  By examining both the divergence of the velocity vector in orthogonal
Cartesian coordinate space $\mathbf{\Gamma} $ of dimension $\R^{\textrm {2fN}}$
and the structure of the Hamiltonian determining a system trajectory, it is
shown that the standard Liouville equation cannot describe anything more than
linear motion for typical Hamiltonians with separated momentum and space
variables and some significant consequences such as the Poincare recurrence
theorem do not obtain for such Hamiltonians. A new stochastic equation which is
everywhere in principle discontinuous is developed for dynamical systems
described by a general Hamiltonian which is a functional of the space and
momentum variables, and where the average trajectory of a system point is
proven to be orthogonal to any constant energy surface consonant with the
system energy at equilibrium. This equation does not assume the presence of
binary collision only, as required in the standard first-order Boltzmann
equation, and is therefore suitable to describe dense systems as well, and may
be viewed as an alternative to the latter. Some new macroscopic variational
principles for non-equilibrium thermodynamical systems are proposed, where one
object for future work would be to relate the microscopic description given
here with the macroscopic principles.
