  We provide new insights into the contact Hamiltonian and Lagrangian
formulations of dissipative mechanical systems. In particular, we state a new
form of the contact dynamical equations, and we review two recently presented
Lagrangian formalisms, studying their equivalence. We define several kinds of
symmetries for contact dynamical systems, as well as the notion of dissipation
laws, prove a dissipation theorem and give a way to construct conserved
quantities. Some well-known examples of dissipative systems are discussed.
