How to linearize ODEs Homework Statement

[Phyisab****]

Homework Statement

A problem in classical mechanics, dealing with an object in space collecting mass as it passes through a dust cloud has led me to the following nonlinear ODE. I'm quite sure it is the correct equation, my old man checked it out. But being the engineer he is, he would just go ahead and solve it numerically, so he didn't have any advice past this point .

$$A\rho(\dot{x})^{2}+m_{0}\ddot{x}+A\rho x \ddot{x}=0$$

The Attempt at a Solution

So my idea was to assume $$\ddot{x}=constant$$ (which left me fewer things to worry about than a second order Taylor expansion). That seems like an ok approximation to me. But I'm still left with a term involving $$(\dot{x})^{2}.$$ That still qualifies as nonlinear right? I rarely encounter nonlinear equations, I'm really out of my area of knowledge here! So even with my approximation I am left with an equation with a form I have never seen before.

$$(\dot{x})^{2}+xa=\frac{m_{0}a}{A\rho}$$Any ideas? This is really quite baffling to me. I don't have slightest idea how to proceed. I don't think a simple integrating factor is going to take care of this thing.

 

[LCKurtz]

So you have an equation of the form

$$\frac{dx}{dt}= \pm\sqrt{a(k-x})}$$

Try separation of variables.

 

[Phyisab****]

Ahhhh of course thanks!

 

[elect_eng]

Any ideas? This is really quite baffling to me. I don't have slightest idea how to proceed. I don't think a simple integrating factor is going to take care of this thing.

It's not correct to assume that $${{d^2x}\over{dt^2}}$$ is constant. You would be forcing a constraint with no physical justification. The physics implied in the equation will dictate the time evolution of $$x$$, $${{dx}\over{dt}}$$ and $${{d^2x}\over{dt^2}}$$. These equations can be solved best using a nonlinear state space representation as follows.

$${{dx}\over{dt}}=v$$
$${{dv}\over{dt}}={{-v^2}\over{m_o/A\rho + x}}$$

From here you need initial conditions for both $$x$$ and $$v$$.

These equations can be solved using numerical integration which propagates the solution forward in time. The critical system variables (x and v, sometimes called the states) fully describe the state of the system. Once you know the time evolution of these two variables, you can calculate anything else from them. Perhaps there is a closed form solution for this system of equations, but I don't see it. In such cases, I'm like your Father and would just plug these equations into Matlab or Simulink for a quick solution.

 

[Phyisab****]

Well clearly it's not ideal, but I'd say it's better than any other option. We don't solve things numerically in undergrad physics lol. So my only choice is to make some kind of approximation. I will see if I can include a higher order term from the taylor expansion, that's the best I'm going to be able to do.

 

[Phyisab****]

From the wording of the problem, a spaceship flying through a dust cloud, expanding about the acceleration is more reasonable than velocity or position. Which are the only two other options. I definitely am supposed to find some approximate form of the equation which can be solved analytically, not just come up with an unsolvable equation.

 

[elect_eng]

Well clearly it's not ideal, but I'd say it's better than any other option. We don't solve things numerically in undergrad physics lol. So my only choice is to make some kind of approximation. I will see if I can include a higher order term from the taylor expansion, that's the best I'm going to be able to do.

I understand what you're saying. The nonlinear state space form I showed is ideal for doing linearization, and higher order expansions too. Often a linear state space system is made from the nonlinear state space system by generating a Jacobian matrix, essentially making the linear coefficients of the matrix from the first partial derivatives.

 

[elect_eng]

From the wording of the problem, a spaceship flying through a dust cloud, expanding about the acceleration is more reasonable than velocity or position. Which are the only two other options.

In my experience, linearization is typically done around the state variables of the system. In this case x and v are the state variables.

I definitely am supposed to find some approximate form of the equation which can be solved analytically, not just come up with an unsolvable equation.

One thing I'd like to be clear about is that I'm not sure that those equations are unsolvable. I can only say that I don't see a solution and my gut feeling is that they aren't solvable in closed form.

 

[Phyisab****]

So you're suggesting I expand v about v0? I know what a Jacobian matrix is but I've never seen it used like this, I can see how it would be used to find a linear approximation for a PDE.

 

[Phyisab****]

By using a Jacobian matrix I would basically be treating x and v and independent variables? How would the resulting approximation be different from expanding with respect to one of these variables?

 

[elect_eng]

By using a Jacobian matrix I would basically be treating x and v and independent variables? How would the resulting approximation be different from expanding with respect to one of these variables?

I think by expanding around all independent variables of the system (i.e. all state variables) your approximate solution is more likely to have physical meaning and to be useful. Basically, you are linearizing around a particular state of the system (say, x=x0 and v=vo). The linearized equations allow an approximate solution for small perturbations around that point. It's also meaningful to look at the stability of the system at that point.

In your problem, one of the state equations is already linear, while the other can be linearized. You can think of the state variables (x,v) as a constant operating point (x0,v0) plus small difference variables (X,V), as follows.

x=x0+X
v=v0+V

Your linearized state equations will be in term of variables X and V, while the linear constant coefficients in the two state equations will be functions of x0 and v0.