I don't understand conservative systems

[Jamin2112]

Homework Statement

Look below.

Homework Equations

So my book says that you have a conservative system when E'(t) = [(1/2)m(x'(t))² + V(x(t))]'(t) = 0. It gives an example of V(x) = -(1/2)x² + (1/4)x⁴, whence setting y = x' gives E = (1/2)y² -(1/2)x² + (1/4)x⁴. According to the book, E = constant. Not sure how they got this since E' ≠ 0 as far as I can tell.

The Attempt at a Solution

Me no understand.

 

[jfgobin]

I found this document with the same example. As soon as my headache leaves, I read it.

 

[vela]

The system obeys Newton's second law, F=ma. It then follows that your expression for E is constant for any V(x). You should be able to show that E' = (ma + dV/dx)v(t), where v(t) is the velocity. Since the force F is given by -dV/dx, the factor in the parentheses vanishes, so E'=0.

However, I'm guessing your book isn't really expecting you to show that E'=0. It's telling you because the system is conservative, E'=0 so that E is a constant.

In the xy-plane, the possible behavior of the system is represented by the family of curves that satisfy $E = \frac{1}{2}my^2 - \frac{1}{2}x^2 + \frac{1}{4}x^4$, where each curve can be associated with a specific value of E.

Take the example of a free particle, where V(x)=0. Then you have $y = \sqrt{2E/m} = \text{constant}$. The particle moves with constant velocity.