The paper investigates various types of transformations in Hamiltonian vector fields on locally conformal symplectic (LCS) manifolds, using a geometric approach. The first part of the paper focuses extensively on some transformations (strictly canonical transformations, non-strictly canonical transformations and canonoid transformations) and some infinitesimal symmetries on Hamiltonian systems, providing a thorough analysis of their structure and significance in the context of Hamiltonian dynamics on LCS manifolds. It examines how these transformations and infinitesimal symmetries affect the system’s geometry and evolution. The paper then turns to the concept of master symmetries and presents a method for finding master symmetries of Hamiltonian systems. Finally, the paper addresses the relationship between the existence of constants of motion (first integrals), infinitesimal symmetries and the properties of infinitesimal canonoid symmetries. To formalize this connection, a family of boundary and coboundary operators is introduced, which help to uncover the deeper geometric and algebraic structures underlying the infinitesimal symmetries and conserved quantities in Hamiltonian systems.
X. Zhao (Fri,) studied this question.