Planar motion for an open string with fixed endpoints.
Consider the motion of a relativistic open string on the \((x, y)\) plane. The
string endpoints are attached to \((x, y)=(0,0)\) and \((x, y)=(a, 0)\), where
\(a>0 .\) As opposed to the relativistic jumping rope, the string now remains in
the \((x, y)\) plane. The motion is described by
$$
\vec{X}(t, \sigma)=\frac{1}{2}(\vec{F}(c t+\sigma)-\bar{F}(c t-\sigma))
$$
where \(F(u)\) is a vector function of a single variable which satisfies
$$
\left|\frac{d \vec{F}}{d u}\right|^{2}=1 \text { and } \vec{F}\left(u+2
\sigma_{1}\right)=\vec{F}(u)+(2 a, 0)
$$
Consider an ansatz of the form
$$
\vec{F}^{\prime}(u) \equiv \frac{d \vec{F}}{d u}=\left(\cos \left[\gamma \cos
\frac{\pi u}{\sigma_{1}}\right], \sin \left[\gamma \cos \frac{\pi
u}{\sigma_{1}}\right]\right)
$$
(a) Is this ansatz consistent with the conditions in \((2) ?\)
(b) Calculate \(\vec{X}^{\prime}(0, \sigma)\). Letting \(\vec{X}(0, \sigma)
\equiv(x(\sigma), y(\sigma))\), give \(d y / d \sigma\) and plot it as a function
of \(\sigma \in\left[0, \sigma_{1}\right]\) assuming, for convenience, that
\(0<\gamma<\pi / 2\). Use this to make a rough sketch of the string position
\(y(\sigma)\) as a function of \(\sigma\) at \(t=0\)
(c) Calculate \(\vec{X}^{\prime}(t, 0)\) and use it to describe the motion of
the string near the origin. What is the interpretation of \(\gamma ?\)
(d) Use the second condition in \((2)\) to find an integral relation between \(a,
\sigma_{1}\) and \(\gamma .\) Assume that \(\gamma\) is small, and find an
approximate explicit relation between these three variables keeping terms of
order \(\gamma^{2}\)
(e) Show that \(a / \sigma_{1}=J_{0}(\gamma)\), where \(J_{0}\) is the Bessel
function of order zero. [Hint: Look up integral representations of Bessel
functions.]
