Examining a differential equation of motion

[JordanGo]

Homework Statement

Examine the equation

x¨ + β(x^2 − 1)x˙ + x = 0

Explain qualitatively the motion in three cases: |x| < 1, |x| > 1, and |x| = 1. In
each case, do you expect the motion to be bounded our unbounded?
Define the energy of the system as E ≡ 1/2 (x^2 + v^2). Show that the time rate of
change of the energy is β (1 − x^2)v^2
Examine for each of the three cases in which case is energy a conserved
quantity? What happens in the other cases?

Homework Equations

x¨ + β(x^2 − 1)x˙ + x = 0

The Attempt at a Solution

I have no idea how to go about this question, the differential equation is not solvable. How do you examine it then?

 

[Mindscrape]

You don't necessarily need a solution to an ode to examine its stability. However, I suppose it may be tough to intuitively be able to describe the motion very well. I mean, besides saying it's a spring with a nonlinear damping term. You could always solve it numerically. Universities always have licenses to MATLAB and mathematica. If you don't have one of those you can use octave or another open source program.

 

[Kurt Peek]

Hi JordanGo,

As Mindscrape noted, your ODE corresponds to a damped harmonic oscillator with a nonlinear damping term. To be specific, the damping term is the coefficient $\beta(x^2-1)$. If this coefficient is positive, then the damping force will be in the opposite direction of the speed $\dot{x}$, and the mass will be "slowed down" or "damped". Conversely, if the coefficient is negative, the mass will be "sped up" and its motion "amplified". Finally, if the coefficient is zero, the motion is undamped. It is easy to see that the coefficient is positive if $|x|>1$.

As for the second part of the question, this is just a matter of filling in $v=\dot{x}$ into the expression for $E$, differentiating using the chain rule, and substituting for $\ddot{x}$ in the resulting expression from the original ODE. (Incidentally, this expression also shows that the system's energy increases if $|x|<1$).

Hope this helps!


Kurt Peek

 

[JordanGo]

Thank you guys, now I understand what to do! I really appreciate it!

 

[paranormal]

As a scientist, it is important to first understand the problem before attempting to solve it. In this case, we are examining a differential equation of motion, which is a mathematical representation of the motion of a system over time. The equation provided is a second-order nonlinear differential equation, which can be difficult to solve analytically. However, we can still gain insight into the motion of the system by examining the equation qualitatively.

First, let's look at the three cases provided: |x| < 1, |x| > 1, and |x| = 1. In each case, the term (x^2 - 1) will be either negative, positive, or zero, respectively. This means that the coefficient β will have a different effect on the motion in each case. In general, we can expect the motion to be bounded when |x| < 1 and unbounded when |x| > 1. This is because the term (x^2 - 1) will act as a restoring force when |x| < 1, keeping the motion within a certain range. On the other hand, when |x| > 1, this term will act as a destabilizing force, allowing the motion to become unbounded.

When |x| = 1, the term (x^2 - 1) will be equal to zero, meaning that the coefficient β will not have any effect on the motion. In this case, the motion will depend solely on the initial conditions of the system. The motion could be bounded or unbounded, depending on the initial conditions.

Next, let's define the energy of the system as E ≡ 1/2 (x^2 + v^2), where v is the velocity of the system. This energy represents the total energy of the system, including both kinetic and potential energy. Now, we can use this energy definition to show that the time rate of change of the energy is β (1 − x^2)v^2. This can be done by taking the time derivative of the energy equation and substituting in the given differential equation for x¨ and x˙. This will result in the given expression for the time rate of change of energy.

Finally, let's examine in which cases the energy is conserved. In general, energy is conserved when the time rate of change is equal to zero. From the previous result