The relativistic constant period potential which yields an energy-independent period at relativistic as well as nonrelativistic energies was investigated both theoretically and numerically. In the theoretical work, the inverse problem of the relativistic electric potential was solved by the Laplace transform method. From the solution of the inverse problem, the relativistic constant-period potential was analytically obtained. In the numerical work, the relativistic constant-period potential was evaluated and its dynamic properties in the presence of a periodic external force were investigated. When the amplitude of the external force is sufficiently large, chaotic behavior is seen to occur. Such a phenomenon was explained by the method based on the Lie transform. The inverse problem of a static gravitational force was solved within the frame of relativity. In the case that the relativistic point particle oscillates with period T(E) under an unknown static gravitational force, the integral equation for the period T(E) was formulated and was solved by the inversion of an Abelian integral equation. Finally the inversion formula with which the component $g_{00}$ of the metric tensor can be constructed from the period T(E) was obtained.
