Perturbation Techniques and Theory for Nonlinear Systems

[sharrington3]

Homework Statement

Given the equation
$$\ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta}$$
Determine a first-order uniform expansion for small but finite theta.

Homework Equations

Other than the equation above, none so far as I am aware.

The Attempt at a Solution

The only thing I could think to do was try to solve this differential equation via the method of undetermined coefficients, which I do not think is right at all. I then planned to expand my solution in a Taylor series about 0. This is from Ali Hasan Nayfeh's Introduction to Perturbation Techniques. My professor gave us a packet of the fourth chapter of the aforementioned text as a basis to solve this and other problems. Nowhere in the text does it give a clear example of what exactly a "first order uniform expansion" is, nor do I even know where to begin. My professor's research interests lie in nonlinear dynamics and chaos, and I fear he is going a little too in depth for my second year physics course. Thank you for any input.

 

[vela]

I'm only making an educated guess here, but I think what you want to do is expand the trig functions using the Taylor series and retain only the lowest-order non-vanishing term. This will leave you with a linear second-order differential equation. Then you want to convert this second-order equation into a system of two first-order equations.

 

[fluidistic]

I think that finding the solution to the original ODE and then expand it using Taylor series is equivalent to solve the "simplified" ODE that vela suggests. Vela's way is much easier for sure.

 

[sharrington3]

That's something along the lines of what I thought of doing. I read up on the subject, and "uniform expansion" only means "without secular terms", so the approximation of my system won't blow up as t→∞. I'm just going to do the Taylor series DE thing. Thanks for your input, guys. It's greatly appreciated.

 

[paranormal]

I would like to provide a response to the above content by explaining the concept of perturbation techniques and theory for nonlinear systems and how it applies to the given homework statement.

Perturbation techniques and theory are methods used to approximate solutions for nonlinear systems, where the equations cannot be solved analytically. These methods involve expanding the solution in a series of small parameter, typically denoted as ε, and then solving the equations order by order. This allows us to obtain an approximate solution that is valid for small values of the parameter ε.

In the given homework statement, we have a nonlinear system described by the equation \ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta}. To apply perturbation techniques, we can introduce a small parameter ε and write the equation as \ddot{\theta}=\epsilon f(\theta), where f(\theta)=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta}. Now, we can solve this equation order by order in ε to obtain an approximate solution for small values of ε.

A first-order uniform expansion refers to expanding the solution in a Taylor series about the point \theta=0, where ε is also assumed to be small. This means that we can write the solution as \theta(\epsilon)=a_0+a_1\epsilon+a_2\epsilon^2+..., where a_0, a_1, a_2, etc. are constants that can be determined by substituting this series into the original equation and equating coefficients of ε to zero.

In summary, the approach to solving this problem using perturbation techniques would involve introducing a small parameter ε, rewriting the equation in terms of this parameter, and then expanding the solution in a Taylor series about \theta=0. The resulting series can then be used to obtain an approximate solution for small values of ε. I would recommend consulting your professor or a textbook on perturbation techniques for further guidance and examples.