# Problem with the harmonic oscillator equation for small oscillations

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• PhillipLammsoose
In summary, the conversation discusses solving a problem about a double pendulum and obtaining two Euler-Lagrange equations. The equations involve a term x'' or y'' that affects the frequency of small oscillations. The usual ansatz for such a problem is to use an eigenvalue problem to determine the frequencies and corresponding eigenvectors. The person discussing the problem found one of the two normal modes but is still unsure if they are the correct equations for the double pendulum.
PhillipLammsoose
TL;DR Summary
I have an extra term in my harmonic oscillator equatiion, does it destroy the solution for small frequencies?
Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations:

1) x''+y''+g/r*x=0
2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi)

the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that bother me. How do these terms effect the endresult for the frequency of small oscillations? I know w^2 (omega) would be w^2=g/r for a classic form of the equation, but what about the x'' or y''?

thanks for helping, I hope this was readable.

I'm a bit puzzled by your equations. Maybe you give some details about how you parametrized the problem?

In any case, for such a problem of linear coupled ODE's with constant coefficients the usual ansatz is
$$x=A \exp(\mathrm{i} \omega t), \quad y=B \exp(\mathrm{i} \omega t).$$
This leads to a eigenvalue problem, which determines ##\omega## (the frequencies of the fundamental modes of the system) and, for each ##\omega##, ##A## and ##B## (the corresponding eigenvectors).

I solved this by seeing that x has to equal y in the 2 equations, so I got a new equation with $$2\ddot{x}+\frac{g}{r}x=0$$ which was the generic harmonic equation I was looking for.

Well, this is one of the two normal modes. You need to find another one. For me it's still not clear that these are the right equations for the double pendulum...

## 1. What is the harmonic oscillator equation for small oscillations?

The harmonic oscillator equation for small oscillations is a second-order differential equation that describes the motion of a simple harmonic oscillator. It can be written as mx'' + kx = 0, where m is the mass of the oscillator and k is the spring constant.

## 2. Why is the harmonic oscillator equation important in physics?

The harmonic oscillator equation is important in physics because it is a fundamental equation that describes the behavior of many physical systems, such as mass-spring systems and pendulums. It also has many real-world applications, such as in oscillating circuits and in the study of molecular vibrations.

## 3. What does "small oscillations" mean in the context of the harmonic oscillator equation?

"Small oscillations" refers to the assumption that the amplitude of the oscillations is small compared to the equilibrium position. This allows for the use of a linear approximation in the equation, making it easier to solve and analyze mathematically.

## 4. How do you solve the harmonic oscillator equation for small oscillations?

The harmonic oscillator equation for small oscillations can be solved using various methods, such as the method of undetermined coefficients or the method of variation of parameters. These methods involve finding a general solution and then applying initial conditions to determine the specific solution for a given system.

## 5. What are some limitations of the harmonic oscillator equation for small oscillations?

One limitation of the harmonic oscillator equation for small oscillations is that it only applies to systems that exhibit simple harmonic motion, where the restoring force is directly proportional to the displacement from equilibrium. It also does not take into account any external forces acting on the system. Additionally, the linear approximation used in the equation may not be accurate for large amplitudes of oscillation.

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