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green-beans
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Homework Statement
A particle of unit mass moves in one dimension with potential V(x) = ½μ2x2 + εx4 (ε>0). Discuss the motion of the particle.
If the particle released from rest at x=a (a>0) express the time period T for the particle to return to a in the form of an integral and show that when ε is small, T is reduced by approximately 3επa2/μ3.
Homework Equations
v = dx/dt = ±√{(2/m) * (E - V(x))}
Time period taken for a particle to move from point x1 to x2:
T = ∫1/{v} dx
The Attempt at a Solution
Discuss the motion:[/B]
If we sketch V(x) it will look alike to parabola with V(0) = 0. Depending on the value of E that the particle has, it will have a simple harmonic motion and will oscillate in the region enclosed between E and the x-axis.
Time period to return to a:
I obtained this and it matched with the answers (sorry for the long expression I wanted to upload a picture but it did not work)
T=(4/μ) ∫(dx)/(√{(a2-x2) +(2/μ2)ε(a4-x4)})
Show that T is reduced by approximately 3επa2/μ3
In this part I am not sure how the fact that ε is small will affect the integral. I am not sure how small the numerator will become and whether the denominator will be greater than numerator. In this case should I just integrate?
Thank you for your help!