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We simulate the low temperature behaviour of an elastic chain in a randompotential where the displacements $u(x)$ are confined to the {\it longitudinal}direction ($u(x)$ parallel to $x$) as in a one dimensional charge densitywave--type problem. We calculate the displacement correlation function $g(x)=<(u(x)-u(0))^2>$ and the size dependent average square displacement$W(L)=<(u(x)-\bar{u})^2>$. We find that $g(x)\sim x^{2\eta}$ with$\eta\simeq3/4$ at short distances and $\eta\simeq3/5$ at intermediatedistances. We cannot resolve the asymptotic long distance dependence of $g$upon $x$. For the system sizes considered we find $g(L/2)\propto W\simL^{2\chi}$ with $\chi\simeq2/3$. The exponent $\eta\simeq3/5$ is in agreementwith the Random Manifold exponent obtained from replica calculations and theexponent $\chi\simeq2/3$ is consistent with an exact solution for the chainwith {\it transverse} displacements ($u(x)$ perpendicular to $x$).Thedistribution of nearest distances between pinning wells and chain-particles isfound to develop forbidden regions.Comment: 19 pages of LaTex, 6 postscript figures available on request,  submitted to Journal of Physics A, MAJOR CHANGE

Elastic chain in a random potential: Simulation of the displacement correlation function and relaxation