The equations of motion of Newtonian mechanics can be written in different ways—depending on the choice of coordinates—and in principle all independent choices have equal rights. However, some choices facilitate the solutions of the equations of motion and others might cause severe problems. It is thus of general interest to find ‘optimal’ coordinates for the description, which is also of practical help, if the system is subject to constraints that require the introduction of ‘coercive forces’, which often are difficult to define. It is thus meaningful to define ‘generalized coordinates’ that fulfill the constraints and also reduce the complexity of the problem by reducing the number of (linear independent) degrees of freedom. The equations of motion in generalized coordinates then are derived from Newton’s equations of motion. It will be found that these equations can also be generated by a variational principle, which specifies a Lagrange function L, which is given by the difference between the kinetic and potential energy in case of conservative forces. An important consequence is that the Lagrange equations of motion can also be applied to other areas of physics. Generalized momenta are defined by the derivative of the Lagrange function with respect to the generalized velocities. Accordingly, if the Lagrange function does not depend on a specific coordinate, e.g. the azimuthal angle \({\varphi }\) , the corresponding generalized momentum (here angular momentum) is a constant of motion. This suggests to transform the formulation to phase-space variables given by coordinates and their associated momenta, which is carried out by a Legendre transformation defining the Hamilton function H. In case of conservative forces the latter just gives the energy of the system in phase-space variables. The variational principle thus can be reformulated in terms of Hamilton’s (equivalent) variational principle which leads to the canonical equations of motion. The latter will be illustrated for a couple of examples. Furthermore, it will be shown again that—for a closed system—the translational invariance leads to the conservation of the total momentum, the rotational invariance to the conservation of total angular momentum, and the invariance with respect to time translations to the conservation of the total energy.