Poincaré-Cartan integral invariants of nonconservative dynamical systems

Traditionally there do not exist integral invariants for a nonconservative system in the phase space of the system. For weak nonconservative systems, whose dynamical equations admit adjoint symmetries, there exist Poincaré and Poincaré-Cartan integral invariants on an extended phase space, where the set of dynamical equations and their adjoint equations are canonical. Moreover, integral invariants also exist for pseudoconservative dynamical systems in the original phase space if the adjoint symmetries satisfy certain condtions.