- #1
ChrisVer
Gold Member
- 3,378
- 464
Well I numerically solved for the potential [itex]V(x)=x^4[/itex], the period:
\begin{equation}
T = \sqrt{8m} \int_0^a \frac{dx}{\sqrt{V(a) - V(x)}}
\end{equation}
where [itex]a[/itex] was the amplitude of the oscillation and [itex]m[/itex] the mass of the particle.
Nevertheless, what I was asked to plot was the above period [itex]T(a)[/itex] for [itex]a\in [0.,2.][/itex].
My problem however is that the period of small values of [itex]a[/itex], diverges. I was able to see how this is the case mathematically, by expanding the square root and obtaining an expression for [itex]T[/itex] that goes for [itex]V(x)=x^p[/itex] as:
[itex]T \sim \frac{1}{a^{p/2-1}} [/itex]
which reproduces also the [itex]V=x^2[/itex] result of constant period.
However I cannot picture what is happening physically. Does it have to do with the flatness of the potential at so small a's and so x's?
Any idea?
\begin{equation}
T = \sqrt{8m} \int_0^a \frac{dx}{\sqrt{V(a) - V(x)}}
\end{equation}
where [itex]a[/itex] was the amplitude of the oscillation and [itex]m[/itex] the mass of the particle.
Nevertheless, what I was asked to plot was the above period [itex]T(a)[/itex] for [itex]a\in [0.,2.][/itex].
My problem however is that the period of small values of [itex]a[/itex], diverges. I was able to see how this is the case mathematically, by expanding the square root and obtaining an expression for [itex]T[/itex] that goes for [itex]V(x)=x^p[/itex] as:
[itex]T \sim \frac{1}{a^{p/2-1}} [/itex]
which reproduces also the [itex]V=x^2[/itex] result of constant period.
However I cannot picture what is happening physically. Does it have to do with the flatness of the potential at so small a's and so x's?
Any idea?