View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 62 (2011) 4742–4757 Contents lists available at SciVerse ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Left time derivatives in mathematics, mechanics and control of motion E.A. Galperin Departement de Mathematiques, Universite du Quebec a Montreal, C.P. 8888, Succ. Centre Ville, Montreal, Quebec H3C 3P8, Canada article info a b s t r a c t Article history: By the traditional representation accepted in mathematics, mechanics and theoretical Received 17 October 2011 physics, the time-derivatives are defined as the right derivatives and are used in this Accepted 24 October 2011 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 Keywords: the positive orientation of time, somehow escaped the attention of scientists. This led to Causality and orientation of time misconceptions and omissions in mechanics, physics and engineering, with unexpected Acceleration assisted control consequences in some cases. All measurements and experiments contain and use only left Forces with the left higher order derivatives 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. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction By an old tradition, the notion of the derivative at a point z is defined as the right limit: f 0.z/ D df =dz D lim Tf .z C 1z/ − f .z/U=1z as 1z ! 0; 1z > 0. If f .z/ is a physical process and z denotes a moment of time, z D t∗, then f 0.t∗/ is non-causal since t∗ C 1t > t∗ is physically inexistent, although in the abstract sense, when f .t/ is supposed to be known for all the future Tt∗; 1/, it is mathematically correct, and rigorous theories in differential equations have been developed E-mail address: [email protected]. 0898-1221/$ – see front matter ' 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.10.066 E.A. Galperin / Computers and Mathematics with Applications 62 (2011) 4742–4757 4743 to describe the evolution of processes f .t/ under tacit assumption that those processes, indeed, follow certain future curve f .t/ obtained from a differential equation postulated for the time interval t 2 Tt∗; t∗ C 1t/ and extended to the future in its original setting for the entire half-axis Tt∗; 1/. Real processes in nature and technology may, however, contain some inputs (controls) from other processes (forces) that may influence the original process under consideration by providing some signals (actions) being measured or directly transmitted onto the original process, without following the tacit assumption for the rigid behavior of the original process f .t/; t 2 Tt∗; t∗ C 1t/. For this reason, the right derivative f 0.t/ written above is known only over the registered past history of the process, and it is physically inexistent at the current moment t∗ since the values f .t∗ C 1t/; 1t > 0, are not yet realized, thus, unknown. It means that feedback controls and other inputs on the right-hand side of equations of motion that depend on the higher order right derivatives are in contradiction with the physical reality, and for that reason they are formally excluded in major works on mechanics [1–5] and in textbooks that are followed in engineering projects. To remove this heavy restriction on the control of motion and processes imposed by the old tradition, the left and delayed time-derivatives are considered in the control of motion and in the process equations to assure the proper functioning and stability, and to improve the vital control systems, such as the autopilot in aviation and flight control systems in spacecrafts, by the acceleration assisted control with the onboard accelerometers. New representations of some physical laws are considered in this paper, and causality of differential systems is studied in relation to the orientation of time. Then, geometry and the time phenomena in classical mechanics are revisited, and the new forms of generalized equations of motion are derived with left and delayed higher order time-derivatives on the right- hand side. The method for their integration is demonstrated by example of a physical pendulum, and an application to the computerized autopilot design is proposed, with the acceleration assisted control to assure the safety of the aircraft in case of a failure of its outboard velocity sensors (Pitot tubes). The paper is organized as follows. In Section2, representations of Newton's second law of motion are considered with the generalization of this law for bodies with variable masses due to Buquoy [6] and later Mestschersky [7] and Levi-Civita [8]. In Section3, the problems related to time orientation and causality are discussed, and in Section4, the forces with left and delayed higher order derivatives of velocity are introduced into the general equation of motion. In Section5, some basic equations of theoretical physics are listed where the consideration of the left and delayed time-derivatives is necessary, with the basic lemma that allows us to reduce the equations with the left time-derivatives to the usual equations currently considered in mathematics. In Section6, the existence of solutions is proved under certain consistency condition related to the left highest order time-derivative on the right-hand side. In Section7, the fields of effective forces are considered, and the parallelogram law is verified for effective forces in linear systems depending on accelerations on right-hand sides. Section8 presents an application to the autopilot design in aviation, to expose the necessity of acceleration assisted control. Section9 presents an application to the reactive motion of a spacecraft with variable mass and acceleration assisted control. In Section 10, the equations in independent coordinates for motion of bodies with variable masses and left higher order derivatives are studied, and the Lagrange and Hamilton equations are presented that include generalized forces with left higher order derivatives. Section 11 presents a method of integration for the equations with left and delayed higher order time-derivatives on the right-hand side on the example of a physical pendulum. In Section 12, some points of interest are summarized, followed by references immediately relative to the problems considered. 2. Representations of Newton's second law of motion The second law of Newton states: ``Law II. The change of motion is proportional to the motive force impressed and is made in the direction of the right line in which that force is impressed'' [1], see also [5, p. 259]. In high school textbooks, this law is written in the form: ma D F, where m is a constant mass, a the acceleration, and F is ``the motive force impressed'' or simply ``a force'', a self-explanatory notion known from life experience. In university textbooks, it is specified in more exact terms: 00 0 00 0 mx D F.t; x.t/; v.t//; v.t/ D x .t/; x .t/ D v .t/ D a.t/; x.0/ D x0; v.0/ D v0; t ≥ 0; (1) which define a particular motion starting at x0; v0, with velocity v.t/ defined as time derivative 0 v.t/ D x .t/ D dx=dt D lim Tx.t C 1t/ − x.t/U=1t as 1t ! 0; 1t > 0: (2) Widely used representations (1)–(2) impose heavy restrictions in mechanics and control theory which restrictions are not necessary and can be removed. The first generalization of Newton's second law for reactive forces was proposed 200 years ago by Buquoy [6]. When m D const, the first formula in (1) can be written as follows: 00 0 mx D mv .t/ D mdv=dt D d.mv/=dt D F.t; x.t/; v.t//; t ≥ 0: (3) If m 6D const, then the last equality in (3) presents a more general form of Law II: d.mv/ D m dv C v dm D F.t; x.t/; v.t//dt; t ≥ 0; dt > 0; (4) where differentials can be viewed as small increments, this leading to the well known interpretation: ``the change of momentum, d.mv/, equals the impulse of force (or simply impulse), Fdt''.