Towards a Unification of the Laws of Physics in Classical Fields Theory

Jean-Paul Caltagirone is Professor Emeritus at the National Polytechnic Institute of the University of Bordeaux, France. His research focuses on the conservation of acceleration in the field of mechanics, the basis of discrete mechanics, which has now been extended to all classical field physics.

• Preface xiList of Symbols xvChapter 1 Objections and Rebuttals of Current Laws 11.1. Discussion on the general concepts 11.1.1. Choosing a three-dimensional space 11.1.2. Galileo’s law of free-falling bodies 21.1.3. Uniform rotation, a Galilean motion 21.1.4. Galileo and Lorentz invariants 51.1.5. Distinction between velocity and celerity 61.1.6. Space–time or space and time? 71.2. Objections to the equations of classical mechanics 81.2.1. Equations of mechanics 81.2.2. Divergence and curl operators 101.2.3. The Stokes relation, an erroneous assumption 111.2.4. Non-relativistic equations 12Chapter 2 A Different View of Space and Time 152.1. Maxwell’s local frame of reference 152.2. Length and time 172.3. The local notion of vector in an n-dimensional space 182.3.1. The example of the gravity vector 202.3.2. Vector precession 212.4. Acceleration, velocity and celerity 222.4.1. Velocity limits 232.4.2. Adding velocities 252.5. Galileo and Lorentz transformations 262.6. The genesis of a unified law 272.6.1. The principle of parsimony or Ockham’s razor 272.6.2. Extension of Galileo’s principle of inertia to rotation 292.6.3. Interpretation of the WEP 322.6.4. Concept of physical homology 362.7. Mass or energy 362.7.1. Energy per unit mass 372.8. The quantities of a unified physics 382.8.1. Historical background 382.8.2. From the international system to the unification of units 392.8.3. Unified variables 392.8.4. Unified potentials 402.8.5. Curvature of physics potentials 432.9. A one-dimensional model of space and time 452.9.1. Representation of space in the equations of physics 45Chapter 3 Unified Law of Motion 493.1. Dynamics of accelerated motions 493.2. Dynamics of uniform expansion and rotational motion 503.3. Kinematics of motion in discrete mechanics 543.3.1. Non-accelerated motion 543.3.2. Accelerated motion 563.4. Laws of conservation of compressive and rotational energy 573.5. Total energy conservation law 593.6. Principle of inertia 623.6.1. The nonlinear terms of inertia 623.6.2. Inertia as the curvature of the Bernoulli potential 633.6.3. Inertia of two superimposed motions 673.6.4. Example of uniform rotational motion 673.7. Helmholtz–Hodge decomposition 693.7.1. HHD of acceleration 693.7.2. HHD of velocity 713.7.3. Orthogonality of the decomposition 723.8. Properties of the law of motion 743.8.1. A relativistic law 743.8.2. A limitless local law 753.8.3. Dissipation of compression and shear energies 763.9. Potential couplings and interaction 773.9.1. Properties of media 773.9.2. Unified law of motion 793.9.3. Law of motion and source potentials 80Chapter 4 Consequences of the Law of Motion 834.1. Weak equivalence principle revisited 834.1.1. The example of a falling heterogeneous body 834.1.2. Violation of the WEP 854.2. Velocity limit 894.2.1. Uniformly accelerated translational motion 894.2.2. Uniformly accelerated rotational motion 944.3. Advection 954.4. Local primal and dual forms of Bernoulli’s law 974.4.1. An equation of motion for incompressible flows of viscous fluid 994.5. Invariances and Noether’s theorem 1014.6. Absence of constitutive laws 105Chapter 5 Fluid Mechanics 1075.1. Inertia, a concept at the heart of mechanics 1075.1.1 Physical meaning of L = V ×∇×V 1095.1.2. Example of a three-dimensional space vector 1115.1.3 Intrinsic Property V⊥∇ × V 1125.1.4. Application of the NS equations to inertia 1165.1.5. The consequences 1185.1.6. A steady solution 1185.2. Incompressible fluid mechanics 1205.2.1. A unified equation of fluid motion 1215.2.2. Poiseuille flow 1235.2.3. Taylor–Green vortex 1255.2.4. Role of vortex stretching on a Taylor–Green vortex 1305.3. Two-phase flows 1335.3.1. Modeling the capillary acceleration 1335.3.2. Geometric curvature 1355.3.3. Classical estimates of normals and curvatures 1365.3.4. Curvature in DM 1385.3.5. Surface energy 1415.3.6. An anisotropic intrinsic superficial tension 1425.3.7. An equation of motion based on capillary acceleration 1445.3.8. Capillary rise between two planar surfaces 1465.4. Compressible flows 1525.4.1. Analysis of redundancies in Euler’s equations 1525.4.2. Proposition for an alternative to Euler’s equations 1555.4.3. Rankine–Hugoniot conditions 1585.4.4. Surface and shock discontinuities 1595.4.5. Some elementary test cases 1615.4.6. Propagation of a surface discontinuity 161Chapter 6 Fluid–Structure Interactions and Porous Media 1756.1. Equation of motion for a solid 1766.2. Connection conditions 1796.3. Some examples 1826.3.1. Fluid–solid monolithic interaction on a simple example 1826.3.2. Periodic shear in a fluid–solid layer 1846.4. Other constitutive laws 1876.4.1. Compression-related properties 1876.4.2. Non-deformable solid 1886.4.3. Viscoelastic model 1896.4.4. Threshold fluid 1906.5. Porous media 1906.5.1. Physical description of flows in porous media 1906.5.2. Discrete approach to flows in porous media 1926.5.3. Darcy’s law 1956.5.4. An examination of media anisotropy 1986.5.5. Darcy–Forchheimer’s law 1986.5.6. Flow in a variable cross-section channel 201Chapter 7 Heat Transfer 2037.1. Introduction 2037.2. Analysis of the heat transfer equations 2047.3. An alternative law of heat propagation 2087.3.1. Maxwell’s local frame of reference 2087.3.2. Modeling radiative transfer 2107.3.3. Modeling diffusion transfer 2137.4. A law of discrete transfer 2157.4.1. Equivalence of discrete distributions 2157.4.2. A unified law of motion 2177.4.3. Advection 2197.4.4. Anisotropy and polarization 2207.4.5. Phase change 2227.4.6. A relativistic equation 2247.4.7. A reduction in the number of variables 2257.5. Test cases 2267.5.1. Radiative transfer between two cylinders 2267.5.2. Heat transfer by diffusion in conductive media 2287.5.3. The Stefan problem for a melting-type phase change 2307.5.4. Simulation of melting in discrete formulation 2337.5.5. Condensation in an undercooled cavity 2357.5.6. Condensation with imposed temperature 2367.5.7. Condensation in an anisothermic system 237Chapter 8 Electromagnetism 2418.1. Introduction 2418.2. A few remarks about Maxwell’s equations 2428.2.1. Maxwell’s model 2428.2.2. Maxwell’s equations in terms of potentials 2448.2.3. Non-existence of monopoles 2458.3. An alternative law of propagation of electromagnetic waves 2458.3.1. Maxwell’s local frame of reference 2468.3.2. Modeling currents 2478.3.3. The discrete law of motion 2538.3.4. Inertia and Lorentz acceleration 2558.3.5. Conservation of charges 2568.3.6. Equations in terms of potential 2588.3.7. A relativistic equation 2598.3.8. The potential existence of monopoles 2608.3.9. A drastic reduction in the number of variables 2628.3.10. Differences and convergences 2638.4. Some examples 2658.4.1. Magnetic field created by a wire of infinite length 2658.4.2. Magnetic field around a permanent magnet 2668.4.3. Induced currents in a cylindrical conductor 2678.4.4. Electromagnet coil 2708.4.5. An example of electromagnetic levitation 2718.5. Propagation of light 2758.5.1. Light propagation equation 2758.5.2. Interference produced from two coherent point sources 2768.5.3. Refraction of a polarized monochromatic wave 278Chapter 9 Relativity, Gravitation 2819.1. An alternative to the theory of relativity 2819.1.1. Alternative relativistic equation 2829.2. Wave–energy duality 2839.3. Photon velocity 2869.3.1. Photon energy in theory of relativity 2869.3.2. Photon energy in discrete mechanics 2889.4. Gravitation 2919.4.1. Physical principles revisited 2919.4.2. Universal law of the fall of bodies with or without mass 2939.4.3. Creation of the energy of bodies 2959.4.4. The mechanism of star accretion 2989.5. Two typical examples 3019.5.1. Gravitational acceleration as a source term 3019.5.2. Gravitational lensing 3019.6. Gravitational redshift 3039.7. Quantification 3099.7.1. Notion of spin 3099.7.2. A potential unification 311References 315Index 327