harmonic oscillator

Examples of harmonic oscillator in the following topics:

• Applications of Second-Order Differential Equations

  • In this atom, we will learn about the harmonic oscillator, which is one of the simplest yet most important mechanical system in physics.
  • In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, $F$, proportional to the displacement, $x$: $\vec F = -k \vec x \,$, where $k$ is a positive constant.
  • If $F$ is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency.
  • Driven harmonic oscillator: Driven harmonic oscillators are damped oscillators further affected by an externally applied force $F(t)$.
  • A solution of damped harmonic oscillator.

• The Integral Test and Estimates of Sums

  • The harmonic series $\sum_{n=1}^\infty \frac1n$ diverges because, using the natural logarithm (its derivative) and the fundamental theorem of calculus, we get:
  • The above examples involving the harmonic series raise the question of whether there are monotone sequences such that $f(n)$ decreases to $0$ faster than $\frac{1}{n}$but slower than $\frac{1}{n^{1 + \varepsilon}}$ in the sense that:
  • The integral test applied to the harmonic series.

• Alternating Series

  • The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).

• Tips for Testing Series

  • The integral test applied to the harmonic series.

• Predator-Prey Systems

  • However, a linearization of the equations yields a solution similar to simple harmonic motion with the population of predators following that of prey by 90 degrees.