Momentum mapping and reduction in contact Hamiltonian systems

Abstract: Due to the emergence of symplectic geometry, the geometric treatment of mechanics underwent a great development during the last century. In this scenario the pressence of symmetries in Hamiltonian systems leads naturally to the existence of conserved quantities. This integrals of motions are described by the well-known momentum mapping. Furthermore, the equations of motion can be simplified by a process known as reduction, if the system is invariant under the action of a certain group. This process can also be considered in the framework of contact geometry, a much more recent field of study. This kind of geometry has proven to be valuable in areas as different as thermodynamics, control theory, or neurogeometry. Our knowledge about contact geometry is much smaller than that of symplectic, and so a process known as symplectificacion is extremely useful, since it allows to study contact problems in the symplectic frame. The aim of the text is to study the commutativity relations between the processes of reduction and symplectification. To do so, we first introduce the basis of symplectic and contact geometry and we show how to perform the reduction via the momentum map in both scenarios. The well-known coisotropic reduction theorem will be crucial in the description. Finally, the symplectification process is analyzed in detail, and its relation with symplectic and contact reduction is studied.