Generalized Hamiltonian Mechanics

In this paper we consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval. More precisely, the operator of our integral equation is supposed to be of the form aI + µ −1 bSµI + K with S the Cauchy integral operator, with piecewise continuous coefficients a and b , with a regular integral operator K , and with a Jacobi weight µ. To the equation [aI + µ −1 bSµI + K]u = f we apply a collocation method, where the collocation points are the Chebyshev nodes of the second kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation in weighted L 2 spaces, we derive necessary and sufficient conditions.

In the context of the Lagrangian and Hamiltonian mechanics, a generalized theory of coordinate transformations is analyzed. On the basis of such theory, a misconception concerning the superiority of the Hamiltonian formalism with respect to the Lagrangian one is criticized. The consequent discussion introduces the relationship between the classical Hamilton action and the covariance properties of equations of motion, at the level of undergraduate teaching courses in theoretical mechanics.

The Hamilton-Jacobi method for constrained systems is discussed. The equations of motion for a free particle constrained to move on the surface of a torus are obtained without using any gauge-fixing conditions. The quantization of this model is also discussed.

The paper studies the geometry of Liouville foliations generated by integrable Hamiltonian systems. It is shown that regular leaves are two-dimensional submanifolds with zero Gaussian curvature and zero Gaussian torsion. There exists a distribution that generates a foliation orthogonal to the Liouville foliation.

The aim of this research is to explore the philosophical position of various scientific theories based on the history and philosophy of science. This is because the philosophy of science, which has usually dealt mainly with epistemology and methodology, is extended to the concern of problems of ontology, that is, metaphysics. Determinism, which is rooted in the metaphysical belief that objective scientific knowledge exists independently of humankind's perception, is comparable to a well-defined mechanism and can be described as "mathematization" of objective scientific knowledge-this is exemplified in the natural laws of dynamics established by Newton, Einstein, and Schrödinger. Conversely, if we move away from determinism, we need anthropomorphic concepts such as "possibility" and "contingency" to define the laws of nature. This paper investigates the shift from classical deterministic thought to the contingently perceived probabilistic theory, changes in scientific theories from a naturalistic viewpoint, and the convergence of theories achieved through this process. Since Darwin announced his theory of evolution, natural sciences have steadily undergone a shift from endeavoring to name, classify, and measure to emphasizing the transience of things, historical interest, and theorization. On the other hand, weak determinism states that things in the world are inevitable but also coincidental. Because there are coincidences, even if we know the current state of an object accurately, we cannot know its future state accurately; we can only know it probabilistically. It seems that things in the world occur through both necessity and coincidence and are not strictly determined. This kind of probabilistic weak determinism can be said to correspond to quantum theory and evolution theory.

A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. As a key feature, the laws are formulated in terms of quantum events rather than of particle states. Temporal and spatial coordinates of a quantum event are treated on equal footing, namely as self-adjoint operators on a Hilbert space. The theory is not based upon Lagrangian or Hamiltonian mechanics, and breaks with the concept of a continuously flowing time. The physical object under consideration is a spinless particle exposed to an external potential. The theory also accounts for particle-antiparticle pair creation and annihilation, and is therefore not a single-particle theory in the usual sense. The Maxwell equations are derived as a straightforward consequence of certain fundamental commutation relations. In the non-relativistic limit and in the limit of vanishing time uncertainty, the Schrödinger equation of a spinless particle exposed to an external electromagnetic field is obtained.

We define common thermodynamic concepts purely within the framework of general Markov chains and derive Jarzynski's equality and Crooks' fluctuation theorem in this setup. In particular, we regard the discrete-time case, which leads to an asymmetry in the definition of work that appears in the usual formulation of Crooks' fluctuation theorem. We show how this asymmetry can be avoided with an additional condition regarding the energy protocol. The general formulation in terms of Markov chains allows transferring the results to other application areas outside of physics. Here, we discuss how this framework can be applied in the context of decision-making. This involves the definition of the relevant quantities, the assumptions that need to be made for the different fluctuation theorems to hold, as well as the consideration of discrete trajectories instead of the continuous trajectories, which are relevant in physics.

The problem of constant-speed ballistics is studied under the umbrella of non-linear non-holonomic constrained systems. The Newtonian approach is shown to be equivalent to the use of Chetaev's rule to incorporate the constraint within the initially unconstrained formulation. Although the resulting equations are not, in principle, obtained from a variational statement, it is shown that the trajectories coincide with those of geometrical optics in a medium with a suitably chosen refractive index, as prescribed by Fermat's principle of least time. This fact gives rise to an intriguing mechano-optical analogy. The trajectories are further studied and discussed.

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include as particular cases the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré and discrete Lagrange-Poincaré equations. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.

Geometric structures underlying commutative and non commutative integrable dynamics are analyzed. They lead to a new characterization of noncommutative integrability in terms of spectral properties and of Nijenhuis torsion of an invariant (1,1) tensor eld. The construction of compatible symplectic structures is also discussed.

A characterization of separability, projectability and integrability of dynamieM systems in terms of the spectral properties of invariant mixed tenser fie.lds with vanishing Nijenhuis tensor is given. In addition, some preliminary results on the inverse problem (from lAouville intcgT'ability to Lax representation) are illustrated.

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.

The construction of a family of real Hamiltonian forms (RHF) for the special class of affine 1 + 1-dimensional Toda field theories (ATFT) is reported. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinitedimensional Hamiltonian systems. The construction method is illustrated on the explicit nontrivial example of RHF of ATFT related to the exceptional algebras E 6 and E 7 . The involutions of the local integrals of motion are proved by means of the classical R-matrix approach.

A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracket such that the anchor is compatible with the Lie bracket of vector fields. We firstly show how most geometrical structures intensively studied in the framework of Lie algebroid can easily be extended in the pre-Lie algebroid context. The principal purpose of this paper is to show that how all these results only depend of the foliated structure and do not depend of the pre-Lie algebroid structure that we can put on a foliated anchored bundle. As application, we obtain a Finsler connection on a foliated anchored bundle which induces the classical Finsler connection on each leaf and a similar result for the Chern connection.

We present a rigorous and rather self-contained analysis of the Verdet constant in graphenelike materials. We apply the gauge-invariant magnetic perturbation theory to a nearestneighbour tight-binding model and obtain a relatively simple and exactly computable formula for the Verdet constant, at all temperatures and all frequencies of sufficiently large absolute value. Moreover, for the standard nearest neighbour tight-binding model of graphene we show that the transverse component of the conductivity tensor has an asymptotic Taylor expansion in the external magnetic field where all the coefficients of even powers are zero.

In this paper of "The Epistemology of Contemporary Physics" series we investigate the epistemological significance and sensibility (and hence interpretability and interpretation) of classical mechanics in its Newtonian and non-Newtonian formulations. As we will see, none of these formulations provide a clear and consistent framework for understanding the physics which they represent and hence they all represent valid formalism without proper epistemology or sensible interpretation.

An essential part in modeling out-of-equilibrium dynamics is the formulation of the irreversible dynamics. In the latter, the main modeling task consists in specifying the relations between thermodynamic forces on the one hand and fluxes on the other hand. As a guardrail to ensure that these relations comply with macroscopic observations one uses, among other principles, the second law of thermodynamics. The latter is considered in this contribution as that the local production of entropy is non-negative. Mainly two major directions have been followed in the literature for the specification of force-flux relations. On the one hand, quasi-linear relations are employed, in which so-called transport coefficients occur, that may depend on the forces themselves in which case we call the relation quasi-linear rather than linear. If the (matrix of) transport coefficients is non-negative, the second law is respected. Such relations have a deeper foundation in the physics of fluctuation-diss...

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a given involution. The resulting subspace is isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate another nonequivalent (real) ones. A crucial role in this construction is played by the assumed analyticity and the invariance of the Hamiltonian under the involution. We show that if the initial system is Liouville integrable, then its complexification and its real forms will be integrable again and this provides a method of finding new integrable systems starting from known ones. We demonstrate our construction by finding real forms of dynamics for the Toda chain and a family of Calogero-Moser models. For these models we also show that the involution of the complexified phase space induces a Cartan-like involution of their Lax representations.

In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems in 2D. We generalize the cosymplectic structures to time-dependent Nambu-Poisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our constructions with various examples.

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

We develop a new method for finding the quantum probability density of arrival at the detector. The evolution of the quantum state restricted to the region outside of the detector is described by a restricted Hamiltonian that contains a non-hermitian boundary term. The non-hermitian term is shown to be proportional to the flux of the probability current operator through the boundary, which implies that the arrival probability density is equal to the flux of the probability current.

The relationships between port-Hamiltonian systems modeling and the notion of monotonicity are explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to maximal cyclically monotone relations, together with their generating functions. This gives rise to new classes of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cyclically monotone relation is shown to be computable through the use of generating functions. Furthermore, connections are discussed with incremental versions of passivity, and it is shown how incrementally port-Hamiltonian systems with strictly convex Hamiltonians are (maximal) equilibrium independent passive. Finally, the results on compositionality of monotone relations are employed for a convex optimization approach to the computation of the equilibrium of interconnected incrementally port-Hamiltonian systems.

This article provides a concise summary of the basic ideas and concepts in port-Hamiltonian systems theory and its use in analysis and control of complex multiphysics systems. It gives special attention to new and unexplored research directions and relations with other mathematical frameworks. Emergent control paradigms and open problems are indicated, including the relation with thermodynamics and the question of uniting the energy-processing view of control, as emphasized by port-Hamiltonian systems theory, with a complementary information-processing viewpoint.

We extend the Jacobi structure from T Q × R and T * Q × R to A × R and A * × R, respectively, where A is a Lie algebroid and A * carries the associated Poisson structure. We see that A * × R possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems, generalizing previous models on T Q × R and g × R.

One of the fundamental objectives of the theory of symplectic singularities is to study the symplectic invariants appearing in various geometrical contexts. In the paper we generalize the symplectic cohomological invariant to the class of generalized canonical mappings. We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and describe the local properties of generic symplectic relations.

We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture), that can be put into correspondence with the usual Hamilton-Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamiltonian-Lagrange picture, defined on the Hamilton-Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton-Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton-Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the the original Hamiltonian motion.

In this paper, we study equilibria of differential equation models for networks. When interactions between nodes are taken to be piecewise constant, an efficient combinatorial analysis can be used to characterize the equilibria. When the piecewise constant functions are replaced with piecewise linear functions, the equilibria are preserved as long as the piecewise linear functions are sufficiently steep. Therefore the combinatorial analysis can be leveraged to understand a broader class of interactions. To better understand how broad this class is, we explicitly characterize how steep the piecewise linear functions must be for the correspondence between equilibria to hold. To do so, we analyze the steady state and Hopf bifurcations which cause a change in the number or stability of equilibria as slopes are decreased. Additionally, we show how to choose a subset of parameters so that the correspondence between equilibria holds for the smallest possible slopes when the remaining param...

A concise but rigorous treatment of variational techniques, focusing primarily on Lagrangian and Hamiltonian systems, this book is ideal for physics, engineering and mathematics students. The book begins by applying Lagrange's equations to a number of mechanical systems. It introduces the concepts of generalized coordinates and generalized momentum. Following this, the book turns to the calculus of variations to derive the Euler-Lagrange equations. It introduces Hamilton's principle and uses this throughout the book to derive further results. The Hamiltonian, Hamilton's equations, canonical transformations, Poisson brackets and Hamilton-Jacobi theory are considered next. The book concludes by discussing continuous Lagrangians and Hamiltonians and how they are related to field theory. Written in clear, simple language, and featuring numerous worked examples and exercises to help students master the material, this book is a valuable supplement to courses in mechanics. Contents I LAGRANGIAN MECHANICS 1 Fundamental concepts 1.1 Kinematics 1.2 Generalized coordinates 1.3 Generalized velocity 1.4 Constraints 1.5 Virtual displacements 1.6 Virtual work and generalized force 1.7 Configuration space 1.8 Phase space 1.9 Dynamics 1.9.1 Newton's laws of motion 1.9.2 The equation of motion 1.9.3 Newton and Leibniz 1.10 Obtaining the equation of motion 1.10.1 The equation of motion in Newtonian mechanics 1.10.2 The equation of motion in Lagrangian mechanics 1.11 Conservation laws and symmetry principles 1.11.1 Generalized momentum and cyclic coordinates 1.11.2 The conservation of linear momentum 1.11.3 The conservation of angular momentum 1.11.4 The conservation of energy and the work function 1.12 Problems v vi Contents 2 The calculus of variations 2.1 Introduction 2.2 Derivation of the Euler-Lagrange equation 2.2.1 The difference between δ and d 2.2.2 Alternate forms of the Euler-Lagrange equation 2.3 Generalization to several dependent variables 2.4 Constraints 2.4.1 Holonomic constraints 2.4.2 Non-holonomic constraints 2.5 Problems 3 Lagrangian dynamics 3.1 The principle of d'Alembert. A derivation of Lagrange's equations 3.2 Hamilton's principle 3.3 Derivation of Lagrange's equations 3.4 Generalization to many coordinates 3.5 Constraints and Lagrange's λ-method 3.6 Non-holonomic constraints 3.7 Virtual work 3.7.1 Physical interpretation of the Lagrange multipliers 3.8 The invariance of the Lagrange equations 3.9 Problems II HAMILTONIAN MECHANICS 4 Hamilton's equations 4.1 The Legendre transformation 4.1.1 Application to thermodynamics 4.2 Application to the Lagrangian. The Hamiltonian 4.3 Hamilton's canonical equations 4.4 Derivation of Hamilton's equations from Hamilton's principle 4.5 Phase space and the phase fluid 4.6 Cyclic coordinates and the Routhian procedure 4.7 Symplectic notation 4.8 Problems 5 Canonical transformations; Poisson brackets 5.1 Integrating the equations of motion 5.2 Canonical transformations Contents vii 5.3 Poisson brackets 5.4 The equations of motion in terms of Poisson brackets 5.4.1 Infinitesimal canonical transformations 5.4.2 Canonical invariants 5.4.3 Liouville's theorem 5.4.4 Angular momentum 5.5 Angular momentum in Poisson brackets 5.6 Problems 6 Hamilton-Jacobi theory 6.1 The Hamilton-Jacobi equation 6.2 The harmonic oscillator-an example 6.3 Interpretation of Hamilton's principal function 6.4 Relationship to Schrödinger's equation 6.5 Problems 7 Continuous systems 7.1 A string 7.2 Generalization to three dimensions 7.3 The Hamiltonian density 7.4 Another one-dimensional system 7.4.1 The limit of a continuous rod 7.4.2 The continuous Hamiltonian and the canonical field equations 7.5 The electromagnetic field 7.6 Conclusion 7.7 Problems Further reading Answers to selected problems Index 3 Be aware that the fields of analytical mechanics and the calculus of variations are vast and this book is limited to presenting some fundamental concepts.

We develop a new method for finding the quantum probability density of arrival at the detector. The evolution of the quantum state restricted to the region outside of the detector is described by a restricted Hamiltonian that contains a non-hermitian boundary term. The non-hermitian term is shown to be proportional to the flux of the probability current operator through the boundary, which implies that the arrival probability density is equal to the flux of the probability current.

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and Lie algebroids which is enough general to simultaneously cover several cases of interest in discrete and continuous descriptions as, for instance, Euler-Lagrange equations, Euler-Poincaré equations, Lagrange-Poincaré equations... The construction of an exact discrete Lagrangian is of considerable interest for the analysis of the error between an exact trajectory and the discrete trajectory derived by a variational integrator.

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include as particular cases the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré and discrete Lagrange-Poincaré equations. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.

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.

The spin-1/2 Ising model on the bow-tie lattice is exactly solved by establishing a precise mapping relationship with its corresponding free-fermion eight-vertex model. Ground-state and finite-temperature phase diagrams are obtained for the anisotropic bow-tie lattice with three different exchange interactions along three different spatial directions.

1. Introduction. Under what conditions on a function f : [0, ∞) → [0, ∞) is it the case that for each metric space (X, d), f • d is still a metric, and, moreover, d and f • d are equivalent metrics? It is well-known that for any metric d, d/(1 + d) is a (bounded) metric that is equivalent to d. On the other hand, d/(1+d 2) need not be a metric; see Proposition 2.7. We call f : [0, ∞) → [0, ∞) metric-preserving (respectively, strongly metric-preserving) if for all metric spaces (X, d), f • d is a metric (respectively, is a metric that is topologically equivalent to d). Although the first reference in the literature to the notion of metric-preserving functions seems to be [19], the first detailed study of these functions was by Sreenivasan in 1947 [16]. Kelley's classic text in general topology mentions some of Sreenivasan's results in an exercise [14, p. 131]: Exercise (Kelley). Suppose f : [0, ∞) → [0, ∞) is continuous, nondecreasing, and subadditive (f (x + y) ≤ f (x) + f (y) for all x, y). Suppose also that f −1 (0) = {0}. Then f is strongly metricpreserving. In the past two decades, a significant literature has developed on the subject of metricpreserving functions. The purpose of this paper is to introduce some of the results and techniques of the field to a broader mathematical audience. We begin our study of metric-preserving functions in the next section, where we build tools to understand these functions and consider some revealing examples. Section 3 examines the important relationship between strongly metric-preserving functions and continuity. In the final section, we survey some of the results on differentiability in the context of metric-preserving functions. Many of the results discussed here have been garnered from papers that have appeared in other languages and in journals unfamiliar to the author; special thanks go to Jozef Doboš for his tremendous help in making so many of these papers available to me. Space constraints prevent me from discussing many avenues of research related to metric-preserving functions that have been pursued by various authors; see [8] for an excellent list of references.

1. Introduction. Under what conditions on a function f : [0, ∞) → [0, ∞) is it the case that for each metric space (X, d), f • d is still a metric, and, moreover, d and f • d are equivalent metrics? It is well-known that for any metric d, d/(1 + d) is a (bounded) metric that is equivalent to d. On the other hand, d/(1+d 2) need not be a metric; see Proposition 2.7. We call f : [0, ∞) → [0, ∞) metric-preserving (respectively, strongly metric-preserving) if for all metric spaces (X, d), f • d is a metric (respectively, is a metric that is topologically equivalent to d). Although the first reference in the literature to the notion of metric-preserving functions seems to be [19], the first detailed study of these functions was by Sreenivasan in 1947 [16]. Kelley's classic text in general topology mentions some of Sreenivasan's results in an exercise [14, p. 131]: Exercise (Kelley). Suppose f : [0, ∞) → [0, ∞) is continuous, nondecreasing, and subadditive (f (x + y) ≤ f (x) + f (y) for all x, y). Suppose also that f −1 (0) = {0}. Then f is strongly metricpreserving. In the past two decades, a significant literature has developed on the subject of metricpreserving functions. The purpose of this paper is to introduce some of the results and techniques of the field to a broader mathematical audience. We begin our study of metric-preserving functions in the next section, where we build tools to understand these functions and consider some revealing examples. Section 3 examines the important relationship between strongly metric-preserving functions and continuity. In the final section, we survey some of the results on differentiability in the context of metric-preserving functions. Many of the results discussed here have been garnered from papers that have appeared in other languages and in journals unfamiliar to the author; special thanks go to Jozef Doboš for his tremendous help in making so many of these papers available to me. Space constraints prevent me from discussing many avenues of research related to metric-preserving functions that have been pursued by various authors; see [8] for an excellent list of references.

The nature and topology of time remains an open question in philosophy, both tensed and tenseless concepts of time appear to have merit. A concept of time including both kinds of time evolution of physical systems in quantum mechanics subsumes the properties of both notions. The linear dynamics defines the universe probabilistically throughout spacetime, and can be seen as the definition of a block universe. The collapse dynamics is the time evolution of the linear dynamics, and is thus of different logical type to the linear dynamics. These two different kinds of time evolution are respectively tensed and tenseless. Ascribing tensed semantics to the collapse dynamics is problematic in the light of special relativity, but this difficulty does not apply to a relational quantum mechanics. In this context, while the linear dynamics is the time evolution of the universe objectively, the collapse dynamics is the time evolution of the universe subjectively, applying solely in the functional frame of reference of the observer.

This paper focuses on investigating soliton and other solutions using three integration schemes to integrate a nonlinear partial differential equation describing the wave propagation in nonlinear low-pass electrical transmission lines. By applying the Kirchhoff's laws and complex transformation, the nonlinear low-pass electrical transmission lines are converted into an equation wave propagation in nonlinear ODE low-pass electrical transmission lines. Later on, mentioned integration schemes viz modified Kudryashov method, sine-Gordon equation expansion method and extended sinh-Gordon equation expansion method are used to carry out new hyperbolic and trigonometric solutions which shows the consistency via computerized symbolic computation package maple. Various types of solitary wave solutions are derived including kink, anti-kink, dark, bright, dark-bright, singular, combined singular, and periodic singular wave soliton solutions. The corresponding three integration schemes are robust and effective for acquiring the new dark, bright, dark-bright, singular or combined singular and optical soliton solutions of the wave propagation in nonlinear low-pass electrical transmission lines. To show the real physical significance of the studied equation, some three dimensional (3D) and two dimensional (2D) figures of obtained solutions are plotted with the use of the Matlab software under the proper choice of arbitrary parameters. Moreover, all derived solutions were verified back into its corresponding equation with the aid of maple program.

In our previous papers [J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton–Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1417–1458; Geometric Hamilton–Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 431–454] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (slicing vector fields) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton–Jacobi theory, by considering special cases like fibered ma...

In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H s 0 (T, R) for any s > −1/2 where H s 0 (T, R) denotes the subspace of the Sobolev space H s (T, R) of elements with mean 0. As an application we show that for any −1/2 < s < 0, the flow map of the Benjamin-Ono equation S t 0 : H s 0 (T, R) → H s 0 (T, R) is nowhere locally uniformly continuous in a neighborhood of zero in H s 0 (T, R).

Energy based approaches have proven to be specially well suited for the modeling and control of mechanical systems. Among these approaches the port-Hamiltonian framework presents interesting properties for the structural modeling of complex systems and for the design of non-linear controllers using passivity. In this paper we use this framework to model a typical micro-robotic contact scenario and to propose a simple but effective globally stabilizing controller. The model and the controller take into account the transitions from a non-contact to a contact state (and the inverse) by the introduction of a non-linear (switching) contact element. A one degree of freedom experimental micro-robotic setup is used to test and illustrate the results.

This paper formulates variational integrators for finite element discretizations of deformable bodies with heat conduction in the form of finite speed thermal waves. The cornerstone of the construction consists in taking advantage of the fact that the Green-Naghdi theory of type II for thermo-elastic solids has a Hamiltonian structure. Thus, standard techniques to construct variational integrators can be applied to finite element discretizations of the problem. The resulting discrete-in-time trajectories are then consistent with the laws of thermodynamics for these systems: for an isolated system, they exactly conserve the total entropy, and nearly exactly conserve the total energy over exponentially long periods of time. Moreover, linear and angular momenta are also exactly conserved whenever the exact system does. For definiteness, we construct an explicit second-order accurate algorithm for affine tetrahedral elements in two and three-dimensions, and demonstrate its performance with numerical examples.