  Since the basic theoretical framework of generalized Hamilton system is not
perfect and complete, there are often some practical problems that can not be
expressed by generalized Hamilton system. The generalized gradient operator is
defined by the structure function on manifold to improve the basic theoretical
framework of the whole generalized Hamilton system. The generalized structural
Poisson bracket \[\left\{ f,g \right\}={{\left\{ f,g \right\}}_{GPB}}+G\left(
s,f,g \right)={{\left\{ f,g \right\}}_{GPB}}+f{{\left\{ s ,g
\right\}}_{GPB}}-g{{\left\{ s ,f \right\}}_{GPB}} \]is defined as well on
manifolds. The geometric bracket is also given, and the covariant extension
form of the generalized Hamilton system directly related to the structure
function, the generalized covariance, is further obtained--generalized
covariant Hamilton system, It includes thorough generalized Hamiltonian system
and S-dynamic system.
