Show that the angle \(\Phi\) between the pericenter and apocenter is equal to the semiperiod of an oscillation in the one-dimensional system with potential energy \(W(x)=U(M / x)+\left(x^{2} / 2\right)\). $$ \Phi=\int_{x_{\min }}^{x_{\max }} \frac{d x}{\sqrt{2(E-W)}} $$

The concept of an oscillation's semiperiod is fundamental in understanding the dynamics of a particle moving in a potential field. In classical mechanics, the semiperiod refers to the time it takes for a particle to travel from one extreme of its motion to the other before reversing direction. For example, in a pendulum's swing, the semiperiod would be the time it takes to go from the highest point on one side to the highest point on the other side.

In the exercise given, the semiperiod is linked to the angle \(\Phi\) between the closest and furthest points in the particle's path, known as the pericenter and apocenter, respectively. This relationship can be visualized in celestial mechanics, where planets exhibit elliptical orbits, and the semiperiod equates to half the time it takes for a planet to orbit from the perihelion to the aphelion.

To establish this link between \(\Phi\) and the semiperiod \(T/2\), integral calculus becomes a vital tool. Specifically, by integrating the motion of the particle over its path, we can directly connect the spatial dynamics with time, demonstrating the practical relevance of calculus in mechanics. This relationship underscores the periodic nature of the system and how an understanding of one-half of the motion can provide insights into the whole oscillatory behavior.

In the scope of classical mechanics, the potential energy function describes the energy stored within a system as a consequence of the position or configuration of the various parts of that system. The case presented in our exercise focuses on a one-dimensional system where the potential energy \(W(x)\) is expressed as a combination of two distinct types of energy terms: the energy due to some central force \(U(M/x)\) and the energy associated with a quadratic term \(x^2/2\).

The first component of the potential energy function resembles that of gravitational or electrostatic potential energy arising from an inverse-square law, where \(M\) is typically a mass or charge constant and \(U\) reflects the strength of the potential. The second term, \(x^2/2\), is indicative of a harmonic oscillator where the energy is proportional to the square of the displacement from an equilibrium position.

To find the turning points of the potential function—pericenter \(x_{\min}\) and apocenter \(x_{\max}\)—the derivative of \(W(x)\) is set to zero. These points are crucial because they represent the boundaries of the oscillatory motion where kinetic energy is zero and potential energy is at a maximum. Thus, understanding the potential energy function is not just about calculating energy values; it's about deducing the behavior and characteristics of the dynamic system in question.

Integral calculus plays a pivotal role in mechanics, providing a means for calculating quantities that are not directly observable or intuitive from the basic laws of motion. In the context of oscillations and energy functions, integrals allow us to bridge the gap between forces and motion over a certain path or period of time.

The integral presented in the problem \(\Phi = \int_{x_{\min}}^{x_{\max}} \frac{dx}{\sqrt{2(E - W)}}\) represents a more abstract concept – the geometrical angle between two points in the phase space of the system, given specific energy levels. This integral also encapsulates the action integral, which is central to the principle of least action in classical mechanics.

The semiperiod \(T/2\) and the integral are intimately connected via energy conservation and Newton's second law of motion. In applying integral calculus to this problem, we can conclude that the integral over the spatial dimensions leads to an expression in time—the semiperiod. Thus, we are translating spatial measurements of the system's configuration into temporal predictions of its dynamics, a principle that finds wide application in various branches of physics, from quantum mechanics to general relativity.