  Geometric mechanics is a branch of mathematical physics that studies
classical mechanics of particles and fields from the point of view of geometry.
In a geometric language, symmetries can be expressed in a natural manner as
vector fields that generate the corresponding symmetry group. Moreover, with
the geometric machinery, the phase space of a mechanical system with symmetries
can be reduced to a space with less dimensions. As it is well-known,
non-conservative forces cannot be written as the gradient of a potential, so
they cannot be absorbed in the Lagrangian or Hamiltonian. The description of a
mechanical system subject to a non-conservative force requires an external
force together with the Lagrangian or Hamiltonian function. In addition,
external forces emerge for the description of certain systems with
non-holonomic constraints.
  In this master's thesis, autonomous Hamiltonian and Lagrangian systems are
studied in the framework of symplectic geometry. After introducing the
geometric tools that will be employed, several results regarding symmetries and
constants of the motion are reviewed. The method of symplectic reduction for
systems with symmetry is also presented. In a second part, non-conservative
systems are presented in a geometric language. A Noether's theorem for
Lagrangian systems subject to external forces is obtained. Other results
regarding symmetries and constants of the motion are derived as well.
Furthermore, a theory for the reduction of forced Lagrangian systems invariant
under the action of a Lie group is presented. These results are particularized
for the so-called Rayleigh dissipation, that is, external forces that can be
written as the derivative of a "potential" with respect to the velocities.
