Although the Lagrange formalism is a convenient method to tackle complex problems, it is of advantage to formulate the dynamics in phase-space variables, i.e. in generalized coordinates and generalized momenta. In this case the time evolution of an observable, that not explicitly depends on time, is given by Poisson brackets which are determined by the derivative of the observable and the Hamiltonian with respect to the phase-space variables. The elementary Poisson bracket between generalized coordinates and generalized momenta will turn out to be unity for associated pairs and their time evolution is given by the Poisson bracket with the Hamilton function, i.e. by the canonical equations of motion. The Poisson brackets thus allow for an algebraic formulation of the dynamics. However, the choice of generalized coordinates is not unique and invertible transformations between the coordinates are allowed, too. But not all transformations are meaningful, since some transformations may lead to equations of motion that are no longer canonical. Allowed transformations then will be given by point transformations and extended canonical transformations, that keep the equations of motion canonical invariant. Furthermore, the elementary Poisson brackets will be shown to be invariant with respect to canonical transformations such that a formulation of classical mechanics is achieved, which is independent on the choice of the generalized coordinates. This will pave the way to quantum mechanics, where the Poisson brackets will be replaced by commutators of operators in an abstract Hilbert space. This also will lead to a rigid formulation of statistical mechanics, where the physical system - in equilibrium - is described by ensembles with properties, that are defined by expectation values of conserved quantities and their fluctuations.