Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

Quasi-optimal synthesis of an adaptive filter in the problem of estimating the state of dynamic systems

The problem of synthesis of filters to estimate the state of dynamical systems is considered based on the condition for the maximum of the generalized power function and stationarity of the generalized Lagrangian and Hamiltonian of the estimated system model. The paper demonstrates that the use of invariants in combination with the decomposition principle makes it possible to simplify the equations of controlled motion and reduce them to a system of independent equations in terms of the number of degrees of freedom. This approach reduces the number of unknown parameters of the motion model, which greatly simplifies the adaptation process when developing filters for quasi-optimal estimation of the state parameters of dynamic systems. Comparative analysis of the results of the mathematical simulation shows that the application of the proposed method increases the efficiency of filters of the Kalman structure.

Role of Quantum Entropy and Establishment of H-Theorems in the Presence of Graviton Sinks for Manifestly-Covariant Quantum Gravity

Based on the introduction of a suitable quantum functional, identified here with the Boltzmann–Shannon entropy, entropic properties of the quantum gravitational field are investigated in the framework of manifestly-covariant quantum gravity theory. In particular, focus is given to gravitational quantum states in a background de Sitter space-time, with the addition of possible quantum non-unitarity effects modeled in terms of an effective quantum graviton sink localized near the de Sitter event horizon. The theory of manifestly-covariant quantum gravity developed accordingly is shown to retain its emergent-gravity features, which are recovered when the generalized-Lagrangian-path formalism is adopted, yielding a stochastic trajectory-based representation of the quantum wave equation. This permits the analytic determination of the quantum probability density function associated with the quantum gravity state, represented in terms of a generally dynamically-evolving shifted Gaussian function. As an application, the study of the entropic properties of quantum gravity is developed and the conditions for the existence of a local H-theorem or, alternatively, of a constant H-theorem are established.

The Inverse Problem of Linear Lagrangian Dynamics

A comprehensive study is reported herein for the evaluation of Lagrangian functions for linear systems possessing symmetric or nonsymmetric coefficient matrices. Contrary to popular beliefs, it is shown that many coupled linear systems do not admit Lagrangian functions. In addition, Lagrangian functions generally cannot be determined by system decoupling unless further restriction such as classical damping is assumed. However, a scalar function that plays the role of a Lagrangian function can be determined for any linear system by decoupling. This generalized Lagrangian function produces the equations of motion and it contains information on system properties, yet it satisfies a modified version of the Euler–Lagrange equations. Subject to this interpretation, a solution to the inverse problem of linear Lagrangian dynamics is provided.