Why is the Harmonic Oscillator so common in physics?

[barnflakes]

I've heard before that it's because when you expand around a minimum point in the potential energy you get a quadratic function, but I can't recall where I read this. Can anyone point me in the right direction, or give their own explanation?

I only ask because I just solved a problem in my research by assuming that the resistive force to something is F=-kx which of course leads to harmonic oscillation - and this was completely unexpected - nobody knew that the resistive force should be, it just happens that it fits the data bang on.

 

[AlephZero]

I've heard before that it's because when you expand around a minimum point in the potential energy you get a quadratic function, but I can't recall where I read this. Can anyone point me in the right direction, or give their own explanation?

When you expand any "smooth" function as a Taylor series about $x = a$ you have
$f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2$ plus higher powers of $(x-a)$

At a minimum (or maximum) the first derivative $f'(a) = 0$ so $f(x)$ is approximately a quadratic.

Any book on optimisation theory or multi-variable calculus should give the corresponding results for functions of more than one variable.

 

[haruspex]

To expand a little on what AlephZero wrote...
If the potential approximates f(a) + f''(a)x² then the restorative force, the derivative of the potential, is linear. At an energy minimum, the coefficient is negative, producing SHM.

 

[barnflakes]

That's really helpful guys, cheers.

 

[pmacias]

The Harmonic Oscillator is a fundamental concept in physics and is commonly seen in many natural phenomena. It is a system where a particle experiences a restoring force that is proportional to its displacement from a fixed point. This restoring force causes the particle to oscillate back and forth around the equilibrium point.

One reason for its prevalence in physics is that many physical systems can be approximated to behave like a Harmonic Oscillator. This is because when a system is in a stable equilibrium, it can be described by a quadratic potential energy function. When this potential energy is expanded around the minimum point, it becomes a quadratic function, which is the defining characteristic of a Harmonic Oscillator. This approximation is often used in mathematical models to simplify the analysis of complex systems.

Additionally, the Harmonic Oscillator is a simple and elegant model that can be applied to a wide range of phenomena, from the motion of atoms in a solid to the vibrations of a guitar string. Its simplicity allows for easy understanding and application, making it a useful tool in many areas of physics.

In regards to your specific situation, the fact that the resistive force in your research was found to be proportional to displacement is not a coincidence. In many systems, the resistive force is proportional to displacement, leading to harmonic oscillations. This is due to the fact that the restoring force in a Harmonic Oscillator is proportional to displacement, and the resistive force acts in the opposite direction to the restoring force. Thus, your unexpected finding is actually in line with the fundamental principles of the Harmonic Oscillator.

In conclusion, the Harmonic Oscillator is a common concept in physics because of its simple and universal nature. Its prevalence can be attributed to its ability to approximate many physical systems and its usefulness in mathematical models. Your discovery in your research is a testament to the wide applicability of this fundamental concept in physics.