Academic literature on the topic 'Post-Hamiltonian systems'

In relativistic celestial mechanics, post-Newtonian (PN) Lagrangian and PN Hamiltonian formulations are not equivalent to the same PN order as our previous work in PRD (2015). Usually, an approximate Lagrangian is used to discuss the difference between a PN Hamiltonian and a PN Lagrangian. In this paper, we investigate the dynamics of compact binary systems for Hamiltonians and Lagrangians, including Newtonian, post-Newtonian (1PN and 2PN), and spin–orbit coupling and spin–spin coupling parts. Additionally, coherent equations of motion for 2PN Lagrangian are adopted here to make the comparison with Hamiltonian approaches and approximate Lagrangian approaches at the same condition and same PN order. The completely opposite nature of the dynamics shows that using an approximate PN Lagrangian is not convincing. Hence, using the coherent PN Lagrangian is necessary for obtaining an exact result in the research of dynamics of compact binary at certain PN order. Meanwhile, numerical investigations from the spinning compact binaries show that the 2PN term plays an important role in causing chaos in the PN Hamiltonian system.

The equations of motion for a Lagrangian mainly refer to the acceleration equations, which can be obtained by the Euler–Lagrange equations. In the post-Newtonian Lagrangian form of general relativity, the Lagrangian systems can only maintain a certain post-Newtonian order and are incoherent Lagrangians since the higher-order terms are omitted. This truncation can cause some changes in the constant of motion. However, in celestial mechanics, Hamiltonians are more commonly used than Lagrangians. The conversion from Lagrangianto Hamiltonian can be achieved through the Legendre transformation. The coordinate momentum separable Hamiltonian can be computed by the symplectic algorithm, whereas the inseparable Hamiltonian can be used to compute the evolution of motion by the phase-space expansion method. Our recent work involves the design of a multi-factor correction map for the phase-space expansion method, known as the correction map method. In this paper, we compare the performance of the implicit algorithm in post-Newtonian Lagrangians and the correction map method in post-Newtonian Hamiltonians. Specifically, we investigate the extent to which both methods can uphold invariance of the motion’s constants, such as energy conservation and angular momentum preservation. Ultimately, the results of numerical simulations demonstrate the superior performance of the correction map method, particularly with respect to angular momentum conservation.

This research relates to a numerical integrator with post-stabilization of several constraints for an autonomous dynamical system. A generally analytical approach shows that the total energy correction is not valid in most cases, while post-stabilization of each independent energy is. As a typical test example, we consider a non-integrable Hamiltonian system of three degrees of freedom, which can be split into two independent pieces, one 1D harmonic oscillator and another 2D non-integrable system, by using a transformation of variables. Phase portraits on Poincaré sections about the 2D system manifest that our analysis is reasonable. In addition, a problem how to compute Lyapunov exponents in constrained systems is proposed. As a suggestion, it is best to stabilize all constraints involving each energy integral and its corresponding variation in order to avoid spurious Lyapunov exponents. Because an appropriately larger time step is acceptable in this sense, it is not expensive to use the fast Lyapunov indicators to describe the global dynamics of phase space for the 3D system, where regions of chaos and order are clearly identified.

Abstract We give a pedagogical introduction to the generalized hydrodynamic approach to inhomogeneous quenches in integrable many-body quantum systems. We review recent applications of the theory, focusing in particular on two classes of problems: bipartitioning protocols and trap quenches, which represent two prototypical examples of broken translational symmetry in either the system initial state or post-quench Hamiltonian. We report on exact results that have been obtained for generic time-dependent correlation functions and entanglement evolution, and discuss in detail the range of applicability of the theory. Finally, we present some open questions and suggest perspectives on possible future directions.

ABSTRACT Since the first detection of gravitational waves by the LIGO/VIRGO team, the related research field has attracted more attention. The spinning compact binaries system, as one of the gravitational-wave sources for broad-band laser interferometers, has been widely studied by related researchers. In order to analyse the gravitational wave signals using matched filtering techniques, reliable numerical algorithms are needed. Spinning compact binaries systems in post-Newtonian (PN) celestial mechanics have an inseparable Hamiltonian. The extended phase-space algorithm is an effective solution for the problem of this system. We have developed correction maps for the extended phase-space method in our previous work, which significantly improves the accuracy and stability of the method with only a momentum scale factor. In this paper, we will add more scale factors to modify the numerical solution in order to minimize the errors in the constants of motion. However, we find that these correction maps will result in a large energy bias in the subterms of the Hamiltonian in chaotic orbits, whose potential and kinetic energy, etc. are calculated inaccurately. We develop a new correction map to reduce the energy bias of the subterms of the Hamiltonian, which can instead improve the accuracy of the numerical solution and also provides a new idea for the application of the manifold correction in other algorithms.

Transient stability analysis and control of power systems with considering flux decay by energy function approach In this paper, transient stability of power systems with structure preserving models is considered. A Hamiltonian function which can be regarded as a Lyapunov function for the system is proposed. Based on this, the influence of flux decay dynamics, especially during a fault, on transient stability is analyzed. With the increase of load power, the variation of stability boundary in the rotor angle/E'q plane is shown. The Energy-based excitation control, aiming at injecting additional damping into the post-fault system may reduce the critical clearing time (CCT). This can be demonstrated by the comparison of different flux decay dynamics in the fault-on condition, and the reason is illustrated by the relationship between rotor angle/E'q and the stability boundary. An improved control strategy is proposed and applied to increase the CCT. Simulation results verify that improvement is obtained both in transient stability and dynamic performance.

Cette thèse porte sur la modélisation du haut-parleur électrodynamique, son émulation et son contrôle. Pour la modélisation, on adopte une approche par composants s’appuyant sur les systèmes hamiltoniens à ports. Plusieurs phénomènes, linéaires ou non, sont ainsi traités puis agrégés dans un cadre multi-physique. Une attention particulière est portée à l’impact des effets thermiques sur les composants électrique et mécanique, pour lesquels on introduit des modèles conservatifs irréversibles originaux. Les simulations montrent que des comportements complexes connus sont régénérés. Un premier contrôle en boucle ouverte est élaboré pour supprimer les distorsions, par platitude différentielle. Afin de fournir au correcteur les paramètres des non-linéarités du modèle, une méthode d’estimation ad hoc est proposée. Celle-ci combine une séparation de la mesure en sous-signaux (organisés par ordre homogène de non-linéarité) et l’optimisation d’une fonction de coût (renforçant le contraste entre les ordres). Après validation numérique, les méthodes d’estimation et de contrôle sont appliquées sur un banc de test. Les paramètres physiques estimés sont cohérents mais les signaux temporels re-simulés indiquent la nécessité d’améliorer le modèle en très basse fréquence et de recourir à des ordres homogènes élevés.Le correcteur temps réel conduit à une réduction mesurable des distorsions sur la pression acoustique. En complément, un autre contrôle en boucle ouverte est développé pour compenser l’effet Doppler dû au mouvement de la membrane. Enfin, des méthodes sur le contrôle en boucle fermée sont proposées. L’une cible l’absorption acoustique en combinant « loi de contrôle en temps fini » (pour l’efficacité) et « passivité » (pour la robustesse). L’autre, plus générale, élabore un procédé de connexion « mi-physique, mi-numérique » entre un système physique et un contrôleur numérique qui rend la passivité insensible au retard introduit par le calculateur temps réel
This thesis concerns electrodynamic loudspeaker modeling, simulation and control. Regarding modeling, we adopt a component-based approach that relies on port-Hamiltonian systems. Several linear and nonlinear phenomena are thus modeled and then aggregated in a multi-physical framework. Particular attention is paid to the impact of thermal effects on electrical and mechanical components, for which we introduce new irreversible conservative models. The simulations regenerate known complex behaviors. A first open-loop control is developed to eliminate distortions by differential flatness. In order to provide the controller with the model's nonlinearity parameters, an ad hoc estimation method is proposed. This combines a separation of the measurement into sub-signals (organized in a homogeneous order of non-linearity) and the optimization of a cost function (improving the contrast between orders). After numerical validation, estimation and control methods are applied on a test bench. The estimated physical parameters are consistent but the re-simulated time signals indicate the need of improvement of the model at very low frequency and to use higher homogeneous orders. The real-time corrector leads to a measurable reduction of the distortions on the sound pressure. In addition, another open-loop control is developed to compensate for the Doppler effect due to the movement of the membrane. Finally, methods on closed-loop control are proposed. One targets acoustic absorption by combining "control law in finite time" (for efficiency) and "passivity" (for robustness). The other, more general, develops an "half-physical, half-digital" method of connection between a physical system and a digital controller that makes the passivity insensitive to the delay introduced by the digital signal processor

In the preceding chapters, the theory for calculations based on the Dirac equation has been laid out in some detail. The discussion of the methods included a comparison with equivalent nonrelativistic methods, from which it is apparent that four-component calculations will be considerably more expensive than the corresponding nonrelativistic calculations—perhaps two orders of magnitude more expensive. For this reason, there have been many methods developed that make approximations to the Dirac equation, and it is to these that we turn in this part of the book. There are two elements of the Dirac equation that contribute to the large amount of work: the presence of the small component of the wave function and the spin dependence of the Hamiltonian. The small component is primarily responsible for the large number of two-electron integrals which, as will be seen later, contain all the lowest-order relativistic corrections to the electron–electron interaction. The spin dependence is incorporated through the kinetic energy operator and the correction to the electronic Coulomb interaction, and also through the coupling of the spin and orbital angular momenta in the atomic 2-spinors, which form a natural basis set for the solution of the Dirac equation. Spin separation has obvious advantages from a computational perspective. As we will show for several spin-free approaches below, a spin-free Hamiltonian is generally real, and therefore real spin–orbitals may be employed for the large and small components. The spin can then be factorized out and spin-restricted Hartree–Fock methods used to generate the one-electron functions. In the post-SCF stage, where the no-pair approximation is invoked, the transformation of the integrals from the atomic to the molecular basis produces a set of real molecular integrals that are indistinguishable from a set of nonrelativistic MO integrals, and therefore all the nonrelativistic correlation methods may be employed without modification to obtain relativistic spin-free correlated wave functions. In most cases, spin–free relativistic effects dominate the relativistic corrections to electronic structure. We will show later that in a perturbation expansion based on the nonrelativistic wave function, the spin-free effects for a closed-shell system enter in first order, whereas the spin-dependent effects make their first contribution in second order.