
#1
Jan213, 02:48 PM

P: 1

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 [itex]\theta_{1}''[/itex][itex]\normalsize = function[/itex] [[itex]\theta_{1}[/itex],[itex]\theta_{2}[/itex],[itex]\theta_{1}'[/itex],[itex]\theta_{2}'[/itex]] (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! 



#2
Jan213, 08:33 PM

Sci Advisor
HW Helper
Thanks
PF Gold
P: 4,502

[tex]f(x,y,z,w) = f(x_0,y_0,z_0,w_0)+\frac{\partial{f}}{\partial{x}}(xx_0)+\frac{\partial{f}}{\partial{y}}(yy_0)+\frac{\partial{f}}{\partial{z}}(zz_0)+\frac{\partial{f}}{\partial{w}}(ww_0)[/tex] where the partials are evaluated at [tex]x_0,y_0,z_0,w_0[/tex] 



#3
Jan313, 05:54 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,896

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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