Linearization of Second Order Differential Equations

[Ruby Tyra]

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!

 

[Chestermiller]

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 $$x_0,y_0,z_0,w_0$$

 

[HallsofIvy]

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".