Motions for systems and structures in space

Journal of Physics and Astronomy Research

JPAR

Vol. 2(3), pp. 070-073, September, 2015. © www.premierpublishers.org, ISSN: 2123-503x

Research Article

Motions for systems and structures in space, described by a set denoted Avd. Theorems for local implosion; Li, dl and angular velocities Lena J-T Strömberg Previously Department of Solid Mechanics, Royal Institute of Technology, KTH, Sweden E-mail: [email protected] In order to describe general motions and matter in space, functions for angular velocity and density are assumed and denoted Avd, as an abbreviation. The framework provides a unified approach to motions at different scales. It is analysed how Avd enters and rules, in terms of results from equations, in field experiments and observations at Earth. Chaos may organize according to Avd, such that more order, Cosmos, appear in complex nonlinear dynamical systems. This reveals that Avd may be governing and that deterministic systems can be created without assuming boundaries and conditions for initial values and forces from outside. A mathematical model for the initiation of Logos (when a paper accelerates into a narrow circular orbit), was described, and denoted local implosion; Li. The theorem for dl, provides discrete solutions to a power law, and this is related to locations of satellites and moons. Key words: Angular velocity, density, Avd, Logos, Local implosion, Discrete locations, Theorems, Deterministic Chaos, Cosmos, unification of kinematics, Lena’s lemma for Riemenn -function INTRODUCTION With the purpose of finding origin of motions, and a proper description without introducing pressure and forces, a small model will be proposed, and denoted Avd, as an abbreviation for angular velocity and density. This is based on functions derived for a non-circular planetary orbit, Strömberg (2014, 2015). Several applications are covered in Strömberg (2014, 2015), e.g. planetary motions, locations, distributed matter, frequencies in acoustics and occurrence of a factor 3/2, also known from Correia and Laskar (2004) as the ratio of sidereal and orbital rotation of Mercury. If assumed valid at different scales, the model is a unification of kinematics, which concerns the geometry of motion in space. Experiments with natural complex motions, not steady, monotonic and linear, but where Avd enter, will be presented and analysed.

Theorems for angular velocities are proposed. To model the paper experiment, with aggregation occurring with a large , denoted as a local implosion, a differential equation is derived. Assuming an iterative solution with transients, a capture into steady states, corresponding to e.g. planetary orbits, is obtained. Within the framework, discrete and quantised solutions are obtained without assuming a differential equation. Model for the origin of motions: Avd Definition. To describe presence of Cosmos, for dynamical systems in space, a set of functions is defined. This will be known as Avd. Avd consists of the functions i) (t)=0 exp(-2(recc/r0)sin(f0t)) where 0 is constant

Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities

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Figure1. Left. Front view of the paper at small oscillations. Right. After entering Li, with a rapid rotational velocity

ii) (t)=0 exp (2(recc/r0)sin(f0t)) where 0 is constant where i) is derived in Strömberg (2015), and ii) is the Ledensity introduced in Strömberg (2014), such that the continuity of mass, is fulfilled, (i.e. point wise no expansion or compaction of mass matter.) Conclusions of physics, for motions, by analysis of Avd Conjecture A-Ph (Avd to Physics)  Elements/Functions in Avd may couple with motion of a material body in space  Functional relations for the elements may provide information on physical laws for motion, and other properties, e.g. invariants and magnitudes,  The coupling requires energy, which is supplied by initial motion of the body in space, and then another motion or process may begin The functions in Avd are derived for an orbiting planet with eccentricity. At Earth, this could be present as a tide, or a memory of a past non-circular orbit. To be applicable at a smaller scale, angular velocities are assumed to be scaled, or that an Avd is created by a motion in the system, interacting with the surroundings. Next the results from experiments for a motion will be presented and analysed. Avd appearing in a free falling paper (A5, thin) To obtain knowledge of pure motion, a field experiment of a natural composed motion not pre-arranged and as little as possible pertubated, will be analysed in terms of Avd. It can be described and ruled by an angular velocity, within short time intervals. 1.First, the paper moves with small horizontal oscillations c.f. Fig. 1 left, given by the functions  2.Then it achieves a large angular acceleration. With this it accelerates into a circular path, where the shape gets much more curved, Fig. 1 right. The space-time when curved will be known as initially, Logos, and then Local implosion, Li. After one lap, the 'internal energy', in the circular shape is released, and the paper achieves its original shape. In the flat shape it moves in a fast translation, in a

different direction. (This may be related to diffusion and isotropic behaviour in space). At this state, it is also possible that it obtains degrees of freedom, d.o.f. that we cannot observe with the eye, (such that presence in other dimension), coupling with the surrounding, e.g. Newtonian gravity, and the oscillation at phase 1, or an own rotational frame created at Logos, however probably not, since only one lap. Performing the experiment, you may note that when too close, the paper approaches towards thee, (in compaction, contraction, by attraction) and then bumps outward again. Sometimes it twists, in oscillations or a twisting lap instead of a Li, and sometimes it moves in an opposite horizontal direction before the fast lap. Detailed description of Logos and Li. Before Logos, as  increases, the curvature increases, and at entire revolution, it forms a small circle. This could be due to either, or all, of the following hypothesis  It copies the motion at larger scale for Earth rotation  An own gravitational field is created  It obeys a Bernoulli’s law, such that a large pressure drop balances the increased velocity, and an isotropic compressive state is prevailed. A mathematical model for the initiation of Li will be formulated Theorem Local implosion: Initiation of the phase 2, for a falling paper, can be modelled with the functionin Avd, since these admits high angular acceleration. Proof. A differentiation acceleration

of

(i),

gives

the

angular

d/dt= f0 0 exp(-2(recc/r0)sin(f0t)) )(-2(recc/r0)cos(f0t)) A linearisation for small f0t close to f0t = gives the differential equation d/dt=0f02(re/r0), which can be integrated exactly, to read =0exp(2(re/r0) f0t). This means that  can increase rapidly, as is the case when entering Li. Another description which gives an increased is this format is obtained by assuming that r0 , i.e. the radius of curvature approaches zero.


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Algebra and Functions on Avd

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1/g(n)2), n are integers or rationals and g(n) is a function of n.

I. Non-dimensional format II. Logarithmic ln of the non-dimensional format, gives a harmonic III. Power expressions. Solutions to nonlinear differential equations, and may be the source of forced vibrations. Result for Earth equator power law Relations for power expressions, in the formatp= Cqwill be known as power laws. With the conditions at Earth, it will be shown that a power law is valid for discrete times, or if a spatial description with t=, discrete angles,. The relation 2=G4/3(1) Is derived from balance of centripetal acceleration and Newtonian law of gravity, and rules both discrete masses (i.e. satellites), and arbitrary mass elements parts of Earth equator considered as a continuum. The result is experimentally valid without introducing the force, by measurement of density and angular velocity. Theorem for discrete (quantised) locations, dl. Theorem dl. With andfrom Avd, (1) implies discrete solutions for the locations. Proof. Identification of the time dependent parts in (1). The results provide a link between a continuum description, and the motions of discrete masses e.g. moons around a planet. If (1) was assumed for Gas Giants at formation, the number of satellites may be compared with the close moons for Jupiter (8), and moons of Saturnus (24). With angular velocity, the angles may be expressed with discrete times. Assuming extension in radial direction provides the orbits and quantisation. To summarize, this will be denoted the Lena-theorem for dl. Lena-theorem for Updated angular velocities, Uav. As an opposite of fast acceleration derived above, we shall consider an iterative format, which stabilize w into a constant proportional to. In conjunction with original format, will also be calculated from previous values, updated with transients. The ratio , will depend on the parameters in the functions of Avd, and the Riemann sum for the Riemann -function. Preliminaries. Consider a subdivision of the logarithmic format ln(), into ln(n+1), where n=n-1exp(-

Theorem Uav: Assume from Avd, the harmonic for ln (n+1)=-2(recc/r0)sin(f0t). The iterative formatn =n2 1exp (-1/n ), where n are e.g. integers 1,2,3, or half integers 3/2, 5/2, 7/2,gives a stable constant solution, such thatn =n-1, for large n. The exact value (for half integers)is given by n =0 exp(-2(recc/r0)sin(f0t)-(2)) , whereis the Riemann-function for half integers. Lena’s Lemma: relates to the complex Riemann -

function (z)=(1–exp(-zln(2))-1((z)+1), where z = a+ib and (z)=(1/nz) Proof of Lemma: Evaluation and identification of terms in the sum. Proof. Insertion and evaluation of ln of products into sums of ln, and identification with the Riemann sum. Remarks.With n being half integers, values more close to the ratios for gas giants and the planets are obtained, also for smaller eccentricities. With the complex Riemann z-function, a harmonic oscillation is obtained, depending on a dimensionless parameter b, and with almost constant frequency for small b. Such couplings may have been used in earlier calculations and when formulating inventing the Riemann hypothesis. Cosmos Subsequently also the word figure will be used for the coupling, to manifest that it is something visible which embodies in a structure. ConjectureA-(Avd-to-cosmos) Functions of Avd may appear in other dynamical systems, e.g. as Cosmos i) to create order when chaos ii) to minimize d.o.f. iii) to obtain a constant energy, or a steady motion Such cosmos can be included equations and modelling, nd as 2 and higher order effects, or as an additional overall principle or constraint. Examples. Two clocks at the same wall achieve a synchronised period. Women living together can achieve the same evaluation-period, however this could be more related to the orientation to the heart as a vertical dumb bell, and the tide.


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Figure 2. The motion of the (metallic) flakes is partly rotational, and downward direction due to gravity from outside, is not so noticeable.

Additional examples of systems in such motions Avd figures in dynamical systems, e.g. a free falling paper. Other examples may be 

Motions of flakes, in a liquid bulb chamber.

When turned upside down, the downward motion due to gravity, is accomplished with (visually almost dominated by) much rotation. When bulb is subjected to motion back and forth around the vertical axis, very much rotation is transferred to the flakes, such that they rotate rapidly, in small arcs, c.f. Figure 2.

and whether this can be a valid model for many systems in complex interaction with the surrounding. If so, chaos may organize according to Avd, and nonlinear deterministic systems can be described without an ‘a priori’ assumption of forces and pressure through boundary conditions. Since applicable at different scales, Avd provides a unification, not of forces, but of kinematics.

CONCLUSION

The same path, but more slow is found for a falling leaf or feather ‘sailing’ in the air, e.g. in the film Forrest Gump. A mathematical model for Logos, when the paper starts accelerates into a narrow circular orbit, denoted Local implosion, Li, was described. This is characterized by a large curvature of the paper. If the kinetic energy is bounded, then the velocity is also bounded, such that a large angular velocity multiplied with a small radius of the orbit, is limited and may be initialized with a finite kinetic energy from previous motion. The word Logos is from th Aristotle (4 century BC). For shape memory alloys Auricchio et al (2008), hard inclusions of martensite are modelled, with small spheres that may unwarp during steady state loading. In this context, it can be mentioned that in classical construction steel, the martensite gives a harder but more brittle behavior. In some applications ductile steels may be preferred, since they can withstand small cracks, and other loads e.g. weld residuals and environment.

In order to describe general motions and matter in space, two functions from Strömberg, 2014, 2015), called Avd were introduced. These were compared with field experiments for multi-d.o.f. motions, and composed systems. It was discussed how these may enter and rule

A theorem for dl, provided discrete solutions to a power law, and this was related to locations of satellites and moons. From an iterative format, stable constant values of angular velocities were obtained from a summary of rationals, derived from results for Riemann -function.

 Smoke rings in a gravity field, rising since lighter than air, and form a spiral, consistent with Avd. 

A whirlwind, created over hot sea.

 Logos occurs in flows on curved surfaces, e.g. the eye globe. More detailed and extensive models of fluid structure interaction with whirls are treated in e.g. Walther and Koumoutsakos (2001) and van Rees et al. (2013).


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ACKNOWLEDGEMENTS To Tech Power Oslo, Stiftelsen Jernkontoret and Dr Rebbah. The author acknowledged the contributions of Prof. Marcos Voelzke, Željko Prebeg, Dr. Cyd Ropp and J. Ponce de Leon for donating their time, critical evaluation, constructive comments, and invaluable assistance toward the improvement of this very manuscript.

REFERENCES Aristotle 4th century B. Rhetoric, Greek treatise on the art of persuasion. Auricchio F, Mielke A, and Stefanelli U (2008).A rateindependent model for the isothermal quasi-static evolution of shape-memory materials, Math. Models Methods Appl. Sci. 18, 125. Correia A, Laskar J (2004).Mercury's capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics. Nature 429, 848-850. Strömberg L (2014). A model for non-circular orbits derived from a two-step linearisation of the Kepler laws. Journal of Physics and Astronomy Research 1(2): 013014. Strömberg L (2015). Models for locations in the solar system. Journal of Physics and Astronomy Research 1(2): 054-058. Strömberg L (2014).Generalized potentials describing orbits in the solar system. Derivation of a ‘close force’ acting on the inner moon Phobos. Journal of Aerospace Science and Technology 1: 48-52. van Rees W.M., Gazzola M., Koumoutsakos P (2013).Optimal shapes for anguilliform swimmers at intermediate Reynolds numbers. Journal of Fluid Mechanics, 722. Walther J.H., Koumoutsakos P (2001).Three-dimensional vortex methods for particle-laden flows with two-way coupling, J. Comput. Physics 167: 39-71. Accepted 15 August, 2015 Citation: Strömberg L (2015). Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities. Journal of Physics and Astronomy Research, 2(3): 070-073.

Copyright: © 2015 Strömberg L. This is an open-access article distributed under the terms of the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited. Motions for systems and structures in space, described by a set denoted Avd.Theorems for local implosion; Li, dl and angular velocities

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