I Problem with the harmonic oscillator equation for small oscillations

AI Thread Summary
The discussion centers on the derivation of Euler-Lagrange equations for a double pendulum, leading to concerns about the impact of second derivatives on the frequency of small oscillations. The equations presented resemble the harmonic oscillator form, prompting questions about how the terms x'' and y'' influence the overall frequency. A common approach to solve such linear coupled ordinary differential equations involves using an ansatz to find the eigenvalues and corresponding eigenvectors, ultimately simplifying the equations. The user identifies that both x and y must be equal in the two equations, resulting in a standard harmonic equation, but acknowledges the need to find a second normal mode. The validity of the equations for the double pendulum remains uncertain, highlighting the complexity of the problem.
PhillipLammsoose
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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.
 
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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...
 
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