Free variable mass nonlocal systems, jerks, and snaps, and their implications in rotating fluids in rockets
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O R I G I NA L PA P E R
Rami Ahmad El-Nabulsi
Free variable mass nonlocal systems, jerks, and snaps, and their implications in rotating fluids in rockets
Received: 10 July 2020 / Revised: 8 September 2020 / Accepted: 23 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Dynamical systems with position varying mass are archetypal examples of classical mechanical systems with rocket engine being a typical realistic model. In the present study, we extend the classical Newtonian mechanics by replacing the kinetic energy by a nonlocal-in-time kinetic energy and the standard velocity by a fractional velocity. These replacements lead to an extension of Newton’s second law of motion which has interesting implications in incompressible fluid dynamics. As an application, we discuss the rotating fluid problem subject to a position varying fluid mass which occurs in rocket dynamics. Several features were observed, mainly the transition from order to disorder in rotating fluids in rockets. PACS 02.30.Tb · 02.70.−c
1 Introduction Nonlocality exhibits significant effects in classical and quantum theories and is not necessarily quantum mechanical in nature [1–7]. Nonlocal effects occur largely in quantum theory and are feebly used in classical mechanics. In classical mechanics, the interest in nonlocal effects has emerged more recently after the interesting work of Suykens [8] where a realistic relationship between classical and quantum mechanics was raised. In fact, in Suykens’s approach, the classical kinetic energy 21 m x˙ x(m ˙ being the body mass and x˙ = d x/dt its x(t+τ ˙ )+x(t−τ ˙ ) 1 where τ is a small parameter relative to the time velocity) is replaced by a nonlocal form 2 m x˙ 2 scale entitled the “nonlocal time parameter”. This parameter is in fact an unknown variable and therefore its interpretation in any dynamical theory must be done carefully. Suykens’s approach was in fact motivated by Feynman’s observation of the kinetic energy term which can be written as a discrete time numerical estimation to velocities 21 m xε + xε − (shifting backward and forward in time for the body position) with x+ = xk+1 − xk , x− = xk − xk−1 and ε = ti = ti+1 −ti in association with a measurement quantum process by means of the k kinetic energy of the particle. Since the following Taylor series expansions: x(t + τ ) ≈ x(t) + nk=1 τk! x (k) (t) )k (k) 1 and x(t − τ ) ≈ x(t) + nk=1 (−τ k! x (t) hold, then the conventional kinetic energy K = 2 m x˙ x˙ may be k 2 replaced by K τ = m2x˙ + m4x˙ nk=1 1+(−1) τ k x (k+1) . Accordingly, jerks and hyperjerk arise. This approach k! has led in fact to a number of interesting properties in different fields and was explored in details in several research papers [9–20]. In this approach, higher-order derivatives terms emerge in agreement with the fundamental aspect of quantum mechanics [21–24] and which are also one of the main features of nonlocality in dynamical systems [25,26]. It is noteworthy that jerks and hyperjerk systems have many interesting
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