Classical Mechanics consists of three parts: Newtonian, Lagrangian and Hamiltonian Mechanics, where each part is a special extension of the previous part. Each part has explicit symmetries (the explicit Laws of Motion), which, in turn, generate implicit or hidden symmetries (like the Law of Conservation of Energy, etc). In this Master\u27s Thesis, different types of hidden symmetries are considered; they are reflected in the Noether Theorem and the Poincare Recurrence Theorem applied to Lagrangian and Hamiltonian Systems respectively.
The Poincare Recurrence Theorem is also applicable to some number theory problems, which can be considered as dynamical systems. In this thesis, we study the problem The first digits of the powers of integers . A dynamical system interpretation for this problem allows to apply the Poincare Recurrence Theorem to find several unexpected hidden symmetries like rotations on the circle and on the torus, as well as hidden arithmetic progressions in the set of all the exponents
