A nonlocal equation of motion with damping is derived by means of a Mori-Zwanzig renormalization process. The treatment is analogous to that of Mori in deriving the Langevin equation. For the case of electrodynamics, a local approximation yields the Lorentz equation; a relativistic generalization gives the Lorentz-Dirac equation. No self-acceleration or self-mass difficulties occur in the classical treatment, although runaway solutions are not eliminated. The nonrelativistic quantum case does not exhibit runaways, however, provided one remains within a weak damping (...) approximation. The correspondence limit shows that a classical limit may be taken, again within the same approximation. (shrink)

Accepted quantum description is stochastic, yet history is nonstochastic, i.e., not representable by a probability distribution. Therefore ordinary quantum mechanics is unsuited to describe history. This is a limitation of the accepted quantum theory, rather than a failing of mechanics in general. To remove the limitation, it would be desirable to find a form of quantum mechanics that describes the future stochastically and the past nonstochastically. For this purpose it proves sufficient to introduce into quantum mechanics, by means of (...) a perfected formal correspondence, certain analogs of the classical initial-condition constants. Through the restoration of such parameters at the quantum level one accomplishes a natural accommodation of time anisotropy, wave-function reduction, and “event” description by quantum mechanical equations of motion alone, without the need for extra postulates (e.g., a projection postulate). This requires a complete restructuring of quantum measurement theory. (shrink)

In this work we review the derivation of Dirac and Weinberg equations based on a “principle of indistinguishability” for the (j,0) and (0,j) irreducible representations (irreps) of the homogeneous Lorentz group (HLG). We generalize this principle and explore its consequences for other irreps containing j≥1. We rederive Ahluwalia–Kirchbach equation using this principle and conclude that it yields $\mathcal{O}(p^{2j} )$ equations of motion for any representation containing spin j and lower spins. We also use the obtained generators of (...) the HLG for a given representation to explore the possibility of the existence of first order equations for that representation. We show that, except for j= $ - \frac{1}{2}$ , there exists no Dirac-like equation for the (j,0)⊕(0,j) representation nor for the ( $ - \frac{1}{2}$ , $ - \frac{1}{2}$ ) representation. We rederive Kemmer–Duffin–Petieau (KDP) equation for the (1,0)⊕( $ - \frac{1}{2}$ , $ - \frac{1}{2}$ )⊕(0,1) representation by this method and show that the (1, $ - \frac{1}{2}$ )⊕( $ - \frac{1}{2}$ ,1) representation satisfies a Dirac-like equation which describes a multiplet of $j = \frac{3}{2}{\text{ and }}j = \frac{1}{2}$ with masses m and m/2, respectively. (shrink)

The problem of motion in general relativity is about how exactly the gravitational field equations, the Einstein equations, are related to the equations of motion of material bodies subject to gravitational fields. This article compares two approaches to derive the geodesic motion of matter from the field equations: the ‘T approach’ and the ‘vacuum approach’. The latter approach has been dismissed by philosophers of physics because it apparently represents material bodies by singularities. (...) I argue that a careful interpretation of the approach shows that it does not depend on introducing singularities at all and that it holds at least as much promise as the T approach. (shrink)

We derive, from the Einstein-Maxwell field equations, the Lorentz equations of motion with radiation reaction for a charged mass particle moving in a background gravitational and electromagnetic field by utilizing a line element for the background space-time in a coordinate system specially adapted to the world line of the particle. The particle is introduced via perturbations of the background space-time (and electromagnetic field) which are singular only on the source world line.

Ibn al-Zarqālluh (al-Andalus, d. 1100) introduced a new inequality in the longitudinal motion of the Moon into Ptolemy’s lunar model with the amplitude of 24′, which periodically changes in terms of a sine function with the distance in longitude between the mean Moon and the solar apogee as the variable. It can be shown that the discovery had its roots in his examination of the discrepancies between the times of the lunar eclipses he obtained from the data of his (...) eclipse observations over a 37-year period in the latter part of the eleventh century and the predictions made on the basis of the lunar theories in the Mumta $$\textit{\d{h}}$$ an zīj (Baghdad, ca. 830) and al-Battānī’s zīj (Raqqa, d. 929), which were available to him at the time. What Ibn al-Zarqālluh found is, in fact, a special case of the annual equation of the Moon, which is applicable in the oppositions and, thus, in the lunar eclipses. The inequality was discovered independently by Tycho Brahe (d. 1601) and Johannes Kepler (d. 1630). As Ibn Yūnus (d. 1009) reports in his $$\textit{\d{H}}$$ ākimī zīj, Ibn al-Zarqālluh’s medieval Middle Eastern predecessors, the Persian astronomers Māhānī (d. ca. 880) and Nayrīzī (d. 922) as well as ‘Alī b. Amājūr (fl. ca. 920), were already acquainted with the problem of the eclipse timing errors, but it had remained unresolved until Ibn Yūnus provided a provisional, and incorrect, solution by reducing the size of the lunar epicycle. As we argue, the diverse ways to tackle the same problem stem from two different methodologies in astronomical reasoning in the traditions developed separately in the Eastern and Western regions of the medieval Islamic domain. (shrink)

We show that particle-antiparticle exchange and covariant motion reversal are two physically different aspects of the same mathematical transformation, either in the prequantal relativistic equation of motion of a charged point particle, in the general scheme of second quantization, or in the spinning wave equations of Dirac and of Petiau-Duffin-Kemmer. While, classically, charge reversal and rest mass reversal are equivalent operations, in the wave mechanical case mass reversal must be supplemented by exchange of the two adjoint (...) equations, implying ψ ⇄ $\bar \psi$ .Denoting by M the rest mass reversal, P the parity reversal, T the Racah time reversal, and Z the ψ ⇄ $\bar \psi$ exchange, the connection with the usual scheme of charge conjugation, parity reversal, and Wigner motion reversal, is with, of course. (shrink)

We present the unification of Riemann–Cartan–Weyl (RCW) space-time geometries and random generalized Brownian motions. These are metric compatible connections (albeit the metric can be trivially euclidean) which have a propagating trace-torsion 1-form, whose metric conjugate describes the average motion interaction term. Thus, the universality of torsion fields is proved through the universality of Brownian motions. We extend this approach to give a random symplectic theory on phase-space. We present as a case study of this approach, the invariant Navier–Stokes (...) class='Hi'>equations for viscous fluids, and the kinematic dynamo equation of magnetohydrodynamics. We give analytical random representations for these equations. We discuss briefly the relation between them and the Reynolds approach to turbulence. We discuss the role of the Cartan classical development method and the random extension of it as the method to generate these generalized Brownian motions, as well as the key to construct finite-dimensional almost everywhere smooth approximations of the random representations of these equations, the random symplectic theory, and the random Poincaré–Cartan invariants associated to it. We discuss the role of autoparallels of the RCW connections as providing polygonal smooth almost everywhere realizations of the random representations. (shrink)

In relativistic theories, the assumption of proper mass constancy generally holds. We study gravitational relativistic mechanics of point particle in the novel approach of proper mass varying under Minkowski force action. The motivation and objective of this work are twofold: first, to show how the gravitational force can be included in the Special Relativity Mechanics framework, and, second, to investigate possible consequences of the revision of conventional proper mass concept (in particular, to clarify a proper mass role in the divergence (...) problem). It is shown that photon motion in the gravitational field can be treated in terms of massless refracting medium, what makes the gravity phenomenon compatible with SR Mechanics framework in the variable proper mass approach.Specifically, the problem of point particle in the spherical symmetric stationary gravitational field is studied in SR-based Mechanics, and equations of motion in the Lorentz covariant form are obtained in the relativistic Lagrangean problem formulation. The dependence of proper mass on potential field strength is derived from the Euler-Lagrange equations as well. One of new results is the elimination of conventional 1/r divergence, which is known to be not removable in Schwarzschild gravitomechanics. Predictions of particle and photon gravitational properties are in agreement with GR classical tests under weak-field conditions; however, deviations rise with potential field strength. The conclusion is made that the approach of field-dependent proper mass is perspective for development of SR gravitational mechanics and further studies of gravitational problems. (shrink)

We examine the problem of deducing the geodesic motion of test particles from Einstein's vacuum field equations and its extension to include gravitational radiation reaction. In the latter case we obtain an equation of motion for a particle which incorporates radiation reaction of the electrodynamical type, but due to shearing radiation, together with a mass-loss formula of the Bondi-Sachs type.

An overlap between the general relativist and particle physicist views of Einstein gravity is uncovered. Noether's 1918 paper developed Hilbert's and Klein's reflections on the conservation laws. Energy-momentum is just a term proportional to the field equations and a "curl" term with identically zero divergence. Noether proved a \emph{converse} "Hilbertian assertion": such "improper" conservation laws imply a generally covariant action. Later and independently, particle physicists derived the nonlinear Einstein equations assuming the absence of negative-energy degrees of freedom (...) for stability, along with universal coupling: all energy-momentum including gravity's serves as a source for gravity. Those assumptions imply that the energy-momentum is only a term proportional to the field equations and a symmetric curl, which implies the coalescence of the flat background geometry and the gravitational potential into an effective curved geometry. The flat metric, though useful in Rosenfeld's stress-energy definition, disappears from the field equations. Thus the particle physics derivation uses a reinvented Noetherian converse Hilbertian assertion in Rosenfeld-tinged form. The Rosenfeld stress-energy is identically the canonical stress-energy plus a Belinfante curl and terms proportional to the field equations, so the flat metric is only a convenient mathematical trick without ontological commitment. Neither generalized relativity of motion, nor the identity of gravity and inertia, nor substantive general covariance is assumed. The more compelling criterion of lacking ghosts yields substantive general covariance as an output. Hence the particle physics derivation, though logically impressive, is neither as novel nor as ontologically laden as it has seemed. (shrink)

Lanczos, C. Einstein's path from special to general relativity.--Balazs, N. L. The acceptability of physical theories: Poincaré versus Einstein.--Ellis, G. F. R. Global and non-global problems in cosmology, by G. F. R. Ellis and D. W. Sciama.--Ehlers, J. The geometry of free fall and light propagation, by J. Ehlers, F. A. E. Pirani and A. Schild.--Trautman, A. Invariance of Lagrangian systems.--Penrose, R. The geometry of impulsive gravitational waves.--Exact solutions of the Einstein-Maxwell equations for an accelerated charge.--Taub, A. H. (...) Plane-symmetric similarity solutions for self-gravitating fluids.--Robinson, I. Equations of motion in the linear approximation, by I. Robinson and J. R. Robinson.--Florides, P. W. Rotating bodies in general relativity.--Chandrasekhar, S. A limiting case of relativistic equilibrium.--Israel, W. The relativistic Boltzmann equation.--Thompson, W. B. The self-consistent test-particle approach to relativistic kinetic theory. (shrink)

The appearance of the time derivative of the acceleration in the equation of motion (EOM) of an electric charge is studied. It is shown that when an electric charge is accelerated, a stress force exists in the curved electric field of the accelerated charge, and in the case of a constant linear acceleration, this force is proportional to the acceleration. This stress force acts as a reaction force which is responsible for the creation of the radiation (instead of the (...) “radiation reaction force” that actually does not exist at low velocities). Thus the initial acceleration should be supplied as an initial condition for the solution of the EOM of an electric charge. (shrink)

In this paper the analysis of the classical dynamics of a charged particle is carried out without considering that the electromagnetic field necessarily goes to zero at infinity. A quite general non-linear equation of motion is obtained for an extended charged particle valid for any distribution of charge in the particle and for an electromagnetic field satisfying any boundary conditions. Some common approximations are analyzed with detail to determine how the usual difficulties arise.

A Finslerian extension of general relativity is examined with particular emphasis on the Finslerian generalization of the equation of motion in a gravitational field. The construction of a gravitational Lagrangian density by substituting the osculating Riemannian metric tensor in the Einstein density is studied. Attention is drawn to an interesting possibility for developing the theory of test bodies against the Finslerian background.

In this paper we study a classical and theoretical system which consists of an elastic medium carrying transverse waves and one point-like high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this system. Afterwards we always consider the case where the concretion is not the wave source any longer. Then the concretion obeys a general and covariant guidance formula, which leads in low-velocity approximation to an (...) equivalent de Broglie-Bohm guidance formula. The concretion moves then as if exists an equivalent quantum potential. A strictly equivalent free Schrödinger equation is retrieved, as well as the quantum stationary states in a linear or spherical cavity. We compute the energy of the concretion, naturally defined from the energy density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation is fulfilled, it strikingly appears that the energy and momentum of the concretion not only are written in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas. (shrink)

In order to understand the rise of runaway solutions in the radiation reaction problem a mechanical model is used. An alternative demonstration of Daboul’s theorem, through Hurwitz’s criterion, is given. The origin of runaway solutions in electrodynamics is discussed. They arise when the particle has a negative mechanical mass or when approximations are used in the equation of motion. In the 1-dimensional mechanical model an exact and linear equation of motion for the particle is obtained, the corresponding exact (...) solution is again runaway when the mechanical mass is negative. The exact solution is not runaway when the mechanical mass is positive. However, the use of approximations leads to an equation of motion which has runaway solutions. It is exhibited that the use of approximations in the 3-dimensional mechanical model is completely necessary because the general equation of motion for the particle is non-linear. The analysis of this case proceeds in a very similar way to the one carried out in electrodynamics. This means that the number of dimensions also plays an important role in the analysis. (shrink)

What is the meaning of general covariance? We learn something about it from the hole argument, due originally to Einstein. In his search for a theory of gravity, he noted that if the equations of motion are covariant under arbitrary coordinate transformations, then particle coordinates at a given time can be varied arbitrarily - they are underdetermined - even if their values at all earlier times are held fixed. It is the same for the values of fields. (...) The argument can also be made out in terms of transformations acting on the points of the manifold, rather than on the coordinates assigned to the points. So the equations of motion do not fix the particle positions, or the values of fields at manifold points, or particle coordinates, or fields as functions of the coordinates, even when they are specified at all earlier times. It is surely the business of physics to predict these sorts of quantities, given their values at earlier times. The principle of general covariance therefore seems untenable. (shrink)

Part IIb presents some of the most important theorems for stable equilibrium states that can be deduced from the four postulates of the unified theory presented in Part I. It is shown for the first time that the canonical and grand canonical distributions are the only distributions that are stable. Moreover, it is shown that reversible adiabatic processes exist which cannot be described by the dynamical equation of quantum mechanics. A number of conditions are discussed that must be satisfied by (...) the general equation of motion which is yet to be discovered. (shrink)

The nature of this book is fourfold: First, it provides comprehensive education in ontology, epistemology, logic, and ethics. From this perspective, it can be treated as a philosophical textbook. Second, it provides comprehensive education in mathematical analysis and analytic geometry, including significant aspects of set theory, topology, mathematical logic, number systems, abstract algebra, linear algebra, and the theory of differential equations. From this perspective, it can be treated as a mathematical textbook. Third, it makes a student and a researcher (...) in philosophy and/or mathematics capable of developing a holistic approach to reality, of undertaking interdisciplinary endeavors, of understanding (and possibly contributing to) advances and research projects in different academic disciplines, and of having more sources of inspiration and pleasure. From this perspective, it can be treated as a contribution to pedagogy and as an attempt to refresh and, indeed, revitalize modern philosophy. Fourth, it seeks to defend, refresh, and enrich philosophical and scientific structuralism and dynamical philosophy (known also as dynamism). From this perspective, this book can be treated as a research monograph on structuralism and dynamism, tackling the fundamental problems of reality, truth, and consciousness. In this context, Nicolas Laos expounds and proposes: (i) the concepts of dynamized time and dynamized space; (ii) a theory and method that he calls the "dialectic of rational dynamicity"; and (iii) his attempt to consider the fundamental problems of philosophy and science from the perspective of the dialectic of rational dynamicity. Thus, this book pertains to every field that is controlled by the function of consciousness, namely, being, knowing, and acting. The philosophy of rational dynamicity, as the author explains in this book, is a way of contemplating the laws of motion of nature, history, and spirit. (shrink)

In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence, one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time philosophy tends (...) to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a \ version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate \-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory, it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors. (shrink)

The analysis of a previous paper (see Ref. 1), in which the possibility of a Finslerian generalization of the equations of motion of gravitational field sources was demonstrated, is extended by developing the Finslerian generalization of the gravitational field equations on the basis of the complete contractionK = K lj lj of the Finslerian curvature tensorK l j hk (x, y). The relevant Lagrangian is constructed by the replacement of the directional variabley i inK by a vector (...) fieldy i (x), so that the notion of osculation may be regarded as the key concept on which the approach is based. The introduction of the auxiliary vector fieldy i (x) is shown to be of physical significance, for the field equations refer not only to the proper field variables but also to a special coordinate system associated withy i (x) through the Clebsch representation of the latter. The status of the conservation laws proves to be similar to that in the theory of the Yang-Mills field. By choosing a special Finslerian metric function we elucidate in detail the structure of the field equations in the static case. (shrink)

What is the meaning of general covariance? We learn something about it from the hole argument, due originally to Einstein. In his search for a theory of gravity, he noted that if the equations of motion are covariant under arbitrary coordinate transformations, then particle coordinates at a given time can be varied arbitrarily - they are underdetermined - even if their values at all earlier times are held fixed. It is the same for the values of fields. (...) The argument can also be made out in terms of transformations acting on the points of the manifold, rather than on the coordinates assigned to the points. So the equations of motion do not fix the particle positions, or the values of fields at manifold points, or particle coordinates, or fields as functions of the coordinates, even when they are specified at all earlier times. It is surely the business of physics to predict these sorts of quantities, given their values at earlier times. The principle of general covariance therefore seems untenable. (shrink)

We consider the relativistic statistical mechanics of an ensemble of N events with motion in space-time parametrized by an invariant “historical time” τ. We generalize the approach of Yang and Yao, based on the Wigner distribution functions and the Bogoliubov hypotheses to find approximate dynamical equations for the kinetic state of any nonequilibrium system, to the relativistic case, and obtain a manifestly covariant Boltzmann- type equation which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU) equation for indistinguishable particles. (...) This equation is then used to prove the H-theorem for evolution in τ. In the equilibrium limit, the covariant forms of the standard statistical mechanical distributions are obtained. We introduce two-body interactions by means of the direct action potential V(q), where q is an invariant distance in the Minkowski space-time. The two- body correlations are taken to have the support in a relative O(2, 1)-invariant subregion of the full spacelike region. The expressions for the energy density and pressure are obtained and shown to have the same forms (in terms of an invariant distance parameter) as those of the nonrelativistic theory and to provide the correct nonrelativistic limit. (shrink)

The relationship between the continuity equation and the HamiltonianH of a quantum system is investigated from a nonstandard point of view. In contrast to the usual approaches, the expression of the current densityJ ψ is givenab initio by means of a transport-velocity operatorV T, whose existence follows from a “weak” formulation of the correspondence principle. Once given a Hilbert-space metricM, it is shown that the equation of motion and the continuity equation actually represent a system in theunknown operatorsH andV (...) T, due to the arbitrariness on the initial condition of the quantum state. The general solution is given in some cases of special interest and a straightforward application to relativistic quantum mechanics is performed. (shrink)

An internationally famous physicist and electrical engineer, the author of this text was a pioneer in the investigation of gravitational waves. Joseph Weber's General Relativity and Gravitational Waves offers a classic treatment of the subject. Appropriate for upper-level undergraduates and graduate students, this text remains ever relevant. Brief but thorough in its introduction to the foundations of general relativity, it also examines the elements of Riemannian geometry and tensor calculus applicable to this field. Approximately a quarter of the (...) contents explores theoretical and experimental aspects of gravitational radiation. The final chapter focuses on selected topics related to general relativity, including the equations of motion, unified field theories, Friedman's solution of the cosmological problem, and the Hamiltonian formulation of general relativity. Exercises. Index. (shrink)

If the ordinary quantal Liouville equation ℒρ= $\dot \rho $ is generalized by discarding the customary stricture that ℒ be of the standard Hamiltonian commutator form, the new quantum dynamics that emerges has sufficient theoretical fertility to permit description even of a thermodynamically irreversible process in an isolated system, i.e., a motion ρ(t) in which entropy increases but energy is conserved. For a two-level quantum system, the complete family of time-independent linear superoperators ℒ that generate such motions is derived; (...) and a physically interesting example is presented in detail. (shrink)

In his general theory of relativity (GR) Einstein sought to generalize the special-relativistic equivalence of inertial frames to a principle according to which all frames of reference are equivalent. He claimed to have achieved this aim through the general covariance of the equations of GR. There is broad consensus among philosophers of relativity that Einstein was mistaken in this. That equations can be made to look the same in different frames certainly does not imply in (...) class='Hi'>general that such frames are physically equivalent. We shall argue, however, that Einstein's position is tenable. The equivalence of arbitrary frames in GR should not be equated with relativity of arbitrary motion, though. There certainly are observable differences between reference frames in GR (differences in the way particles move and fields evolve). The core of our defense of Einstein's position will be to argue that such differences should be seen as fact-like rather than law-like in GR. By contrast, in classical mechanics and in special relativity (SR) the differences between inertial systems and accelerated systems have a law-like status. The fact-like character of the differences between frames in GR justifies regarding them as equivalent in the same sense as inertial frames in SR. (shrink)

The equations of motion are obtained for closed systems of charged particles interacting with either an electric or a magnetic field. In each case they include constraints, expressed by the laws of induction, which are of importance in giving a complete specification of the systems usually treated in thermal physics. There are equivalent alternative formulations of the equations of motion, which permit designation of different subsystems; a discussion of these subsystems, their interrelationship, and their external parameters (...) clarifies the significance and applicability of the various expressions for electrical and magnetic work that appear in the general literature. Quantum mechanical treatment of particular subsystems provides a basis for application of statistical mechanics, leading to the identification of appropriate thermodynamic potentials. (shrink)

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by (...) an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation. (shrink)

According to history texts, philosophers searched for a unifying natural law whereby natural phenomena and numbers are related. More than 2300 years ago, Aristotle postulated that nature requires minimum energy. More than 220 years ago, Euler applied the minimum energy postulate. More than 200 years ago, Lagrange provided a mathematical “proof” of the postulate for conservative systems. The resulting Principle of Least Action served only to derive the differential equations of motion of a conservative system. Then, 170 years (...) ago, Hamilton presented what he claimed to be a “general method in dynamics.” Hamilton's resulting “Law of Varying Action” was supposed to apply to both conservative and non-conservative systems and was supposed to yield either the differential equations of motion or the integrals of those differential equations. However, no direct evaluation of the integrals of motion ever resulted from Hamilton's law of varying action. In 1975, a scant 29 years ago, following five years of controversy with engineer mechanicians, Dr. Wolfgang Yourgrau, Editor, Foundations of Physics, published my first paper based on Aristotle's postulate, without mathematical proof. That and subsequent papers present, through applications, a true “general method in dynamics.” In this essay, I present the mathematical proof that is missing from my 1975 and subsequent papers. Six fundamental integrals of analytical mechanics are derived from Aristotle's postulate. First, however, Hamilton must be revisited to show why his H function and his “force function” prevents the law of varying action from being the general method in dynamics that he claimed it to be. I have found that Hamilton’s Law of Varying Action (HLVA), as Hamilton presented it, cannot be applied to systems for which the force function is non-integrable. In 1972, Dr. B.E. Gatewood and Dr. D.P. Beres (then a graduate student) discovered that the end-point term associated with the principle of least action does not vanish. I named the new equation, “the general energy equation.” In 1973, because I was doing with it what Hamilton claimed could be done with HLVA, I simply assumed that this new equation was HLVA. I gave the new equation the misnomer HLVA. In 2001, I learned that I had made a grave mistake. I found that HLVA is at most a special case of the general energy equation. My interpretation of Aristotle's postulate permits one to by-pass the differential equations of motion completely for both conservative and non-conservative systems (no calculus of variations). (shrink)

Some unusual relations between stress tensors, conservation and equations of motion are briefly reviewed.

A Schrödinger-like relativistic wave equation of motion for the Lorentz-scalar potential is formulated based on a Lagrangian formalism of relativistic mechanics with a scaled time as the evolution parameter. Applications of this Schrödinger-like formalism for the Lorentz-scalar potential are given: For the square-step potential, the predictions of this formalism are free from the Klein paradox, and for the Coulomb potential, this formalism yields the exact bound-state eigenenergies and eigenfunctions.

The use of the axial vector representing a three-dimensional rotation makes the rotation representation much more compact by extending the trigonometric functions to vectorial arguments. Similarly, the pure Lorentz transformations are compactly treated by generalizing a scalar rapidity to a vector quantity in spatial three-dimensional cases and extending hyperbolic functions to vectorial arguments. A calculation of the Wigner rotation simplified by using the extended functions illustrates the fact that the rapidity vector space obeys hyperbolic geometry. New representations bring a Lorentz-invariant (...) fundamental equation of motion corresponding to the Galilei-invariant equation of Newtonian mechanics. (shrink)

The parametrized post-Newtonian (PPN) approximation is generalized to accommodate Rastall's modification of Einstein's theory of gravity, which allows nonzero divergence of the energy-momentum tensor. Rastall's theory is then shown to have consistent field equations, gauge conditions, and the correct Newtonian limit of the equations of motion. The PPN parameters are obtained and shown to agree experimentally with those for the Einstein theory. In light of the nonzero divergence condition, integral conservation laws are investigated and shown to yield (...) conserved energy-momentum and angularmomentum. We conclude that the above generalization of metric theories, within the PPN framework, is a natural extension of the concept of metric theories. (shrink)

Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata and walks starting from very general principles. The intrinsic discreteness of this framework is reconciled with the continuous Lorentz symmetry by reformulating the notion of inertial reference frame in terms of the constants of motion of the quantum walk dynamics. (...) In particular, among the symmetries of the quantum walk which recovers the Weyl equation—the so called Weyl walk—one finds a non linear realisation of the Poincaré group, which recovers the usual linear representation in the small wave-vector limit. In this paper we characterise the full symmetry group of the Weyl walk which is shown to be a non linear realization of a group which is the semidirect product of the Poincaré group and the group of dilations. (shrink)

In this work we study the relativistic mechanics of continuous media on a fundamental level using a manifestly covariant proper time procedure. We formulate equations of motion and continuity (and constitutive equations) that are the starting point for any calculations regarding continuous media. In the force free limit, the standard relativistic equations are regained, so that these equations can be regarded as a generalization of the standard procedure. In the case of an inviscid fluid we (...) derive an analogue of the Bernoulli equation. For irrotational flow we prove that the velocity field can be derived from a potential. If in addition, the fluid is incompressible, the potential must obey the d'Alembert equation, and thus the problem is reduced to solving the d'Alembert equation with specific boundary conditions (in both space and time). The solutions indicate the existence of light velocity sound waves in an incompressible fluid (a result known in previous literature (19) ). Relaxing the constraints and allowing the fluid to become linearly compressible one can derive a wave equation, from which the sound velocity can again be computed. For a stationary background flow, it has been demonstrated that the sound velocity attains its correct values for the incompressible and nonrelativistic limits. Finally viscosity is introduced, bulk and shear viscosity constants are defined, and we formulate equations for the motion of a viscous fluid. (shrink)

Classical mechanics is empirically successful because the probabilistic mean values of quantum mechanical observables follow the classical equations of motion to a good approximation (Messiah 1970, 215). We examine this claim for the one-dimensional motion of a particle in a box, and extend the idea by deriving a special case of the ideal gas law in terms of the mean value of a generalized force used to define "pressure." The examples illustrate the importance of probabilistic averaging as (...) a method of abstracting away from the messy details of microphenomena, not only in physics, but in other sciences as well. (shrink)

This paper examines the problem of founding irreversibility on reversible equations of motion from the point of view of the Brussels school's recent developments in the foundations of quantum statistical mechanics. A detailed critique of both their 'subdynamics' and 'transformation' theory is given. It is argued that the subdynamics approach involves a generalized form of 'coarse-graining' description, whereas, transformation theory cannot lead to truly irreversible processes pointing to a preferred direction of time. It is concluded that the Brussels (...) school's conception of microscopic temporal irreversibility, as such, is tacitly assumed at the macroscopic level. Finally a logical argument is provided which shows, independently of the mathematical formalism of the theory concerned, that statistical reasoning alone is not sufficient to explain the arrow of time. (shrink)

A unified axiomatic theory that embraces both mechanics and thermodynamics is presented in three parts. It is based on four postulates; three are taken from quantum mechanics, and the fourth is the new disclosure of the existence of quantum states that are stable (Part I). For nonequilibrium and equilibrium states, the theory provides general original results, such as the relation between irreducible density operators and the maximum work that can be extracted adiabatically (Part IIa). For stable equilibrium states, it (...) shows for the first time that the canonical and grand canonical distributions are the only stable distributions (Part IIb). The theory discloses the incompleteness of the equation of motion of quantum mechanics not only for irreversible processes but, more significantly, for reversible processes (Part IIb). It establishes the operational meaning of an irreducible density operator and irreducible dispersions associated with any state, and reveals the relationship between such dispersions and the second law (Part III). (shrink)

In the present paper, we introduce new models of pendulum motions for two cases: the first model consists of a pendulum with mass M moving at the end of a string with a suspended point moving on an ellipse and the second one consists of a pendulum with mass M moving at the end of a spring with a suspended point on an ellipse. In both models, we use the Lagrangian functions for deriving the equations of motions. The derived (...) equations are reduced to a quasilinear system of the second order. We use a new mathematical technique named a large parameter method for solving both models’ systems. The analytical solutions are obtained in terms of the generalized coordinates. We use the numerical techniques represented by the fourth-order Runge–Kutta method to solve the autonomous system for both cases. The stabilities of the obtained solutions are studied using the phase diagram procedure. The obtained numerical solutions and analytical ones are compared to examine the accuracy of the mathematical and numerical techniques. The large parameter technique gives us the advantage to obtain the solutions at infinity in opposite with the famous Poincare’s method which was used by many outstanding scientists in the last two centuries. (shrink)

Relativistic quantum mechanics has been formulated as a theory of the evolution ofevents in spacetime; the wave functions are square-integrable functions on the four-dimensional spacetime, parametrized by a universal invariant world time τ. The representation of states with spin is induced with a little group that is the subgroup of O(3, 1) leaving invariant a timelike vector nμ; a positive definite invariant scalar product, for which matrix elements of tensor operators are covariant, emerges from this construction. In a previous study (...) a second-order equation was introduced similar to the second-order Dirac equation, based on a quadratic function of two operators which are the self-adjoint parts, in this new scalar product, of γμpμ. It is shown in this paper that one of these operators, in fact the one from which the gyromagnetic moment is obtained, can be used to construct a first-order equation. The corresponding quantum theory is somewhat analogous to Dirac's spinor form; the Hamilton equations appear to describe dynamical degrees of freedom in a spacelike hyperplane orthogonal to nμ (in Dirac's theory the motion appears to be lightlike). It is shown that the integration over nμ required by unitarity results in timelike motion (as in the expectation value of Dirac's α). Explicit forms are obtained for the wave functions and currents for free motion. The general form of the theory is written for the (five-dimensional) pre-Maxwell fields required by gauge invariance. (shrink)

In this paper I examine the foundations of Laplace's famous statement of determinism in 1814, and argue that rather than derived from his mechanics, this statement is based on general philosophical principles, namely the principle of sufficient reason and the law of continuity. It is usually supposed that Laplace's statement is based on the fact that each system in classical mechanics has an equation of motion which has a unique solution. But Laplace never proved this result, and in (...) fact he could not have proven it, since it depends on a theorem about uniqueness of solutions to differential equations that was only developed later on. I show that the idea that is at the basis of Laplace's determinism was in fact widespread in enlightenment France, and is ultimately based on a re-interpretation of Leibnizian metaphysics, specifically the principle of sufficient reason and the law of continuity. Since the law of continuity also lies at the basis of the application of differential calculus in physics, one can say that Laplace's determinism and the idea that systems in physics can be described by differential equations with unique solutions have a common foundation. (shrink)

It is argued that time's arrow is present in all equations of motion. But it is absent in the point particle approximations commonly made. In particular, the Lorentz-Abraham-Dirac equation is time-reversal invariant only because it approximates the charged particle by a point. But since classical electrodynamics is valid only for finite size particles, the equations of motion for particles of finite size must be considered. Those equations are indeed found to lack time-reversal invariance, thus ensuring (...) an arrow of time. Similarly, more careful considerations of the equations of motion for gravitational interactions also show an arrow of time. The existence of arrows of time in quantum dynamics is also emphasized. (shrink)

A Lagrangian description is presented which can be used in conjunction with particle interpretations of quantum mechanics. A special example of such an interpretation is the well-known Bohm model. The Lagrangian density introduced here also contains a potential for guiding the particle. The advantages of this description are that the field equations and the particle equations of motion can both be deduced from a single Lagrangian density expression and that conservation of energy and momentum are assured. After (...) being developed in a general form, this Lagrangian formulation is then applied to the special case of the Bohm model as an example. It is thereby demonstrated that such a Lagrangian description is compatible with the predictions of quantum mechanics. (shrink)

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes (...) the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q. (shrink)

Dissipative systems are described by a Hamiltonian, combined with a “dynamical matrix” which generalizes the simplectic form of the equations of motion. Criteria for dissipation are given and the examples of a particle with friction and of the Lotka-Volterra model are presented. Quantization is first introduced by translating generalized Poisson brackets into commutators and anticommutators. Then a generalized Schrödinger equation expressed by a dynamical matrix is constructed and discussed.