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The classical Lagrange formalism is generalized to the case of arbitrarystationary (but not necessarily conservative) dynamical systems. It is shownthat the equations of motion for such systems can be derived in the standardways from the Lagrange functions which are linear in system velocities, have noexplicit dependence on time, and do not require the introduction of anyadditional degrees of freedom. We show that time-symmetry of such Lagrangefunctions leads to the integrals of motion naturally generalizing the notion ofenergy but not coinciding with it in non-conservative cases. Thenon-conservative analogs of Hamilton equations, Poisson brackets,Hamilton-Jacobi equations, Liouville theorem and Principle of Stationary Actionare discussed as well. As an example, we consider two cases demonstrating thework of the proposed schema: the classical model of one-dimensional dampedharmonic oscillator and a special class of non-physical exactly-solvablemulti-dimensional dynamical models.

Classical Lagrange formalism for non-conservative dynamical systems