# Link between harmonic functions and harmonic oscillators?

• Navierstoked
In summary, harmonic functions and harmonic oscillating systems refer to things that are periodic in both mathematics and physics. In math, they are often described using sines and cosines, while in physics, they are discussed in relation to pendulums and similar objects. Although all solutions of the Laplace equation in complex analysis are by definition harmonic, not all of them are necessarily periodic. It is important to distinguish between mathematics and physics when discussing these concepts. For more details, researching on Wikipedia may be helpful.

#### Navierstoked

I'm a bit confused wether or not there is a link between harmonic functions (solutions of the Laplace pde) and harmonic oscillating systems? What is the meaning of "harmonic" in these cases? Thanks!

Harmonic in both physics and math usually refer to things which are periodic. In math it is usually used in descriptions involving sines and cosines. In physics it is usually used in discussing pendulums and such things.

mathman said:
Harmonic in both physics and math usually refer to things which are periodic. In math it is usually used in descriptions involving sines and cosines. In physics it is usually used in discussing pendulums and such things.
Ok.. So are all the solutions of the Laplace equation (in complex analysis) periodic? By definition, they are harmonic. So is a harmonic function and a periodic function the same thing?

Harmonic functions are more general. I suggest looking things up on Wikipedia will give you a lot more details. The main point I was trying to make is that you need to separate mathematics and physics to clarify the understanding.

I can confirm that there is indeed a link between harmonic functions and harmonic oscillators. Both of these concepts involve the concept of harmonicity, which refers to a repetitive or oscillatory motion or pattern.

In the case of harmonic functions, they are solutions to the Laplace partial differential equation, which describes a type of harmonic motion in two or more dimensions. This motion is characterized by a steady oscillation between two extremes, without any external forces acting on the system.

Similarly, harmonic oscillators are physical systems that exhibit harmonic motion, where the restoring force is directly proportional to the displacement from equilibrium. This type of motion can be seen in various systems, such as a mass attached to a spring or a pendulum swinging back and forth.

The term "harmonic" in both cases refers to the fact that the motion or function follows a harmonic or sinusoidal pattern. This is due to the fact that the solutions to the Laplace equation and the equations of motion for harmonic oscillators involve sine and cosine functions.

In summary, the link between harmonic functions and harmonic oscillators lies in their shared characteristic of harmonicity, which describes a repetitive or oscillatory motion or pattern. Both concepts are important in understanding various physical phenomena and have applications in fields such as engineering, physics, and mathematics.

## 1. What is a harmonic function?

A harmonic function is a function that satisfies the Laplace's equation. In simpler terms, it is a function whose value at any point is equal to the average of its neighboring points.

## 2. What is a harmonic oscillator?

A harmonic oscillator is a type of mechanical system that follows the laws of simple harmonic motion. It consists of a mass attached to a spring and is characterized by its oscillatory motion around a fixed equilibrium point.

## 3. What is the relationship between harmonic functions and harmonic oscillators?

The motion of a harmonic oscillator can be described by a harmonic function. This is because the displacement of the mass from its equilibrium point at any given time can be represented by a sine or cosine function, which are examples of harmonic functions.

## 4. What are some real-life examples of harmonic oscillators?

Some common examples of harmonic oscillators include a pendulum, a mass-spring system, and a tuning fork. They can also be found in many electronic devices, such as clocks, watches, and radios.

## 5. How are harmonic functions and harmonic oscillators used in science and engineering?

Harmonic functions and harmonic oscillators have many applications in science and engineering. They are used in fields such as physics, mathematics, and electrical engineering to model and study various systems and phenomena, such as vibrations, waves, and resonance.