Topics in contact Hamiltonian systems: analytical and numerical perspectives

The work of this thesis explores contact Hamiltonian systems as a
geometrical setting to study physical systems with dissipation. Unlike
symplectic dynamical systems, contact Hamiltonian systems do not
conserve energy, allowing the description of systems with different
types of dissipation and forcing. The thesis is divided into three
parts: the first part provides background knowledge on contact
manifolds and introduces contact Hamiltonian systems with examples.
The second part focuses on numerical methods for contact Hamiltonian
systems, including geometry preserving integrators and deep learning
techniques. The third part presents analytical results: the
computation of the Baker-Campbell-Hausdorff formula for certain
algebras and the study of symmetry and integrability in contact
Hamiltonian systems. The thesis builds on previously published work
and includes unpublished work in progress.