We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell’s equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations St $${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{$2$}}$$ S x 2 =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are (...) observed in the terms of these expansions and this suggests that S(x, t) should be written as a product of the elementary solution x2/(2t) and a function f=f(φ) where φ=φ(x, |t|) owing to the symmetry property which is that S(x, −t)=−S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of φ(x, |t|)=φ0 |t/t0|κ (x/x 0)λ≥0 with κ, λ∈ ℝ results in a soluble ordinary differential equation, of first order in u=ln φ and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called “trivial.” The phase plane (fu, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the “trivial” fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid. (shrink)

In this paper we show how the dynamics of the Schrödinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{C}}_{0,1}$\end{document}. Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be (...) completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time. (shrink)

It is shown that the admissible solutions of the continuity and Bernoulli or Burgers' equations of a perfect one-dimensional liquid are conditioned by a relation established in 1949–1950 by Pauli, Morette, and Van Hove, apparently, overlooked so far, which, in our case, stipulates that the mass density is proportional to the second derivative of the velocity potential. Positivity of the density implies convexity of the potential, i.e., smooth solutions, no shock. Non-elementary and symmetric solutions of the above equations are given (...) in analytical and numerical form. Analytically, these solutions are derived from the original Ansatz proposed in Ref. 1 and from the ensuing operations which show that they represent a particular case of the general implicit solutions of Burgers' equation. Numerically and with the help of an ad hoc computer program, these solutions are simulated for a variety of initial conditions called “compatible” if they satisfy the Morette–Van Hove formula and “anti-compatible” if the sign of the initial velocity field is reversed. In the latter case, singular behaviour is observed. Part of the theoretical development presented here is rephrased in the context of the Hopf–Lax formula whose domain of applicability for the solution of the Cauchy problem of the homogeneous Hamilton–Jacobi equation has recently been enlarged. (shrink)

Bellman optimality principle for the stochastic dynamic system on time scales is derived, which includes the continuous time and discrete time as special cases. At the same time, the Hamilton–Jacobi–Bellman equation on time scales is obtained. Finally, an example is employed to illustrate our main results.

Cancer chemotherapy has been the most common cancer treatment. However, it has side effects that kill both tumor cells and immune cells, which can ravage the patient’s immune system. Chemotherapy should be administered depending on the patient’s immunity as well as the level of cancer cells. Thus, we need to design an efficient treatment protocol. In this work, we study a feedback control problem of tumor-immune system to design an optimal chemotherapy strategy. For this, we first propose a mathematical model (...) of tumor-immune interactions and conduct stability analysis of two equilibria. Next, the feedback control is found by solving the Hamilton–Jacobi–Bellman equation. Here, we use an upwind finite-difference method for a numerical approximate solution of the HJB equation. Numerical simulations show that the feedback control can help determine the treatment protocol of chemotherapy for tumor and immune cells depending on the side effects. (shrink)

We observe that Schrödinger’s equation may be written as two real coupled Hamilton–Jacobi (HJ)-like equations, each involving a quantum potential. Developing our established programme of representing the quantum state through exact free-standing deterministic trajectory models, it is shown how quantum evolution may be treated as the autonomous propagation of two coupled congruences. The wavefunction at a point is derived from two action functions, each generated by a single trajectory. The model shows that conservation as expressed through a (...) continuity equation is not a necessary component of a trajectory theory of state. Probability is determined by the difference in the action functions, not by the congruence densities. The theory also illustrates how time-reversal symmetry may be implemented through the collective behaviour of elements that individually disobey the conventional transformation (T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{T}$$\end{document}) of displacement (scalar) and velocity (reversal). We prove that an integral curve of the linear superposition of two vectors can be derived algebraically from the integral curves of one of the constituent vectors labelled by integral curves associated with the other constituent. A corollary establishes relations between displacement functions in diverse trajectory models, including where the functions obey different symmetry transformations. This is illustrated by showing that a (T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{T}$$\end{document}-obeying) de Broglie-Bohm trajectory is a sequence of points on the (non-T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{T}$$\end{document}) HJ trajectories, and vice versa. (shrink)

"The choice we have is not between reasonable proposals and an unreasonable utopianism. Utopian thinking does not undermine or discount real reforms. Indeed, it is almost the opposite: practical reforms depend on utopian dreaming."--Russell Jacoby, _Picture Imperfect_ Utopianism suffers from an image problem: A recent exhibition on utopias in Paris and New York included photographs of Hitler's _Mein Kampf_ and a Nazi concentration camp. Many observers judge utopians and their sympathizers as foolhardy dreamers at best and murderous totalitarians at worst. (...) However, as noted social critic and historian Russell Jacoby argues in this salient, polemical, and innovative work, not only has utopianism been unfairly characterized, a return to an iconoclastic utopian spirit is vital for today's society. Shaped by the works of Theodor Adorno, Walter Benjamin, Ernst Bloch, Gustav Landauer, and other predominantly Jewish thinkers, iconoclastic utopianism revives society's dormant political imagination and offers hope for a better future. Writing against the grain of history, Jacoby reexamines the anti-utopian mindset and identifies how utopian thought came to be regarded with such suspicion. He challenges standard readings of such anti-utopian classics as _1984_ and _Brave New World_ and offers stinging critiques of the influential liberal and anti-utopian theorists Hannah Arendt, Isaiah Berlin, and Karl Popper. He argues that these thinkers mistakenly equate utopianism with totalitarianism. The reputation of utopian thought has also suffered from the failures of, what Jacoby terms, the blueprint utopian tradition and its oppressive emphasis on detailing all aspects of society and providing fantastic images of the future. In contrast, the iconoclastic utopians, like those who follow God's prohibition against graven images, resist both the blueprinters' obsession with detail and the modern seduction of images. Jacoby suggests that by learning from the hopeful spirit of iconoclastic utopians and their willingness to accept new possibilities for society, we open ourselves to new and more imaginative ideas of the future. (shrink)

"The choice we have is not between reasonable proposals and an unreasonable utopianism. Utopian thinking does not undermine or discount real reforms. Indeed, it is almost the opposite: practical reforms depend on utopian dreaming."--Russell Jacoby, _Picture Imperfect_ Utopianism suffers from an image problem: A recent exhibition on utopias in Paris and New York included photographs of Hitler's _Mein Kampf_ and a Nazi concentration camp. Many observers judge utopians and their sympathizers as foolhardy dreamers at best and murderous totalitarians at worst. (...) However, as noted social critic and historian Russell Jacoby argues in this salient, polemical, and innovative work, not only has utopianism been unfairly characterized, a return to an iconoclastic utopian spirit is vital for today's society. Shaped by the works of Theodor Adorno, Walter Benjamin, Ernst Bloch, Gustav Landauer, and other predominantly Jewish thinkers, iconoclastic utopianism revives society's dormant political imagination and offers hope for a better future. Writing against the grain of history, Jacoby reexamines the anti-utopian mindset and identifies how utopian thought came to be regarded with such suspicion. He challenges standard readings of such anti-utopian classics as _1984_ and _Brave New World_ and offers stinging critiques of the influential liberal and anti-utopian theorists Hannah Arendt, Isaiah Berlin, and Karl Popper. He argues that these thinkers mistakenly equate utopianism with totalitarianism. The reputation of utopian thought has also suffered from the failures of, what Jacoby terms, the blueprint utopian tradition and its oppressive emphasis on detailing all aspects of society and providing fantastic images of the future. In contrast, the iconoclastic utopians, like those who follow God's prohibition against graven images, resist both the blueprinters' obsession with detail and the modern seduction of images. Jacoby suggests that by learning from the hopeful spirit of iconoclastic utopians and their willingness to accept new possibilities for society, we open ourselves to new and more imaginative ideas of the future. (shrink)

Aesthetics and Music is a rich and interesting study. Hamilton's approach is innovative. He interleaves chapters on the history of philosophical thought about music with more theoretical discussions of music, sound, rhythm and improvisation, but does not cover the work–performance relation, depiction or expression. He draws on an atypically broad range of examples, including avant-garde, medieval, non-Western and jazz. The assumptions are humanist: ‘I wish to argue for an aesthetic conception of music as an art … according to which (...) music is a human activity grounded in the body and bodily movement and interfused with human life’.The historical chapters are valuable and not without analysis and criticism. Hamilton shows how the ancient Greek theorists were more interested in music's mathematical properties as reflecting the underlying harmony of relations between cosmic bodies than in the practice of musicians. While they equated the value of art with its contribution to an education for citizenship and while their concept of the arts differed …. (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)

The potential term in the Schrödinger equation can be eliminated by means of a conformal transformation, reducing it to an equation for a free particle in a conformally related fictitious configuration space. A conformal transformation can also be applied to the Klein–Gordon equation, which is reduced to an equation for a free massless field in an appropriate (conformally related) spacetime. These procedures arise from the observation that the Jacobi form of the least action principle and (...) the Hamilton–Jacobi equation of classical non-relativistic mechanics can be interpreted in terms of conformal transformations. (shrink)

BackgroundCORE (Clarity and Openness in Reporting: E3-based) Reference (released May 2016 by the European Medical Writers Association [EMWA] and the American Medical Writers Association [AMWA]) is a complete and authoritative open-access user’s guide to support the authoring of clinical study reports (CSRs) for current industry-standard-design interventional studies. CORE Reference is a content guidance resource and is not a CSR Template.TransCelerate Biopharma Inc., an alliance of biopharmaceutical companies, released a CSR Template in November 2018 and recognised CORE Reference as one of (...) two principal sources used in its development.MethodsThe regulatory medical writing and statistical professionals who developed CORE Reference conducted a critical review of the TransCelerate CSR Template. We summarise our major findings and recommendations in this communication. We also re-examined and edited the Version 1 CORE Reference Terminology Table that we first published in 2016, and we present this as Version 2 in this communication.ResultsOur major critical review findings indicate that opportunities remain to refine the CSR Template structure and instructional text, enhance content clarity, add web links to referenced guidance documents, improve transparency to support the broad readership of CSRs, and develop supporting resources.The CORE Reference ‘Terminology Table’ Version 2 includes estimand as a defined term and an adaptation of the original ‘worked study example’ to incorporate the recently evolved concept of ‘estimands’.ConclusionsAs TransCelerate’s CSR Template represents an important milestone in authoring CSRs, we offer CSR authors advice and recommendations on its use, similarities, and differences with CORE Reference and advise them to consider shared interpretations between the two.RegistrationCORE Reference is registered with the EQUATOR Network. The TransCelerate CSR Template is not registered with any external organisation to the knowledge of the authors of this paper. (shrink)

We show that in classical mechanics the momentum may depend only on the coordinates and can thus be considered as a field. We formulate a special Lagrangian formalism as a result of which the momenta satisfy differential equations which depend only on the coordinates. The solutions correspond to all possible trajectories. As a bonus the Hamilton-Jacobi equation results in a very simple way.

Hamilton-Jacobi theory is applied to find appropriate canonical transformations for the calculation of the phase-space path integrals of the relativistic particle equations. Hence, canonical transformations and Hamilton-Jacobi theory are also introduced into relativistic quantum mechanics. Moreover, from the classical physics viewpoint, it is very interesting to find and to solve the Hamilton-Jacobi equations for the relativistic particle equations.

This paper gives a technically elementary treatment of some aspects of Hamilton -Jacobi theory, especially in relation to the calculus of variations. The second half of the paper describes the application to geometric optics, the optico-mechanical analogy and the transition to quantum mechanics. Finally, I report recent work of Holland providing a Hamiltonian formulation of the pilot-wave theory.

Some postulates are introduced to go from the classical Hamilton-Jacobi theory to the quantum one. We develop two approaches in order to calculate propagators, establishing the connection between them and showing the equivalence of this picture with more known ones such as the Schrödinger's and the Feynman's formalisms. Applications of the above-mentioned approaches to both the standard case of the harmonic oscillator and to the harmonic oscillator with time-dependent parameters are made.

Concentrating upon applications that are most relevant to modern physics, this valuable book surveys variational principles and examines their relationship to dynamics and quantum theory. Stressing the history and theory of these mathematical concepts rather than the mechanics, the authors provide many insights into the development of quantum mechanics and present much hard-to-find material in a remarkably lucid, compact form. After summarizing the historical background from Pythagoras to Francis Bacon, Professors Yourgrau and Mandelstram cover Fermat's principle of least time, the (...) principle of least action of Maupertuis, development of this principle by Euler and Lagrange, and the equations of Lagrange and Hamilton. Equipped by this thorough preparation to treat variational principles in general, they proceed to derive Hamilton's principle, the Hamilton-Jacobi equation, and Hamilton's canonical equations. An investigation of electrodynamics in Hamiltonian form covers next, followed by a resume of variational principles in classical dynamics. The authors then launch into an analysis of their most significant topics: the relation between variational principles and wave mechanics, and the principles of Feynman and Schwinger in quantum mechanics. Two concluding chapters extend the discussion to hydrodynamics and natural philosophy. Professional physicists, mathematicians, and advanced students with a strong mathematical background will find this stimulating volume invaluable reading. Extremely popular in its hardcover edition, this volume will find even wider appreciation in its first fine inexpensive paperbound edition. (shrink)

The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and (...) a Hamilton-Jacobi equation is obtained. (shrink)

Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; the Hamilton–Jacobi equation loaded by the (...) quantum potential. This equation in pair with the continuity equation leads to getting the Schrödinger equation describing a state of the superfluid quantum medium ; gravitomagnetic wave equations loaded by forces acting on the outer space. These waves obey to the Planck’s law of radiation. (shrink)

Is quantum mechanics about ‘states’? Or is it basically another kind of probability theory? It is argued that the elementary formalism of quantum mechanics operates as a well-justified alternative to ‘classical’ instantiations of a probability calculus. Its providing a general framework for prediction accounts for its distinctive traits, which one should be careful not to mistake for reflections of any strange ontology. The suggestion is also made that quantum theory unwittingly emerged, in Schrödinger’s formulation, as a ‘lossy’ by-product of a (...) quantum-mechanical variant of the Hamilton-Jacobi equation. As it turns out, the effectiveness of quantum theory qua predictive algorithm makes up for the computational impracticability of that master equation. (shrink)

The Hamilton–Jacobi (HJ) equation for the action function plays a fundamental role in classical mechanics. A known consequence of the HJ equation is a blow-up of a disturbed free-particle solution. Following the idea of Sivashinsky, we formulate an extension of the HJ equation in which perturbations eventually evolve into a finite autosoliton associated with an elementary particle. A novel element of the model is stability of the autosoliton. We link uncertainties in the position and momentum (...) of a particle to a width and amplitude of the autosoliton. We formulate restrictions on the coefficients of the model and compare the model with some existing theories of extended elementary objects. (shrink)

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \ that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply (...) each other. Jacobi’s theorem generates a microstate-dependent time parametrization \ even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics. (shrink)

The special and general relativity theories are used to demonstrate that the velocity of an unradiative particle in a Schwarzschild metric background, and in an electrostatic field, is the group velocity of a wave that we call a “particle wave,” which is a monochromatic solution of a standard equation of wave motion and possesses the following properties. It generalizes the de Broglie wave. The rays of a particle wave are the possible particle trajectories, and the motion equation of (...) a particle can be obtained from the ray equation. The standing particle wave equation generalizes the Schrödinger equation of wave amplitudes. The particle wave motion equation generalizes the Klein–Gordon equation; this result enables us to analyze the essence of the particle wave frequency. The equation of the eikonal of a particle wave generalizes the Hamilton–Jacobi equation; this result enables us to deduce the general expression for the linear momentum. The Heisenberg uncertainty relation expresses the diffraction of the particle wave, and the uncertainty relation connecting the particle instant of presence and energy results from the fact that the group velocity of the particle wave is the particle velocity. A single classical particle may be considered as constituted of geometrical particle wave; reciprocally, a geometrical particle wave may be considered as constituted of classical particles. The expression for a particle wave and the motion equation of the particle wave remain valid when the particle mass is zero. In that case, the particle is a photon, the particle wave is a component a classical electromagnetic wave that is embedded in a Schwarzschild metric background, and the motion equation of the wave particle is the motion equation of an electromagnetic wave in a Schwarzschild metric background. It follows that a particle wave possesses the same physical reality as a classical electromagnetic wave. This last result and the fact that the particle velocity is the group velocity of its wave are in accordance with the opinions of de Broglie and of Schrödinger. We extend these results to the particle subjected to any static field of forces in any gravitational metric background. Therefore we have achieved a synthesis of undulatory mechanics, classical electromagnetism, and gravitation for the case where the field of forces and the gravitational metric background are static, and this synthesis is based only on special and general relativity. (shrink)

Analytical models describing the motion of colloidal particles in given force fields are presented. In addition to local approaches, leading to well known master equations such as the Langevin and the Fokker–Planck equations, a global description based on path integration is reviewed. A new result is presented, showing that under very broad conditions, during its evolution a dissipative system tends to minimize its energy dissipation in such a way to keep constant the Hamiltonian time rate, equal to the difference between (...) the flux-based and the force-based Rayleigh dissipation functions. In fact, the Fokker–Planck equation can be interpreted as the Hamilton–Jacobi equation resulting from such minumum principle. At steady state, the Hamiltonian time rate is maximized, leading to a minimum resistance principle. In the unsteady case, we consider the relaxation to equilibrium of harmonic oscillators and the motion of a Brownian particle in shear flow, obtaining results that coincide with the solution of the Fokker–Planck and the Langevin equations. (shrink)

In recent years, the non-relativistic quantum dynamics derived from three assumptions; probability current conservation, average energy conservation, and an epistemic momentum uncertainty. Here we show that, these assumptions can be derived from a natural extension of classical statistical mechanics.

The time-dependent Schrödinger equation has been derived from three assumptions within the domain of classical and stochastic mechanics. The continuity equation isnot used in deriving the basic equations of the stochastic theory as in the literature. They are obtained by representing Newton's second law in a time-inversion consistent equation. Integrating the latter, we obtain the stochastic Hamilton-Jacobi equation. The Schrödinger equation is a result of a transformation of the Hamilton-Jacobi equation (...) and linearization by assigning the arbitrary constant ħ=2mD. An experiment is proposed to determine ħ and to test a hypothesis of the theory directly. A mathematical apparatus is formulated from the Jacobian formalism to derive physical parameters from ψ(x, t) and to obtain operators for the boundary cases of the theory. The operator formalisms are compared by means of a well-known solution in the quantum theory. (shrink)

In this paper a new trajectory-based representation to non-relativistic quantum mechanics is formulated. This is ahieved by generalizing the notion of Lagrangian path which lies at the heart of the deBroglie-Bohm “ pilot-wave” interpretation. In particular, it is shown that each LP can be replaced with a statistical ensemble formed by an infinite family of stochastic curves, referred to as generalized Lagrangian paths. This permits the introduction of a new parametric representation of the Schrödinger equation, denoted as GLP-parametrization, and (...) of the associated quantum hydrodynamic equations. The remarkable aspect of the GLP approach presented here is that it realizes at the same time also a new solution method for the N-body Schrödinger equation. As an application, Gaussian-like particular solutions for the quantum probability density function are considered, which are proved to be dynamically consistent. For them, the Schrödinger equation is reduced to a single Hamilton–Jacobi evolution equation. Particular solutions of this type are explicitly constructed, which include the case of free particles occurring in 1- or N-body quantum systems as well as the dynamics in the presence of suitable potential forces. In all these cases the initial Gaussian PDFs are shown to be free of the spreading behavior usually ascribed to quantum wave-packets, in that they exhibit the characteristic feature of remaining at all times spatially-localized. (shrink)

This paper suggests an online solution for the optimal tracking control of robotic systems based on a single critic neural network -based reinforcement learning method. To this end, we rewrite the robotic system model as a state-space form, which will facilitate the realization of optimal tracking control synthesis. To maintain the tracking response, a steady-state control is designed, and then an adaptive optimal tracking control is used to ensure that the tracking error can achieve convergence in an optimal sense. To (...) solve the obtained optimal control via the framework of adaptive dynamic programming, the command trajectory to be tracked and the modified tracking Hamilton-Jacobi-Bellman are all formulated. An online RL algorithm is the developed to address the HJB equation using a critic NN with online learning algorithm. Simulation results are given to verify the effectiveness of the proposed method. (shrink)

A deterministic equation of the Hamilton-Jacobi type is proposed for a single particle:S t+(1/2m)(∇S)2+U{S}=0, whereU{S} is a certain operator onS, which has the sense of the potential of the self-generated field of a free particle. Examples are given of potentials that imply instability of uniform rectilinear motion of a free particle and yieldrandom fluctuations of its trajectory. Galilei-invariant turbulence-producing potentials can be constructed using a single universal parameter—Planck's constant. Despite the fact that the classical trajectory concept is (...) retained, the mechanics of the particle then admits quantum-type effects: an uncertainty relation, de Broglie-type waves and their interference, discrete energy levels, and zero-point fluctuations. (shrink)

Adaptive dynamic programming, which belongs to the field of computational intelligence, is a powerful tool to address optimal control problems. To overcome the bottleneck of solving Hamilton–Jacobi–Bellman equations, several state-of-the-art ADP approaches are reviewed in this paper. First, two model-based offline iterative ADP methods including policy iteration and value iteration are given, and their respective advantages and shortcomings are discussed in detail. Second, the multistep heuristic dynamic programming method is introduced, which avoids the requirement of initial admissible control (...) and achieves fast convergence. This method successfully utilizes the advantages of PI and VI and overcomes their drawbacks at the same time. Finally, the discrete-time optimal control strategy is tested on a power system. (shrink)

Kottler, Cartan, and van Dantzig independently uncovered a key property of the Maxwell equations, which, in retrospect, is instrumental for treating noninertial situations. The essence of this KCD procedure is outlined. Present traditions incompatible with the KCD procedure are identified. KCD predicts a rotation-induced magnetoelectric effect in vacuum, as verified by the experiments of Kennard and Pegram. The description of nonvacuum situations still has some unresolved differences awaiting further experimental delineation. Explicit calculations and technical specifications of experiments receive references to (...) the literature. The emphasis of presentation stresses the conceptual reorganization necessary for lifting the Kennard-Pegram experiments out of their present state of obscurity. The question of whether or not KCD is a physically viable counterpart of the Lagrange-Hamilton-Jacobi procedure in mechanics is contingent on the outcome of more detailed experimentation of the Kennard-Pegram type. (shrink)

I commemorate David Lewis by discussing an aspect of modality within analytical mechanics, which is closely related to his work on counterfactuals. This concerns the way Hamilton ‐Jacobi theory uses ensembles, i.e. sets of possible initial conditions.

The Addendum of this note presents a brief perspective and an additional development pertaining to a previous paper. The even forceK n as well as the odd forceK 0 of the time-inversion-covariant (TIC) equation of motion in the presence of a magnetic field are derived from the results of the previous paper by a hint of generalization in classical physics. Then, by following identical steps as in the previous paper, the Addendum completes the derivation of the stochastic Hamilton- (...) class='Hi'>Jacobi and the Schrödinger equations in the electromagnetic field. The Erratum of our note points out minor typographical errors plus one clarification of a definition. (shrink)

The discussion of the recently derived quantum Hamilton equations for a spinning particle is extended to spin measurement in a Stern–Gerlach experiment. We show that this theory predicts a continuously changing orientation of the particles magnetic moment over the course of its motion across the Stern–Gerlach apparatus. The final measurement results agree with experiment and with predictions of the Pauli equation. Furthermore, the Einstein–Podolsky–Rosen–Bohm thought experiment is investigated, and the violation of Bells’s inequalities is reproduced within this stochastic (...) mechanics approach. The origin of the violation of Bell’s inequalities is traced to the the non-local nature of the velocity fields for an entangled state in the stochastic formalism, which is a result of a non-separable probability distribution of the considered particles. (shrink)

In an incendiary 2010 Nature article, M. A. Nowak, C. E. Tarnita, and E. O. Wilson present a savage critique of the best-known and most widely used framework for the study of social evolution, W. D. Hamilton’s theory of kin selection. More than a hundred biologists have since rallied to the theory’s defence, but Nowak et al. maintain that their arguments ‘stand unrefuted’. Here I consider the most contentious claim Nowak et al. defend: that Hamilton’s rule, the core (...) explanatory principle of kin selection theory, ‘almost never holds’. I first distinguish two versions of Hamilton’s rule in contemporary theory: a special version (HRS) that requires restrictive assumptions, and a general version (HRG) that does not. I then show that Nowak et al. are most charitably construed as arguing that HRS almost never holds, while HRG buys its generality at the expense of explanatory power. While their arguments against HRS are fairly uncontroversial, their arguments against HRG are more contentious, yet these have been largely overlooked in the ensuing furore. I consider the arguments for and against the explanatory value of HRG, with a view to assessing what exactly is at stake in the debate. I suggest that the debate hinges on issues concerning the causal interpretability of regression coefficients, and concerning the explanatory function Hamilton’s rule is intended to serve. (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, (...) class='Hi'>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)

Hamilton's principle and Hamilton's law are discussed. Hamilton's law is then applied to achieve direct solutions to time-dependent, nonconservative, initial value problems without the use of the theory of differential or integral equations. A major question has always plagued competent investigators who use “energy methods,” viz., “Why is it that one can derive the differential equations for a system from Hamilton's principle and then solve these equations (at least in principle) subject to applicable initial and boundary (...) conditions; but one cannot obtain a solution directly from Hamilton's principle except in very special cases?” This paper provides the answer to that question. In Hamilton's own words, “... the peculiar combination it [i.e., Hamilton's law] involves, of the principles of variations with those of partial differentials, for the determination and use of an important class of integrals, may constitute, when it shall be matured by the future labors of mathematicians, a separate branch of analysis.”. (shrink)

The idea of a canonical transformation emerged in 1837 in the course of Carl Jacobi's researches in analytical dynamics. To understand Jacobi's moment of discovery it is necessary to examine some background, especially the work of Joseph Lagrange and Siméon Poisson on the variation of arbitrary constants as well as some of the dynamical discoveries of William Rowan Hamilton. Significant figures following Jacobi in the middle of the century were Adolphe Desboves and William Donkin, while the (...) delayed posthumous publication in 1866 of Jacobi's full dynamical corpus was a critical event. François Tisserand's doctoral dissertation of 1868 was devoted primarily to lunar and planetary theory but placed Hamilton–Jacobi mathematical methods at the forefront of the investigation. Henri Poincaré's writings on celestial mechanics in the period 1890–1910 succeeded in making canonical transformations a fundamental part of the dynamical theory. Poincaré offered a mathematical vision of the subject that differed from Jacobi's and would become influential in subsequent research. Two prominent researchers around 1900 were Carl Charlier and Edmund Whittaker, and their books included chapters devoted explicitly to transformation theory. In the first three decades of the twentieth century Hamilton–Jacobi theory in general and canonical transformations in particular would be embraced by a range of researchers in astronomy, physics and mathematics. (shrink)

Recent work on the history of General Relativity by Renn, Sauer, Janssen et al. shows that Einstein found his field equations partly by a physical strategy including the Newtonian limit, the electromagnetic analogy, and energy conservation. Such themes are similar to those later used by particle physicists. How do Einstein's physical strategy and the particle physics derivations compare? What energy-momentum complex did he use and why? Did Einstein tie conservation to symmetries, and if so, to which? How did his work (...) relate to emerging knowledge of the canonical energy-momentum tensor and its translation-induced conservation? After initially using energy-momentum tensors hand-crafted from the gravitational field equations, Einstein used an identity from his assumed linear coordinate covariance x'=Mx to relate it to the canonical tensor. Usually he avoided using matter Euler-Lagrange equations and so was not well positioned to use or reinvent the Herglotz-Mie-Born understanding that the canonical tensor was conserved due to translation symmetries, a result with roots in Lagrange, Hamilton and Jacobi. Whereas Mie and Born were concerned about the canonical tensor's asymmetry, Einstein did not need to worry because his Entwurf Lagrangian is modeled not so much on Maxwell's theory as on a scalar theory. Einstein's theory thus has a symmetric canonical energy-momentum tensor. But as a result, it also has 3 negative-energy field degrees of freedom. Thus the Entwurf theory fails a 1920s-30s a priori particle physics stability test with antecedents in Lagrange's and Dirichlet's stability work; one might anticipate possible gravitational instability. This critique of the Entwurf theory can be compared with Einstein's 1915 critique of his Entwurf theory for not admitting rotating coordinates and not getting Mercury's perihelion right. One can live with absolute rotation but cannot live with instability. Particle physics also can be useful in the historiography of gravity and space-time, both in assessing the growth of objective knowledge and in suggesting novel lines of inquiry to see whether and how Einstein faced the substantially mathematical issues later encountered in particle physics. This topic can be a useful case study in the history of science on recently reconsidered questions of presentism, whiggism and the like. Future work will show how the history of General Relativity, especially Noether's work, sheds light on particle physics. (shrink)

In the many worlds community there seems to exist a belief that the physics of quantum theory is completely defined by it’s Hamilton operator given in an abstract Hilbert space, especially that the position basis may be derived from it as preferred using decoherence techniques.We show, by an explicit example of non-uniqueness, taken from the theory of the KdV equation, that the Hamilton operator alone is not sufficient to fix the physics. We need the canonical operators $\hat{p}$ (...) , $\hat{q}$ as well. As a consequence, it is not possible to derive a “preferred basis” from the Hamilton operator alone, without postulating some additional structure like a “decomposition into systems”. We argue that this makes such a derivation useless for fundamental physics. (shrink)

In 1975, the English evolutionist William Donald Hamilton held in Brazil a series of lectures entitled “Population genetics and social behaviour”. The unpublished notes of these conferences—written by Hamilton and recently discovered at the British Library—offer an opportunity to reflect on some of the author’s ideas about evolution. The year of the conference is particularly significant, as it took place shortly after the applications of the Price equation with which Hamilton was able to build a model (...) that included several levels of selection. In this paper I mainly analyse the inaugural lecture in which Hamilton proposes a simple model to disprove the hypothesis supported by the British zoologist C. Vero Wynne-Edwards regarding mechanisms to prevent “over-exploitation of the food supply” in “the interests of the survival of the group”. The document presented here is of great historical interest. Not only because manuscript offers a model that—since it was intended for teaching purposes—had never before appeared in the published version, but also because of the general index of the lectures that accompanies it. The latter allows us to make some hypothetical considerations on the relationship and differences between kin-selection, group-selection and inclusive fitness that Hamilton wanted to present to the attentive, well-prepared audience of the foreign university that had invited him. (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)

Aesthetics and Music is a rich and interesting study. Hamilton's approach is innovative. He interleaves chapters on the history of philosophical thought about music with more theoretical discussions of music, sound, rhythm and improvisation, but does not cover the work–performance relation, depiction or expression. He draws on an atypically broad range of examples, including avant-garde, medieval, non-Western and jazz. The assumptions are humanist: ‘I wish to argue for an aesthetic conception of music as an art … according to which (...) music is a human activity grounded in the body and bodily movement and interfused with human life’ .The historical chapters are valuable and not without analysis and criticism. Hamilton shows how the ancient Greek theorists were more interested in music's mathematical properties as reflecting the underlying harmony of relations between cosmic bodies than in the practice of musicians. While they equated the value of art with its contribution to an education for citizenship and while their concept of the arts differed …. (shrink)