Variational method for nonconservative field theories: Formulation and twoPT-symmetric case examples

Among the problems C. D. Bailey has questioned in a recent paper (Ref. 1) are a precise and general formulation of Hamilton's variational principle and the establishment of a sufficiency criterion for this to be a minimum principle. In this paper, we will try to answer these questions using the geometric theory of classical mechanics.

This chapter presents extended forms of Hamilton's Principle and the phase space Hamilton's principle based on the extended Lagrangian and Hamiltonian methods. Hamilton's Principle has already been treated in the context of traditional Lagrangian and Hamiltonian mechanics. Noether's theorem, a method for using symmetries of the extended Lagrangian to identify quantities that are conserved during the motion of the system, is also presented here. Noether's theorem is a powerful technique for discovering conserved quantities in complex Lagrangian systems. The chapter presents the basics of the method in the simple context of Lagrangian systems with a finite number of degrees of freedom. Both the traditional and the extended Hamilton's Principles are an application of the calculus of variations to mechanics. The traditional Hamilton's Principle uses the coordinate parametric method, while the extended Hamilton's Principle uses the general parametric method.

Design of auxetic plates with only one degree of freedom

The stochastic thermodynamics of systems with a few degrees of freedom has been studied extensively so far. We would like to extend the study to systems with more degrees of freedom and even further-continuous fields with infinite degrees of freedom. The simplest case for a continuous stochastic field is the Edwards-Wilkinson elastic manifold. It is an exactly solvable model of which the heat statistics in the relaxation process can be calculated analytically. The cumulants require a cutoff spacing to avoid ultraviolet divergence. The scaling behavior of the heat cumulants with time and the system size as well as the large deviation rate function of the heat statistics in the large size limit is obtained.

Quantum effects in non-inertial frames and curved space-time have been studied for decades. One of the most intriguing discoveries is the existence of virtual particles in the quantum vacuum. One example in which such particles can be observed is through the Unruh effect, which predicts that a uniformly accelerated observer would see a thermal bath of (virtual) particles in the quantum vacuum. Initial doubts on the existence of these virtual particles were allayed by the Unruh De-Witt (UDW) detector, which promoted these virtual particles to real excitations of the detector. Ever since, semiclassical approaches to relativistic quantum field theory (QFT) in flat and curved space-time have been predominantly analysed through a two-level Unruh De-Witt detector utilizing perturbation theory. It is noted, however, that the change of relativistic frame preserves Gaussian properties (e.g. Gaussian distribution of the Wigner function) of the system. In this thesis, we specifically explore the effects of relativistic (non-inertial) reference frames on QFT by considering relativistic QFT in flat-space time. These systems are analysed from the perspective of Gaussian quantum information, via homodyne detection and UDW detectors with a harmonic oscillator degree of freedom.Our project starts by reviewing and developing bipartite/multi-partite entanglement measures (specifically, entanglement of formation), in the hope that we can utilize these measures to understand the entanglement properties of the quantum vacuum. In general, these measure have infinite degrees of freedom and are incomputable. In this project the multi-partite entanglement measure is applied to the Gaussian regime, which reduces the number of degrees of freedom to a finite one, hence making it a computable measure.In our next project, we attempt to understand how relativistic reference frames affect the communication between different observers. To obtain an intuition of its effect, a passive (time-delay) signal sent from a uniformly accelerated observer to a stationary observer was analysed. When we naively implemented the self-homodyne detection scheme, the signal appeared noisy. This effect was referred to as apparent decoherence, and its effect could be traced back to our naive assumption that all of the signal’s information would be stored in the time-delayed mode; the vacuum entanglement that pre-existed before the (time-delay) signal was created must be accounted for. We follow by analysing the effects that emerge due to communication between different relativistic reference frames. We then develop a communication technique referred to as ideal-homodyne which is a homodyne detection scheme robust to these effects. We used this detection scheme to analyse the correlation/entanglement properties of an accelerated signal (created via accelerated mirror, squeezer and phase shifter) sent to an inertial observer.In our last project, we utilize the knowledge that was gained through the prior projects to analyse (subcycle) electro-optic sampling. In the literature, there were doubts as to whether electro-optic sampling were truly detecting virtual particles in the quantum vacuum, or particle excitations that were created as a by-product of the detection mechanism. We were able to pin-point some similarities between electro-optic sampling and subcycle probe via the UDW detector. The regime in which the electro-optic sampling directly maps virtual particles from the vacuum into real excitations of the probe field was identified by establishing an equivalence to the UDW detector in certain parameter regimes.

We review basic aspects of the mathematical theory of abstract supersymmetric quantum mechanics, which can be applied to supersymmetric quantum mechanics both with finite degrees of freedom and with infinite degrees of freedom, including supersymmetric quantum field theory.

In the types of vibration considered in previous chapters it has always been possible to assume, without introducing undue errors, that the system consisted of one or more rigid bodies connected by massless elastic elements. Following from this it became relatively straightforward to formulate the equations of motion in terms of the chosen displacement co-ordinates, and to solve for the natural frequencies according to the number of degrees of freedom. For various reasons this approach cannot be applied directly to the vibrations of beams and shafts. Both mass and elasticity are generally ‘distributed’ along the length and mathematically there are then infinite degrees of freedom. Even when ‘point’ masses predominate and the mass of the beam is neglected it is not possible (except in the single-mass case) to write down the restoring force on any mass in terms of its displacement at any instant. In consequence, ‘approximate’ methods have been developed for general use, ‘exact’ analysis being applied only to a few standard cases. These approximate methods are usually limited to the determination of the fundamental (that is, first mode) frequency which, as in most multi-degree-of-freedom vibrations, is of prime practical importance.

In this paper, for studying the influence of the randomness of structural parameters of high-speed elevator lifting system (HELS) caused by manufacturing error and installation error, a continuous time-varying model of HELS was constructed, considering the compensation rope mass and the tension of the tensioning system. The Galerkin weighted residual method is employed to transform the partial differential equation with infinite degrees of freedom (DOF) into the ordinary differential equation. The five-order polynomial is used to fit the actual operation state curve of elevator, and input as operation parameters. The precise integration method of time-varying model of HELS is proposed. The determination part and the random part response expression of the longitudinal dynamic response of HELS are derived by the random perturbation method. Using the precise integration method, the sensitivity of random parameters is determined by solving the random part response expression of time-varying model of HELS, and the digital characteristics of the acceleration response are analyzed. It is found that the line density of the hoisting wire rope has the maximum sensitivity on longitudinal vibration velocity response, displacement response and acceleration response, and the sensitivity of the elastic modulus of the wire rope is smallest.

Planar soft space robotic manipulators: Dynamic modeling and control

Toward the quantitative theory of the instanton liquid (I).: Phenomenology and the method of collective coordinates

Non-universal finite size effects with universal infinite-size free energy

In this chapter, continuous body models (i.e. systems with infinite degrees of freedom, governed by partial differential equations, functions of both time and space) are introduced. As is known, closed form solutions can only be obtained in some simple applications. For this reason, the study was conducted mainly for educational purposes, in preparation for the discretization methods described in the following chapter. In particular, vibrations in continuous systems, using both the direct (the “propagative” solution) and modal approach in principal coordinates (the “stationary” solution), have been studied. At the end of the chapter, the bending vibrations of cables, beams and “taut beams” are shown as examples.

The experimental positioning of a flexible arm by a human operator is described. The flexible arm is characterized by nonlinearity and infinite degrees of freedom. The human operator interprets a given task and finds a procedure for the operation, devising an effective way of achieving the goal, based on his or her experience and knowledge about the task. The question of how these results can be modeled and applied to automatic control systems is considered, and the use of a reference input filter is proposed. The filter gives a satisfactory overall response if the time constant is chosen appropriately.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A notion of stochastic deformation is introduced and the corresponding algebraic deformation procedure is developed. This procedure is analogous to the deformation of an algebra of observables like deformation quantization, but for an imaginary deformation parameter (the Planck constant). This method is demonstrated on diverse relativistic and nonrelativistic models with finite and infinite degrees of freedom. It is shown that under stochastic deformation the model of a nonrelativistic particle interacting with the electromagnetic field on a curved background passes into the stochastic model described by the Fokker-Planck equation with the diffusion tensor being the inverse metric tensor. The first stochastic correction to the Newton equations for this system is found. The Klein-Kramers equation is also derived as the stochastic deformation of a certain classical model. Relativistic generalizations of the Fokker-Planck and Klein-Kramers equations are obtained by applying the procedure of stochastic deformation to appropriate relativistic classical models. The analog of the Fokker-Planck equation associated with the stochastic Lorentz-Dirac equation is derived too. The stochastic deformation of the models of a free scalar field and an electromagnetic field is investigated. It turns out that in the latter case the obtained stochastic model describes a fluctuating electromagnetic field in a transparent medium.

Soft robotics is an area where the robots are designed by using soft and compliant modules which provide them with infinite degrees of freedom. The intrinsic movements and deformation of such robots are complex, continuous and highly compliant because of which the current modelling techniques are unable to predict and capture their dynamics. This paper describes a machine learning based actuation and system identification technique to discover the governing dynamics of a soft bodied swallowing robot. A neural based generator designed by using Matsuoka’s oscillator has been implemented to actuate the robot so that it can deliver its maximum potential. The parameters of the oscillator were found by defining and optimising a quadratic objective function. By using optical motion tracking, time-series data was captured and stored. Further, the data were processed and utilised to model the dynamics of the robot by assuming that few significant non-linearities are governing it. It has also been shown that the method can generalise the surface deformation of the time-varying actuation of the robot.