In this work, it is formulated and analyzed the static equilibrium and the dynamics for three dimensional deformation of elastic rods. The intrinsically one-dimensional theory that is employed, which may be called the special Cosserat theory of rods, is geometrically exact, namely, it is not based upon geometrical approximations or mechanical assumptions. For the rod deformation, it is adopted the Bernoulli hypotheses and for simplicity, the linear constitutive relations are employed. The deformed configuration of the rod is described by the displacement vector of the deformed centroid curve and an orthonormal moving frame, rigidly attached to the cross-section of the rod. The orientation of the moving frame, relative to the inertial one, is related by the rotation matrix, parameterized by three elemental rotations. In the sense of Cosserat theory, the equations of motion are nonlinear partial dfferential equations, which are functions of time and one space variable. For the static equilibrium, however, the equations become nonlinear ordinary differential equations with one space variable, which can be solved approximately using standard techniques like the perturbation method. After the static equilibrium equation are solved, the displacement functions are obtained. These nonlinear displacement functions, which are functions of generic nodal displacements and rotations, are used for dynamical analysis. To obtain the dynamics of the Cosserat rod, it is used the Lagrangian approach, formed from the kinetic and strain energy expressions. Furthermore, the equations of motion, which are nonlinear ordinary dfferential equations, are solved numerically using the Newmark method. As an application, a curved rod, constrained to rotate inside a hole, is investigated numerically and experimentally. When using the Cosserat rod approach, that take into account all the geometric nonlinearities in the rod, the higher accuracy of the dynamic responses is achieved by dividing the system into a few elements, which is much less than in the traditional FEM