad \text{for } |x| \leq l.\]
Once the particle moves beyond this range, the string is either compressed or stretched, and therefore, the system stores elastic potential energy. This state is modeled by the quadratic potential as follows: \[V(x) = \frac{1}{2} k(|x|-l)^2, \quad \text{for } |x| > l.\]
The constant k refers to the string's stiffness and directly influences how much potential energy is stored for a given displacement from the equilibrium position. It is crucial for students to realize that potential energy landscapes help illuminate the behavior of physical systems—particles will move in a way to minimize their potential energy, which provides insight into the dynamics of such systems.

Analyzing the motion of a system in classical mechanics can be elegantly represented through phase-plane trajectories. These are graphical plots showing how a system evolves with respect to its position and momentum over time.

A point on this phase-plane represents the state of the system at any given moment. For the described particle attached to the elastic string, if the total energy E of the system is represented by a constant value, then the phase-plane trajectory is simply a curve on the graph that shows all possible states the system can be in with that amount of energy.

Key Insights from Phase-Plane Trajectories

• Each closed loop represents a periodic motion, indicating that the system oscillates back and forth in position and momentum.
• For energies less than \(\frac{1}{2}kl^2\), the trajectories remain bound within vertical lines at \(x = l\) and \(x = -l\), showing that the particle stays within the natural length of the string.
• At energy levels equal to or greater than \(\frac{1}{2}kl^2\), the trajectories pass beyond these vertical lines, indicating that the particle stretches the string on either side.

The elegance of phase-plane trajectories lies in their simplicity; they provide a complete description of a mechanical system's motion without solving the differential equations of motion. This visual representation is a powerful tool for students to understand and predict system behavior.

When delving into more advanced concepts of classical mechanics, such as action-angle variables, we are stepping into the realm of analytical mechanics, where the motion of systems is described in terms of periodic orbits and conserved quantities.

The action, denoted by I, is one of these conserved quantities and is particularly useful in systems with periodic motion. It is calculated over one complete cycle of motion, and, intuitively, it can be thought of as the 'amount' of motion, capturing the aggregate effect of the system's dynamics over a period.

Interpreting the Action

• The action variable simplifies the treatment of complex systems, by boiling down the infinite possibilities of motion into a single descriptive value.
• It is especially useful in systems with symmetries, where it can assist in reducing the number of variables to consider.

The associated angle variable, represented by \(\phi\), complements the action. It quite literally describes the 'angle' of the system within its cyclical motion. As in the example given in the exercise, finding the angle variable often involves integration over time to discern how a system's position ties into its periodicity.
These variables are important in the study of mechanics because they provide a clearer understanding of periodic behaviors and allow for simplifications that make both qualitative, and quantitative predictions possible. For instance, understanding action-angle variables illuminates how we can capture complex motions such as the orbits of planets or the vibrations of molecules. As students dive into these concepts, they gain valuable insights into the fundamental laws that govern dynamic systems.