Motion along a line

zebrastripes

1. The problem statement, all variables and given/known data

A particle P of unit mass moves along an x-axis under the influence of the force

F(x)=2(x[itex]^{3}[/itex]-x)

Firstly, I find V(x)=x[itex]^{2}[/itex]-x[itex]^{4}[/itex]/2.
Where equilibrium points are F(x)=0 so x=0 with energy V(0)=0, x=1 with energy V(1)=V(-1)=1/2.
And I have also sketched the graph.

So here are the parts I'm stuck on:

1) Initially P is projected from the point x=1/2 with speed U. Using conservation of energy, find the turning points (where x'=0) as a function of U. Find the maximum value of U for which the resultant motion will be bounded.

2)Stating from Newton's second law, prove that a particle displaced by a small amount from x=0 will perform periodic oscillations with a frequency of [itex]\sqrt{2}[/itex]



2. Relevant equations

T=mx'[itex]^{2}[/itex]/2

T+V=E

3. The attempt at a solution

So for 1), I start with

U[itex]^{2}[/itex]/2+(1/2)[itex]^{2}[/itex]-(1/2)[itex]^{4}[/itex]/2=E

Giving E=U[itex]^{2}[/itex]+7/16

Then, because energy is conserved and x'=0 at turning points:

x[itex]^{2}[/itex]-x[itex]^{4}[/itex]/2=U[itex]^{2}[/itex]+7/16

And now I'm really stuck for how to find the turning points as functions of U?

I'm guessing I'm going about this the completely wrong way??

And I have no idea what I'm supposed to be doing for 2)

Sorry this is long, but any help will be greatly appreciated

Ray Vickson

If v = velocity (= dx/dt), then by conservation of energy we have
[tex] \frac{1}{2} v^2 + V(x) = \frac{1}{2} U^2 + V(1/2) =
\frac{1}{2} U^2 + \frac{7}{32},[/tex] since [itex] V(1/2) = 7/32.[/itex] At turning points, [itex]v = 0,[/itex] so you get an equation in [itex]x.[/itex]

For small |x| the potential V(x) is almost linear in x. If you neglect the x⁴ term in V(x) you get a simple harmonic oscillator.

RGV

zebrastripes

Thanks RGV, but under what assumptions can I say for small |x|?

And would you mind giving me a hint about what to do for part 2)?

Thanks again!