  The classical Lagrange formalism is generalized to the case of arbitrary
stationary (but not necessarily conservative) dynamical systems. It is shown
that the equations of motion for such systems can be derived in the standard
ways from the Lagrange functions which are linear in system velocities, have no
explicit dependence on time, and do not require the introduction of any
additional degrees of freedom. We show that time-symmetry of such Lagrange
functions leads to the integrals of motion naturally generalizing the notion of
energy but not coinciding with it in non-conservative cases. The
non-conservative analogs of Hamilton equations, Poisson brackets,
Hamilton-Jacobi equations, Liouville theorem and Principle of Stationary Action
are discussed as well. As an example, we consider two cases demonstrating the
work of the proposed schema: the classical model of one-dimensional damped
harmonic oscillator and a special class of non-physical exactly-solvable
multi-dimensional dynamical models.
