A methodology for deriving dual variational principles for the classical Newtonian mechanics of mass points in the presence of applied forces, interaction forces, and constraints, all with a general dependence on particle velocities and positions, is presented. Methods for incorporating constraints are critically assessed. General theory, as well as explicitly worked out variational principles for a dissipative system (Lorenz) and a system with anholonomic constraints (Pars) are demonstrated. Conditions under which a (family of) dual Hamiltonian flows, as well as constants of motion, may be associated with a dissipative, and possibly constrained, primal system are specified.