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)

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)

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)

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)

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.

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.

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)

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)

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)

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)

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)

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)

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)

We review the relativistic classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelbert–Schrödinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields with compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through (...) the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann–Robertson–Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson–Walker scale factor. Using the Einstein equations, one see that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann–Robertson–Walker dynamical model and the configurations in the conformal coordinates. (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)

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)

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)

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)

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)

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.

The time-scale dynamic equations play an important role in modeling complex dynamical processes. In this paper, the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations are studied. The definition and criterion of the Mei symmetry of the Birkhoffian system on time scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. As a special case, the Mei symmetry of time-scale Hamilton canonical (...) class='Hi'>equations is discussed and new conserved quantities for the Hamiltonian system on time scales are derived. Two examples are given to illustrate the application of results. (shrink)

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system describing in terms of ray trajectories, for a stationary refractive medium, a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the geometrical optics (“eikonal”) approximation, which is contained as a simple limiting case. Due to the fact, moreover, that the time independent Schrödinger equation is itself a Helmholtz-like equation, the same mathematics holding for a classical optical beam (...) turns out to apply to a quantum particle beam moving in a stationary force field, and leads to a system of Hamiltonian equations providing exact and deterministic particle trajectories and dynamical laws, and containing the laws of Classical Mechanics in the eikonal limit. (shrink)

The contemporary scientific and technological progress builds on the accomplishments of classical mechanics from the 19th century when the so-called ‘European scientific method and values’ were accepted practically by the whole educated world. Most scientific results and conclusions were reached based on the causal ontological approach proposed in principle already by Plato’s Socrates and developed further by Aristotle. Despite the late-modern paradigm shift in science, the topicality of the ontological approach proposed by Aristotle remains. On the other hand, 19th and (...) 20th century philosophers, mainly positivists such as Mach and Avenarius but also Schlick and Carnap, attempted to change this approach to unify scientific knowledge in accordance with an ideological, i.e. positivist outlook on reality. The authors place a special emphasis on the contribution of Rudolf Carnap and his interaction with Martin Heidegger. Three very different theories are applied to physical reality in the present: classical mechanics in the standard macroscopic realm, Copenhagen quantum mechanics in the microscopic realm, and special theory of reality in both realms in the case of systems consisting of objects having higher velocity values. Any explanation or description of transitions between different realms and theories had not been provided until now. Our paper describes the corresponding evolution in the modern period and identifies the underlying false philosophical assumptions and statements existing in today’s scientific systems. We will then demonstrate that one common theory for all realms of reality may exist; one that will be based fully on Hamilton equations. Only time change of particle impulse is to be determined by a corresponding force. All necessary characteristics of physical reality may be derived in such a case. Direct correlations of such physical approach to philosophy will be drawn. (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.

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)

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 theories of inclusive fitness and multilevel selection provide alternative perspectives on social evolution. The question of whether these perspectives are of equal generality remains a divisive issue. In an analysis based on the Price equation, Queller argued (by means of a principle he called the separation condition) that the two approaches are subject to the same limitations, arising from their fundamentally quantitative-genetical character. Recently, van Veelen et al. have challenged Queller’s results, using this as the basis for a broader (...) critique of the Price equation, the separation condition, and the very notion of inclusive fitness. Here we show that the van Veelen et al. model, when analyzed in the way Queller intended, confirms rather than refutes his original conclusions. We thereby confirm (i) that Queller’s separation condition remains a legitimate theoretical principle and (ii) that the standard inclusive fitness and multilevel approaches are indeed subject to the same limitations. (shrink)

The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton’s equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.

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 PhD dissertation examines the conceptual and theoretical foundations of the most general and most widely used framework for understanding social evolution, W. D. Hamilton's theory of kin selection. While the core idea is intuitive enough (when organisms share genes, they sometimes have an evolutionary incentive to help one another), its apparent simplicity masks a host of conceptual subtleties, and the theory has proved a perennial source of controversy in evolutionary biology. To move towards a resolution of these controversies, (...) we need a careful and rigorous analysis of the philosophical foundations of the theory. My aim in this work is to provide such an analysis. I begin with an examination of the concepts behavioural ecologists employ to describe and classify types of social behaviour. I stress the need to distinguish concepts that are often conflated: for example, we need to distinguish simple cooperation from collaboration in collective tasks, behaviours from strategies, and control from manipulation and coercion. I proceed from here to the formal representation of kin selection via George R. Price’s covariance selection mathematics. I address a number of interpretative issues the Price formalism raises, including the vexed question of whether kin selection theory is ‘formally equivalent’ to multi-level selection theory. In the second half of the dissertation, I assess the uses and limits of Hamilton’s rule for the evolution of social behaviour; I provide a precise statement of the conditions under which the rival neighbour-modulated fitness and inclusive fitness approaches in contemporary kin selection theory are equivalent (and describe cases in which they are not); and I criticize recent formal attempts to establish the controversial claim that kin selection leads to organisms behaving as if maximizing their inclusive fitness. (shrink)

The levels of selection problem was central to Maynard Smith’s work throughout his career. This paper traces Maynard Smith’s views on the levels of selection, from his objections to group selection in the 1960s to his concern with the major evolutionary transitions in the 1990s. The relations between Maynard Smith’s position and those of Hamilton and G.C. Williams are explored, as is Maynard Smith’s dislike of the Price equation approach to multi-level selection. Maynard Smith’s account of the ‘core Darwinian (...) principles’ is discussed, as is his debate with Sober and Wilson (1998) over the status of trait-group models, and his attitude to the currently fashionable concept of pluralism about the levels of selection. (shrink)

Maxwellian electrodynamics genesis is considered in the light of the author’s theory change model previously tried on the Copernican and the Einstein revolutions. It is shown that in the case considered a genuine new theory is constructed as a result of the old pre-maxwellian programmes reconciliation: the electrodynamics of Ampere-Weber, the wave theory of Fresnel and Young and Faraday’s programme. The “neutral language” constructed for the comparison of the consequences of the theories from these programmes consisted in the language of (...) hydrodynamics with its rich content of analogous models ranging from the uncompressible fluid up to molecular vortices. The programmes’ meeting led to construction of the whole hierarchy of crossbred objects beginning from the displacement current and up to common hybrids. After that the interpenetration of the pre-maxwellian programmes began that marked the beginning of theoretical schemes of optics and electromagnetism unification. Maxwell’s programme did assimilate some ideas of the Ampere-Weber programme, as well as the presuppositions of the programmes of Fresnel and Faraday; and the significance of this fact for further methodology of scientific research programmes development is discussed. It is argued that the core of Maxwell’s unification strategy was formed by Kantian epistemology looked through the prism of William Whewell and such representatives of Scottish Enlightenment as Thomas Reid and William Hamilton. All these enabled Maxwell to start to unify not only optics and electromagnetism, but British and continental research traditions as well. Maxwell’s programme did supersede the Ampere-Weber one because Maxwell did put forward as a synthetic principle the idea, that differed from that of Ampere-Weber by its flexible and contra-ontological, strictly epistemological, Kantian character. For Maxwell, ether was not the last building block of physical reality, from which fields and charges should be constructed . “Action at a distance”, “incompressible fluid”, “molecular vortices” were only analogies for Maxwell, capable to direct the researcher on the “right” mathematical relations. From the “representational” point of view all this hydrodynamical models were doomed to failure efforts to describe what can not be described in principle – things in themselves, the “nature” of electrical and magnetic phenomena. On the contrary, Maxwell aimed his programme to find empirically meaningful mathematical relations between the electrodynamics basic objects, i.e. the creation of inter - coordinated electromagnetic field equations system. Namely the application of this epistemology enabled Hermann von Helmholtz and his pupil Heinrich Hertz to arrive at such a version of Maxwell’s theory that served a heuristical basis for the radiowaves discovery. (shrink)

The \\)-Robertson–Walker spacetime is under investigation. With the derived Hamilton operator, we are solving the Wheeler–De Witt Equation and its Schrödinger-like extension, for physically important forms of the effective potential. The closed form solutions, expressed in terms of Heun’s functions, allow us to comment on the occurrence of Universe from highly probable quantum states.

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)

The symplectic structures (brackets, Hamilton's equations, and Lagrange's equations) for the Dirac electron and its classical model have exactly the same form. We give explicitly the Poisson brackets in the dynamical variables (x μ,p μ,v μ,S μv). The only difference is in the normalization of the Dirac velocities γμγμ=4 which has significant consequences.

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.

This paper defends hedonic intentionalism, the view that all pleasures, including bodily pleasures, are directed towards objects distinct from themselves. Brentano is the leading proponent of this view. My goal here is to disentangle his significant proposals from the more disputable ones so as to arrive at a hopefully promising version of hedonic intentionalism. I mainly focus on bodily pleasures, which constitute the main troublemakers for hedonic intentionalism. Section 1 introduces the problem raised by bodily pleasures for hedonic intentionalism and (...) some of the main reactions to it. Sections 2 and 3 rebut two main approaches equating bodily pleasures with non- intentional episodes. More precisely, section 2 argues that bodily pleasures cannot be purely non-intentional self-conscious feelings, by relying on Brentano’s objection to Hamilton’s theory of pleasure. Section 3 argues that bodily pleasures cannot be non-intentional sensory qualities by relying on Brentano’s objections to Stumpf’s theory of pleasure. Section 4 develops a brentanian view of the intentionality of bodily pleasures by claiming bodily pleasures are directed at a sui generis class of sensory qualities. Section 5 presents an objection to Brentano’s later theory of pleasure according to which all sensory pleasures are directed at sensing acts. (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. Attention to the gauge generator G of Rosenfeld, Anderson, Bergmann, Castellani et al., a specially _tuned sum_ of first-class constraints, facilitates seeing that a solitary first-class constraint in fact generates not a gauge transformation, but a bad physical change in electromagnetism or (...) General Relativity. The change spoils the Lagrangian constraints, Gauss's law or the Gauss-Codazzi relations describing embedding of space into space-time, in terms of the physically relevant velocities rather than auxiliary canonical momenta. But the resemblance between the gauge generator G and the Hamiltonian H leaves still unclear where objective change is in GR. Insistence on Hamiltonian-Lagrangian equivalence, a theme emphasized by Castellani, Sugano, Pons, Salisbury, Shepley and Sundermeyer among others, holds the key. Taking objective change to be ineliminable time dependence, one recalls that there is change in vacuum GR just in case there is no time-like vector field xi^a satisfying Killing's equation L_xi g_mn=0, because then there exists no coordinate system such that everything is independent of time. Throwing away the spatial dependence of GR for convenience, one finds explicitly that the time evolution from Hamilton's equations is real change just when there is no time-like Killing vector. The inclusion of a massive scalar field is simple. No obstruction is expected in including spatial dependence and coupling more general matter fields. Hence change is real and local even in the Hamiltonian formalism. The considerations here resolve the Earman-Maudlin standoff over change in Hamiltonian General Relativity: the Hamiltonian formalism is helpful, and, suitably reformed, it does not have absurd consequences for change and observables. Hence the classical problem of time is resolved. The Lagrangian-equivalent Hamiltonian analysis of change in General Relativity is compared to Belot and Earman's treatment. The more serious quantum problem of time, however, is not automatically resolved due to issues of quantum constraint imposition. (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)

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 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)

The “land community” (or “biotic community”) that features centrally in Aldo Leopold’s Land Ethic has typically been equated with the concept of “ecosystem.” Moreover, some have challenged this central Leopoldean concept given the multitude of meanings of the term “ecosystem” and the changes the term has undergone since Leopold’s time (see, e.g., Shrader-Frechette 1996). Even one of Leopold’s primary defenders, J. Baird Callicott, asserts that there are difficulties in identifying the boundaries of ecosystems and suggests that we recognize that their (...) boundaries are determined by scientific questions ecologists pose (Callicott 2013). I argue that we need to rethink Leopold’s concept of land community in the following ways. First, we should recognize that Leopold’s views are not identical to those of his contemporaries (e.g., Clements, Elton), although they resemble those of some subsequent ecologists, including some of our contemporaries (e.g., O’Neill 2001, Post et al. 2007, Hastings and Gross 2012). Second, the land community concept does not map cleanly onto the concept of “ecosystem”; it also incorporates elements of the “community” concept in community ecology by emphasizing the interactions between organisms and not just the matter/energy flow of the ecosystem concept. Third, the boundary question can be illuminated by considering some of the recent literature on the nature of biological individuals (in particular, Odenbaugh 2007; Hamilton, Smith, and Haber 2009; Millstein 2009), focusing on concentrations of causal relations as determinative of the boundaries of the land community qua individual. There are challenges to be worked out, particularly when the interactions of community members do not map cleanly onto matter/energy flows, but I argue that these challenges can be resolved. The result is a defensible land community concept that is ontologically robust enough to be a locus of moral obligation while being consistent with contemporary ecological theory and practice. (shrink)

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the set (...) of phase space paths determined by solutions of Hamilton’s equations. The MaxEnt inference algorithm with Shannon’s concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes’ formulation of predictive statistical mechanics, and the concept of macroscopic reproducibility for time dependent processes. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation, and leads to corresponding loss of information. In accordance with the original idea of Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the 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)

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)

Quaternions consist of a scalar plus a vector and result from multiplication or division of vectors by vectors. Division of vectors is equivalent to multiplication divided by a scalar. Quaternions as used here consist of the scalar product with positive sign plus the vector product with sign determined by the right-hand rule. Units are specified by the multiplication process. Trigonometric functions are quaternions with units that can satisfy Hamilton's requirements. The square of a trigonometric quaternion is a real number (...) provided that the product of the scalar number and the vector is not commutative. Maxwell's electromagnetic equations for empty space can be represented by a single quaternion equation. (shrink)

A language for quantum physics is derived from set theory by replacing the classical predicate algebra (Boolean) by a certain quantum predicate algebra (rational projective), time space and the Hamilton-Schroedinger dynamics by a Feynman-like graph dynamics, and the Dirac spin operators by topological switching operators on the graph. The development is described from the basic level of elementary monadic processes to the level of the free Dirac equation.

This memoir consists of two parts, of which the first deals with the foundations of the theory of the struggle for existence, and begins with the introduction of the important concept of quantity of life, besides that of population. The fundamental equations are then established for the case where the individuals of a biological association mutually devour each other, the reasoning being based on the principle of encounters and on the fundamental hypothesis of the existence of equivalents of the (...) individuals constituting the different species which form the association.Having now obtained the equations which determine the rates of change of the numbers of individuals and of the quantities of life of the species, the memoir proceeds to find the equations relating to the stationary state, by establishing theorems which are afterwards applied for the purpose of obtaining the three general laws of the fluctuations. This investigation of the equilibrium state is then followed by the determination of the integrals of the fluctuation equations, and the deduction from these of their most important consequences.The second part of the memoir contains the enumeration and demonstration of the general laws of the struggle for existence, which flow from the discussion of the principles and the integrals obtained in the first part. In this way there is established a new sort of dynamics, demographic dynamics, which, though substantially different from the dynamics of material systems, is developed from an analogous point of view.The second part begins with the principle of the conservation of demographic energy, according to which there are two sorts of energy, one actual and one potential, which transform mutually the one into the other. This principle is the analogue of the principle of conservation of mechanical energy. It is followed by the enunciation of the three laws relating to the biological fluctuations, the experimental verification of which has been investigated by several naturalists. The success which has attended their efforts is well known.Everybody knows the importance ofHamilton's principle in mechanics and in all the domains of physical science. An analogous variation principle can be found in biology, and from it one can deduce the fluctuation equations in the canonical Hamiltonian form and also in the form of a Jacobian partial differential equation. Their integrals in involution from the subject next studied, and in this way are found cases where the integration is reducible to quadratures.Hamilton's principle leads to the principle of least action . There exists also in biology a closely related principle, which may be called the principle of least vital action. Its analytical form is such that it requires the existence of a true minimum, a state of affairs which does not always hold good in the analogous case in mechanics.Die Abhandlung besteht aus zwei Teilen: Der erste ist den Grundlagen der Theorie des Kampfes ums Dasein gewidmet. Man beginnt damit, neben der Bevölkerung die Menge des Lebens einzuführen, welche eine sehr wichtige Rolle spielt. Man geht dann dazu über, die Grundgleichungen für den Fall aufzustellen, in welchem die Individuen einer biologischen Assoziation sich gegenseitig auffressen. Man wendet das Prinzip der Begegnungen an und stützt sich auf die Grundhypothese der Existenz von Individuenäquivalenten der verschiedenen Arten, welche die Assoziation bilden.Wenn man die Gleichungen, welche die Variationen der Individuenzahlen und der Lebensmengen der Arten liefern, gefunden hat, so schreitet man zu den statischen oder Gleichgewichtsgleichungen fort und findet so die Theoreme, die man alsdann benutzt, um die drei allgemeinen Gesetze der Fluktuationen zu erhalten.Nach dem Studium des Gleichgewichtes geht man dazu über, die verschiedenen Integrale der Fluktuationengleichungen zu finden und zieht endlich aus ihnen die wichtigsten Folgerungen.Im zweiten Teile werden die allgemeinen Gesetze des Kampfes ums Dasein, welche aus der Diskussion der Prinzipien und der im ersten Teil gewonnenen Integrale stammen, ausgesprochen und bewiesen. Man baut so eine neue Dynamik auf: die demographische Dynamik, welche, obschon sie wesentlich verschieden von derjenigen der materiellen Systeme ist, sich nach einem analogen Gesichtspunkt entwickelt.Man beginnt in diesem zweiten Teil mit dem Prinzip der Erhaltung der demographischen Energie, nach welchem zwei Energien, die aktuelle und die potentielle, sich ineinander verwandeln. Wir haben hier eine Analogie zum Prinzip der Erhaltung der mechanischen Energie. Dann werden die drei Gesetze der biologischen Fluktuationen formuliert. Für sie haben verschiedene Naturforscher experimentelle Bestätigungen gesucht. Man kennt den Erfolg ihrer Bemühungen.Alle Welt kennt die Bedeutung des Prinzips vonHamilton in der Mechanik und in allen Gebieten der Physik. Man kann ein dem Variationskalkül analoges Prinzip in der Biologie finden. Aus ihm kann man die Fluktuationsgleichungen in der kanonischen Form vonHamilton und in der Gestalt einer Gleichung mit partiellen Differentialquotienten vonJacobi herleiten. Ihre Integrale bilden den Gegenstand der folgenden Studien. Man findet Fälle, in denen die Integration sich auf Quadraturen reduziert.Das Prinzip vonHamilton zieht das Prinzip der kleinsten Wirkung oder das Prinzip vonMaupertuis nach sich. Auch in der Biologie gibt es ein mit dem vorgenannten zusammenhängendes Prinzip, welches man das Prinzip der kleinsten vitalen Wirkung nennen kann. Seine analytische Form ist derart, dass es sich um ein wahres Minimum handelt, das im analogen Fall in der Mechanik nicht immer zutrifft. (shrink)

The mechanical aspect of momentum, basically its role as a tangent vector of the trajectory of the particle, is related to properties of the momentum found in the contexts of Hamilton's optico-mechanical analogy, de Broglie's matter waves, and quantum mechanics. These properties are treated in a systematic way by considering an approximation of the particle mechanical action of the particle by a step function. A special method of discretizing partial differential equations is shown to be required. Using this (...) method, a discrete dynamics is developed. It is shown that particle dynamics can be regarded as the limit case of the discrete dynamics as the step functions tend to the continuous ones. The equation of motion of a free particle in an arbitrary reference system is deduced in two ways: (i) in continuous dynamics by making use of the invariance of action within changes of reference systems, and (ii) by taking the mentioned limit in discrete dynamics of an equation which expresses that the mechanical and wave-theoretical aspects of the momentum are interrelated in specific way. (shrink)