Left time derivatives in mathematics, mechanics and control of motion

• Authors: E. A. Galperin
• Venue: Computers & Mathematics with Applications
• Year: 2011

By the traditional representation accepted in mathematics, mechanics and theoretical physics, the time-derivatives are defined as the right derivatives and are used in this way in differential equations describing processes in nature and technology. The fact that even for infinitely smooth x(t), the right time-derivatives do not physically exist, due to the positive orientation of time, somehow escaped the attention of scientists. This led to misconceptions and omissions in mechanics, physics and engineering, with unexpected consequences in some cases. All measurements and experiments contain and use only left time-derivatives, thereby with time delays. All processes require some kind of transmittal of information (forces, actions) which takes time, so the expressions that define their evolution from a current state actually contain the left and delayed time derivatives, even if they are written with the exact right time-derivatives, according to the classical tradition. In this paper, the causal representations of physical processes by differential equations with the left time-derivatives on the right-hand side are considered for some basic problems in classical mechanics, physics and technology. The use of the left time-derivatives explicitly takes into account the causality of processes depending on the transmission of information and defines the motions subject to external forces that may depend 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 they are also involved in the manual control of aircraft or spacecraft that depends on accelerations of the craft itself. The consistency condition is presented, and the existence of solutions for equations of motion driven by forces with higher order derivatives of velocity is proved. The inclusion of such forces in the autopilot design is proposed to assure the safety of the aircraft in case of a failure of its outboard velocity sensors. It is demonstrated that the classical form of the 2nd law of Newton is preserved with respect to the effective forces for which the parallelogram law of addition is valid. Then the Lagrange and Hamilton equations are extended to include the generalized forces with the left higher order derivatives, and a method for the solution of such equations with the left and delayed higher order derivatives is presented with the example of a physical pendulum. The results open new avenues in science and technology providing the basis for correct design in the projects sensitive to information transmittal.