# Linear approximation of non-linear equation.

I am trying to understand what it means to make a linear approximation but will get more specific in a minutes. First, some context.

The problem I am working on is more of a physics but the question I have is a purely mathematical one (I think). I am trying to derive the linear equations of motion of a system. It is a 2 DOF, so two equations. I came up with: $$I_x*\ddot{q_x} + K_1*q_x = -r*F_1*sin(q_x)+r*F_2*cos(q_y)$$ and $$I_y*\ddot{q_y} + K_2*q_y = -r*F_1*cos(q_x)-r*F_2*cos(q_y)$$ From above, there are two variables, $$q_x$$ and $$q_y$$. You can ignore all other letters since I want to focus on the trigonometric parts, on the right side. The problem is that there are two non-linear elements, the cos(q) and the sin(q).

I have two questions now:

Question 1: For the specific example (assuming that the equations are correct), the linear approximation of this system would be to use Taylor series, and as a result say $$cos(q)=1$$ and $$sin(q)=q$$ ? I have seens that many times, when using Taylor, for the first one, they would right $$cos(q)=1-\frac{q^2}{q!}$$ because it is a better aproximation. But that is non-linear so it will not do for what I want to do, correct?

Question 2: This is a more general question about linearization. I have heard some times, in the context of engineering, something along the lines of "this system is to non-linear for us to use a linear version of it". I especially remember that being said when having ln(). Does this basically mean that there is not good linear (Taylor) approximation for the system that would be useful?

• It only makes sense, in general, to linearise about some trajectory (typically a stable equilibrium point). – copper.hat Jan 9 at 19:18
• The answer is part physics. The well known linear equation for a pendulum works well when the amplitude of the swing is small. If it is large then the linear equation is not so good. – user121049 Jan 9 at 19:35