DOCSLIB.ORG

Appendix a the CALCULUS of VARIATIONS Introduction A.L

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix a the CALCULUS of VARIATIONS Introduction A.L Appendix A THE CALCULUS OF VARIATIONS Introduction The calculus of variations provides the mathematical foundation for the study of analytical mechanics as well as a number of other branches of physical sciences. In this appendix we study the essentials of the calculus of variations for applications to problems of analytical dynamics. We begin our study by introducing the notion of a functional and by way of motivation pose a number of classical problems concemed with the extremum of func­ tionals. We then show that the extremum conditions of a functional appear as solu­ tions of a differential equation - the so-called Euler-Lagrange equation. Next we con­ sider the extremization of functionals subject to different types of constraints and show that the Lagrange multiplier method provides a versatile technique for taking account of such constraints. Finally we consider functionals with variable end points and derive associated extremum conditions. A.l Functions and Functionals A function, for example u = u (x ,y), establishes a relationship between two or more variables e.g. u, and x and y . Generally the dependent variable, u, is expres sed in terms of one or more independent variables, in this case x and y, and when the numerical values of the independent variables are specified, that of the dependent vari­ able can be calculated. A functional establishes a relationship between one variable and several functions. Consider, for instance, s = fX 2 F (x, u(x), u' (x), v(x), v'(x» dx (A.l.1) XI Here in order to calculate a numerical value for s one must specify the functions u (x), and v(x). 310 For only one function, we may write X 2 s = J F (x, y (x ), y' (x )) dx (A. 1.2) XI y / Y I 2 B Yl ........ / A x O a b Figure A.I. Distance between two points in a plane Example A.l.l What is the shortest distance between two points in a plane? The distance between points A and B may be expressed as (see Figure A.I) s = J ds (a) But (b) Thus (c) Clearly when y, as a function of x, takes different forms, the functional s will take different values. Hence in equation (c) we have the integrand F = ~, and we ask which particular form of y , as a function of x , minimizes s . The study of finding the minima andlor maxima of functionals is referred to as the cal­ culus of variations. We shall develop procedures for finding such extremum condi­ tions of functionals in due course. For example A.1.1 at hand it is worth noting that: 1) Different forms of y which compete to minimize s in (c) are aH functions of x, i.e. the independent variable is not altered. 311 2) AU forms of y which are allowed to compete for minimization of s should pass through points A and B, where x = a and x = b respectively. 3) The forms of y considered must be continuous in the interval AB so that y in the integrand should remain finite. Thus there are some restrictions to be imposed on competing functions. In other prob­ lems the restrictions may take different forms. Such restrictions on the competing functions are referred to as the admissibility requirements. Problem A.I.I The brachystochrone problem: A particle m subject to gravitation g on a fric­ tionless path s of arbitrary shape z = z (x), is to move between points (0,0) and (x 2, z 2) in the shortest time possible. Determine the integrand F in X2 t=fFdx o x O x z !g Z2 .......... ........ Z A.2 Review of Extremum Values of Functions The maximum (minimum) points of functions are distinguished by the property that slight changes of the independent variables will result in a decrease (increase) in the value of the dependent variable whether the changes in the independent variable are positive or negative. For instance for a function such as y = y (x), if there exists a minimum point at x = o, we have that y (o + E) - Y (a) > O (A.2.1) Where E is a small change in x and it may be positive or negative. 312 Let us expand y (a + E) by TayIor series near the point x = a : y(a + E) = y(a) + y , Ia E + TI1" y Ia E2 + 3!1 Y '" I a E3 + (A.2.2) The derivatives y', y" etc. are alI evaluated at x = a and hence the function y = y (x) and its derivatives must be continuous at x = a. From equation (A.2.2) it can be seen that if the sign of [ y (a + E) - Y (a) ] is to be independent of the sign of E, as E approaches zero, the coefficient of y' must vanish since the sign of this term does depend upon the sign of e. Further, it is apparent from equation (A.2.2) that at a maximum point the coefficient of e2, nameIy ;! y", must be non-positive while at a minimum point it must be non-negative. Thus summarizing y' = O and y" < O maximum point (A. 2.3) y = O and y" > O minimum point It is of course possible for y' to vanish at a point where y" = O. In that case condi­ tions in relations (A.2.3) are not sufficient to determine the nature of the point and one must examine the coefficients of higher order terms. Thus at x = O of the function y = x 3 the coefficient of e3 does not vanish and since the value of this term will depend upon the sign of E one must conclude that x = O is neither a maximum nor a minimum point of the function. On the other hand x = O for the function y = x 4 can be readiIy seen to be a minimum point. It is to be noted that our definition of a maximum or minimum point, in effect describes a "tuming point" of the function. Now a given function may possess several tuming points in its range and while the conditions stated in relations (A.2.3) bold IocalIy at every tuming point, they do not yield information as to whether a given point is the global maximum or the global minimum point of the function. Thus for determining global extremum points a systematic search must be carried out amongst alI the tuming points of the function in the domain of the independent variabIe. It is also possible for a function to attain its extremum value at the boundaries of its range. In that case the extremum point need not be a tuming point and such conditions in relations (A.2.3) may not hold. In the case of a function of two independent variables such as z = z(x,y) (A.2.4) we have a surface which can be pictured as a terrain. We are now interested in finding the peaks of the hills and the Iowest points of the valIeys in this terrain. At a point x = a, y = b the following possibilities may arise i) Maximum Then z (a + E, b + a) - z (a, b) < O 313 independent of the sign of small increments e and a and as e ~ O and a ~ O. ii) Minimum Then z (a + e, b + a) - z (a , b) > o independent of the sign of e and a and as e ~ O and a ~ O. iii) A saddle point (see Figure A.2) z y Figure A.2. A saddle point Then for some values of e and a, again independent of their signs, we may have z (a + e, b + a) - z (a, b) > O z (a + e, b + a) - z (a , b) = O z (a + e, b + a) - z (a , b) < O as e ~ O and a ~ O. Again let us expand z (a + e, b + a) br. a Taylor serirS about x = a, y = b. , z (a + e, b + a) = z (a , b) + z 'x e + z 'y a + a,b a,b 'xx 2 'yy 2 'xy 'xxx 3 2\. z Ia,b e + 2\. z Ia,b a + z leaa,b + 3\. z Ia,b e + ... (A.2.5) where, ( )'x implies partial differentiation with respect to x etc., and once again alI the partial derivatives are evaluated at x = a, y = b. Hence it is assumed that z (x , y) is continuous and possesses continuous derivatives at (a, b). 314 By a similar reasoning for the independence of the sign of [z (a + e, b + a) - z (a , b)] from the signs of e and a we conclude that for the cases i, ii, and iii, cited above, we must have z'x = O Z'y = O at (a, b) (A.2.6) To find out whether the point (a, b) is a maximum or minimum point we examine the second order terms. These form a quadratic function Q (e, a) which we write as 1 [ z 'Xl: z 'xy 1[ e 1 Q (e, a) = 2![e a] z 'xy z 'yy a (A.2.7) For a maximum or minimum point the sign of Q must be independent of the signs of e and a. Thus for a minimum point, Q should remain non-negative for alI real values and either signs of e and a. Likewise, for a maximum point, Q should remain non­ positive for alI real values and signs of e and a. This property of Q, namely possessing only one sign, independent of signs of e andlor a, depends upon the form of the symmetric coefficient matrix in equation (A.2.7).
Load more

Recommended publications
  • Notes on Calculus II Integral Calculus Miguel A. Lerma
    Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus.
    [Show full text]
  • Leibniz' Dynamical Metaphysicsand the Origins
    LEIBNIZ’ DYNAMICAL METAPHYSICSAND THE ORIGINS OF THE VIS VIVA CONTROVERSY* by George Gale Jr. * I am grateful to Marjorie Grene, Neal Gilbert, Ronald Arbini and Rom Harr6 for their comments on earlier drafts of this essay. Systematics Vol. 11 No. 3 Although recent work has begun to clarify the later history and development of the vis viva controversy, the origins of the conflict arc still obscure.1 This is understandable, since it was Leibniz who fired the initial barrage of the battle; and, as always, it is extremely difficult to disentangle the various elements of the great physicist-philosopher’s thought. His conception includes facets of physics, mathematics and, of course, metaphysics. However, one feature is essential to any possible understanding of the genesis of the debate over vis viva: we must note, as did Leibniz, that the vis viva notion was to emerge as the first element of a new science, which differed significantly from that of Descartes and Newton, and which Leibniz called the science of dynamics. In what follows, I attempt to clarify the various strands which were woven into the vis viva concept, noting especially the conceptual framework which emerged and later evolved into the science of dynamics as we know it today. I shall argue, in general, that Leibniz’ science of dynamics developed as a consistent physical interpretation of certain of his metaphysical and mathematical beliefs. Thus the following analysis indicates that at least once in the history of science, an important development in physical conceptualization was intimately dependent upon developments in metaphysical conceptualization. 1.
    [Show full text]
  • THE EARTH's GRAVITY OUTLINE the Earth's Gravitational Field
    GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY OUTLINE The Earth's gravitational field 2 Newton's law of gravitation: Fgrav = GMm=r ; Gravitational field = gravitational acceleration g; gravitational potential, equipotential surfaces. g for a non–rotating spherically symmetric Earth; Effects of rotation and ellipticity – variation with latitude, the reference ellipsoid and International Gravity Formula; Effects of elevation and topography, intervening rock, density inhomogeneities, tides. The geoid: equipotential mean–sea–level surface on which g = IGF value. Gravity surveys Measurement: gravity units, gravimeters, survey procedures; the geoid; satellite altimetry. Gravity corrections – latitude, elevation, Bouguer, terrain, drift; Interpretation of gravity anomalies: regional–residual separation; regional variations and deep (crust, mantle) structure; local variations and shallow density anomalies; Examples of Bouguer gravity anomalies. Isostasy Mechanism: level of compensation; Pratt and Airy models; mountain roots; Isostasy and free–air gravity, examples of isostatic balance and isostatic anomalies. Background reading: Fowler §5.1–5.6; Lowrie §2.2–2.6; Kearey & Vine §2.11. GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY FIELD Newton's law of gravitation is: ¯ GMm F = r2 11 2 2 1 3 2 where the Gravitational Constant G = 6:673 10− Nm kg− (kg− m s− ). ¢ The field strength of the Earth's gravitational field is defined as the gravitational force acting on unit mass. From Newton's third¯ law of mechanics, F = ma, it follows that gravitational force per unit mass = gravitational acceleration g. g is approximately 9:8m/s2 at the surface of the Earth. A related concept is gravitational potential: the gravitational potential V at a point P is the work done against gravity in ¯ P bringing unit mass from infinity to P.
    [Show full text]
  • A Brief Tour of Vector Calculus
    A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0.
    [Show full text]
  • Differentiation Rules (Differential Calculus)
    Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below.
    [Show full text]
  • A Conjecture of Thermo-Gravitation Based on Geometry, Classical
    A conjecture of thermo-gravitation based on geometry, classical physics and classical thermodynamics The authors: Weicong Xu a, b, Li Zhao a, b, * a Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, Ministry of Education of China, Tianjin 300350, China b School of mechanical engineering, Tianjin University, Tianjin 300350, China * Corresponding author. Tel: +86-022-27890051; Fax: +86-022-27404188; E-mail: [email protected] Abstract One of the goals that physicists have been pursuing is to get the same explanation from different angles for the same phenomenon, so as to realize the unity of basic physical laws. Geometry, classical mechanics and classical thermodynamics are three relatively old disciplines. Their research methods and perspectives for the same phenomenon are quite different. However, there must be some undetermined connections and symmetries among them. In previous studies, there is a lack of horizontal analogical research on the basic theories of different disciplines, but revealing the deep connections between them will help to deepen the understanding of the existing system and promote the common development of multiple disciplines. Using the method of analogy analysis, five basic axioms of geometry, four laws of classical mechanics and four laws of thermodynamics are compared and analyzed. The similarity and relevance of basic laws between different disciplines is proposed. Then, by comparing the axiom of circle in geometry and Newton’s law of universal gravitation, the conjecture of the law of thermo-gravitation is put forward. Keywords thermo-gravitation, analogy method, thermodynamics, geometry 1. Introduction With the development of science, the theories and methods of describing the same macro system are becoming more and more abundant.
    [Show full text]
  • Introduction to the Modern Calculus of Variations
    MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result.
    [Show full text]
  • Finite Elements for the Treatment of the Inextensibility Constraint
    Comput Mech DOI 10.1007/s00466-017-1437-9 ORIGINAL PAPER Fiber-reinforced materials: finite elements for the treatment of the inextensibility constraint Ferdinando Auricchio1 · Giulia Scalet1 · Peter Wriggers2 Received: 5 April 2017 / Accepted: 29 May 2017 © Springer-Verlag GmbH Germany 2017 Abstract The present paper proposes a numerical frame- 1 Introduction work for the analysis of problems involving fiber-reinforced anisotropic materials. Specifically, isotropic linear elastic The use of fiber-reinforcements for the development of novel solids, reinforced by a single family of inextensible fibers, and efficient materials can be traced back to the 1960s are considered. The kinematic constraint equation of inex- and, to date, it is an active research topic. Such materi- tensibility in the fiber direction leads to the presence of als are composed of a matrix, reinforced by one or more an undetermined fiber stress in the constitutive equations. families of fibers which are systematically arranged in the To avoid locking-phenomena in the numerical solution due matrix itself. The interest towards fiber-reinforced mate- to the presence of the constraint, mixed finite elements rials stems from the mechanical properties of the fibers, based on the Lagrange multiplier, perturbed Lagrangian, and which offer a significant increase in structural efficiency. penalty method are proposed. Several boundary-value prob- Particularly, these materials possess high resistance and lems under plane strain conditions are solved and numerical stiffness, despite the low specific weight, together with an results are compared to analytical solutions, whenever the anisotropic mechanical behavior determined by the fiber derivation is possible. The performed simulations allow to distribution.
    [Show full text]
  • Visual Differential Calculus
    Proceedings of 2014 Zone 1 Conference of the American Society for Engineering Education (ASEE Zone 1) Visual Differential Calculus Andrew Grossfield, Ph.D., P.E., Life Member, ASEE, IEEE Abstract— This expository paper is intended to provide = (y2 – y1) / (x2 – x1) = = m = tan(α) Equation 1 engineering and technology students with a purely visual and intuitive approach to differential calculus. The plan is that where α is the angle of inclination of the line with the students who see intuitively the benefits of the strategies of horizontal. Since the direction of a straight line is constant at calculus will be encouraged to master the algebraic form changing techniques such as solving, factoring and completing every point, so too will be the angle of inclination, the slope, the square. Differential calculus will be treated as a continuation m, of the line and the difference quotient between any pair of of the study of branches11 of continuous and smooth curves points. In the case of a straight line vertical changes, Δy, are described by equations which was initiated in a pre-calculus or always the same multiple, m, of the corresponding horizontal advanced algebra course. Functions are defined as the single changes, Δx, whether or not the changes are small. valued expressions which describe the branches of the curves. However for curves which are not straight lines, the Derivatives are secondary functions derived from the just mentioned functions in order to obtain the slopes of the lines situation is not as simple. Select two pairs of points at random tangent to the curves.
    [Show full text]
  • A Variational Approach to Lagrange Multipliers
    JOTA manuscript No. (will be inserted by the editor) A Variational Approach to Lagrange Multipliers Jonathan M. Borwein · Qiji J. Zhu Received: date / Accepted: date Abstract We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied. Keywords Lagrange multiplier Variational method Convex duality Constrained optimiza- tion Nonsmooth analysis · · · · Mathematics Subject Classification (2000) 90C25 90C46 49N15 · · 1 Introduction The Lagrange multiplier method is fundamental in dealing with constrained optimization prob- lems and is also related to many other important results. There are many different routes to reaching the fundamental result. The variational approach used in [1] provides a deep under- standing of the nature of the Lagrange multiplier rule and is the focus of this survey. David Gale's seminal paper [2] provides a penetrating explanation of the economic meaning of the Lagrange multiplier in the convex case. Consider maximizing the output of an economy with resource constraints. Then the optimal output is a function of the level of resources. It turns out the derivative of this function, if exists, is exactly the Lagrange multiplier for the constrained optimization problem. A Lagrange multiplier, then, reflects the marginal gain of the output function with respect to the vector of resource constraints. Following this observation, if we penalize the resource utilization with a (vector) Lagrange multiplier then the constrained optimization problem can be converted to an unconstrained one. One cannot emphasize enough the importance of this insight. In general, however, an optimal value function for a constrained optimization problem is nei- ther convex nor smooth.
    [Show full text]
  • Universal Gravitational Constant EX-9908 Page 1 of 13
    Universal Gravitational Constant EX-9908 Page 1 of 13 Universal Gravitational Constant EQUIPMENT 1 Gravitational Torsion Balance AP-8215 1 X-Y Adjustable Diode Laser OS-8526A 1 45 cm Steel Rod ME-8736 1 Large Table Clamp ME-9472 1 Meter Stick SE-7333 INTRODUCTION The Gravitational Torsion Balance reprises one of the great experiments in the history of physics—the measurement of the gravitational constant, as performed by Henry Cavendish in 1798. The Gravitational Torsion Balance consists of two 38.3 gram masses suspended from a highly sensitive torsion ribbon and two1.5 kilogram masses that can be positioned as required. The Gravitational Torsion Balance is oriented so the force of gravity between the small balls and the earth is negated (the pendulum is nearly perfectly aligned vertically and horizontally). The large masses are brought near the smaller masses, and the gravitational force between the large and small masses is measured by observing the twist of the torsion ribbon. An optical lever, produced by a laser light source and a mirror affixed to the torsion pendulum, is used to accurately measure the small twist of the ribbon. THEORY The gravitational attraction of all objects toward the Earth is obvious. The gravitational attraction of every object to every other object, however, is anything but obvious. Despite the lack of direct evidence for any such attraction between everyday objects, Isaac Newton was able to deduce his law of universal gravitation. Newton’s law of universal gravitation: m m F G 1 2 r 2 where m1 and m2 are the masses of the objects, r is the distance between them, and G = 6.67 x 10-11 Nm2/kg2 However, in Newton's time, every measurable example of this gravitational force included the Earth as one of the masses.
    [Show full text]
  • RELATIVISTIC GRAVITY and the ORIGIN of INERTIA and INERTIAL MASS K Tsarouchas
    RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS K Tsarouchas To cite this version: K Tsarouchas. RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS. 2021. hal-01474982v5 HAL Id: hal-01474982 https://hal.archives-ouvertes.fr/hal-01474982v5 Preprint submitted on 3 Feb 2021 (v5), last revised 11 Jul 2021 (v6) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Relativistic Gravity and the Origin of Inertia and Inertial Mass K. I. Tsarouchas School of Mechanical Engineering National Technical University of Athens, Greece E-mail-1: [email protected] - E-mail-2: [email protected] Abstract If equilibrium is to be a frame-independent condition, it is necessary the gravitational force to have precisely the same transformation law as that of the Lorentz-force. Therefore, gravity should be described by a gravitomagnetic theory with equations which have the same mathematical form as those of the electromagnetic theory, with the gravitational mass as a Lorentz invariant. Using this gravitomagnetic theory, in order to ensure the relativity of all kinds of translatory motion, we accept the principle of covariance and the equivalence principle and thus we prove that, 1.
    [Show full text]