Generalized equations of motion for mechanical systems with variable masses and forces depending on higher order derivatives

The Buquoy generalization of Newton's second law of motion for systems of bodies with variable masses driven by reactive forces produced by ejected burnt fuel (Mestschersky) is considered, with its extension for motions subject to external forces depending on accelerations and higher order derivatives of velocities. Such forces are exhibited in Weber's electro-dynamic law of attraction; they are produced by the Kirchhoff-Thomson adjoint fluid acceleration resistance acting on a body moving in a fluid and are also involved in manual control of aircrafts and spacecrafts that depends on acceleration of the craft itself. The causality of systems driven by such forces is assured by consideration of the left higher order derivatives in the right-hand sides of the equations of motion. The consistency condition and a new solution method are presented, and the existence and uniqueness of solutions for equations of motion driven by such forces is proved. The notion of effective forces is discussed, and the parallelogram law is verified for the effective forces in mechanical systems with left higher order derivatives in controls. On this basis, the new autopilot design is proposed for added security in civil aviation, independent of the currently used Pitot tubes which may fail or render the local measurements of wind gusts instead of the correct estimates for the average relative velocity of the aircraft with respect to the wind in flight or to the airstrip at landing.