Linearization of Second Order Differential Equations

Ruby Tyra
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I'm having some difficulties figuring out how to linearize second order differential equations for a double pendulum.

I have an equation that is in the form of

\theta_{1}''\normalsize = function [\theta_{1},\theta_{2},\theta_{1}',\theta_{2}']

(The original equation is found at http://www.myphysicslab.com/dbl_pendulum.html, the equations inside the orange rectangle.)

I was told to replace that function by a linear function of all four variables but I don't know where to start with that since the original equation is much more complex than the simple pendulum example we were given.

Thank you!
 
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Ruby Tyra said:
I'm having some difficulties figuring out how to linearize second order differential equations for a double pendulum.

I have an equation that is in the form of

\theta_{1}''\normalsize = function [\theta_{1},\theta_{2},\theta_{1}',\theta_{2}']

(The original equation is found at http://www.myphysicslab.com/dbl_pendulum.html, the equations inside the orange rectangle.)

I was told to replace that function by a linear function of all four variables but I don't know where to start with that since the original equation is much more complex than the simple pendulum example we were given.

Thank you!

If f = f(x,y,z,w) and you want to linearize, use the chain rule:

f(x,y,z,w) = f(x_0,y_0,z_0,w_0)+\frac{\partial{f}}{\partial{x}}(x-x_0)+\frac{\partial{f}}{\partial{y}}(y-y_0)+\frac{\partial{f}}{\partial{z}}(z-z_0)+\frac{\partial{f}}{\partial{w}}(w-w_0)

where the partials are evaluated at x_0,y_0,z_0,w_0
 
Note that what Chestermiller is saying is essentially the same as replacing the function by a Taylor polynomial in all variables, then dropping all but the linear terms. And that, in turn, is the same as replacing the "surface" by its "tangent plane".
 
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