This thesis investigates the design and analysis of state observers for nonlinear mechanical systems. It focuses on systems with measurable positions and the reconstruction of the unknown velocities, which can be used in state feedback control. The basis for the observers design is modelling, therefore the Euler-Lagrange and the Hamilton equations have been used. Two approaches from the literature in terms of state estimation are discussed. Firstly, an observer based on the immersion and invariance method. It is a reduced observer and its utilization is limited to a certain class of systems which are characterised through the existence of a special coordinate transformation. Therefore, in general, a set of nonlinear PDEs has to be solved. Nevertheless, these equations have been solved for a few examples. For instance, the system cart and pendulum, hence it will be the example of application in the following. The second state estimation approach is based on the sliding mode method. Two different observers, the super twisting and the fixed time observer are discussed. Both are distinguished by the fact that the estimation error converges to zero in finite time. The stated observers are designed for the system cart and pendulum. Simulation results and measurement of the laboratory setup are shown. Finally existing control strategies for swinging up and stabilisation at the upper equilibrium are extended by the stated observers.