Lagrangian Formulation Research Articles

On the adaptation of the Lagrange formalism to continuous time stochastic optimal control: A Lagrange-Chow redux

We show how the classical Lagrangian approach to solving constrained optimization problems from standard calculus can be extended to solve continuous time stochastic optimal control problems. Connections to mainstream approaches such as the Hamilton-Jacobi-Bellman equation and the stochastic maximum principle are drawn. Our approach is linked to the stochastic maximum principle, but more direct and tied to the classical Lagrangian principle, avoiding the use of backward stochastic differential equations in its formulation. Using infinite dimensional functional analysis, we formalize and extend the approach first outlined in Chow (1992) within a rigorous mathematical setting using infinite dimensional functional analysis. We provide examples that demonstrate the usefulness and effectiveness of our approach in practice. Further, we demonstrate the potential for numerical applications facilitating some of our key equations in combination with Monte Carlo backward simulation and linear regression, therefore illustrating a completely different and new avenue for the numerical application of Chow's methods.

Nonlocal Lagrangian fields and the second Noether theorem. Non-commutative U(1) gauge theory

This article focuses on three main contributions. Firstly, we provide an in-depth overview of the nonlocal Lagrangian formalism. Secondly, we introduce an extended version of the second Noether’s theorem tailored for nonlocal Lagrangians. Finally, we apply both the formalism and the extended theorem to the context of non-commutative U(1) gauge theory, including its Hamiltonian and quantization, showcasing their practical utility.

Wave run-up characteristic on armour layer breakwater using the smoothed particle hydrodynamics (SPH) method

Coastal structures, such as breakwater, play a critical role in protecting shorelines from the erosive forces of waves and currents. Understanding the wave run-up phenomenon on these structures is essential for designing effective coastal defense systems. This research aims to investigate the wave run-up characteristics on an armour layer breakwater using the Smoothed Particle Hydrodynamics (SPH) method (Meshless particle method based on the Lagrangian formulation), a numerical method that captures complex fluid-structure interactions. The SPH method’s inherent ability to handle free-surface flows and dynamic interactions makes it particularly suited for simulating wave run-up. The research employs numerical testing and modelling of a non-overtopping breakwater using the smoothed particle hydrodynamics method simulated with the DualSPHysics program. The wave characteristic data used in this research are the waves on the south coast of Java. The results of this research indicate that the correlation between wave run-up characteristics (Ru/H) and the Iribarren number, both using the numerical SPH method and theory, tends to exhibit a similar pattern. Remarkably, the results from both approaches exhibit a striking similarity, revealing that as the Iribarren number increases, the incremental growth of Ru/H diminishes, eventually stabilizing at a critical threshold. The Coefficient of Determination (R2) between wave run-up characteristics (Ru/H) obtained from numerical simulations using the SPH method and Günbak’s theory is 0.8824.

An Eulerian–Lagrangian particle method for weakly compressible viscous flows using peridynamic differential operator

This study proposes a peridynamic differential operator (PDDO)-based Eulerian–Lagrangian hybrid particle method for weakly compressible viscous flows. The PDDO is utilized to transform the governing partial differential equations into their integral form; hence, the issue of local non-differentiability is eliminated. Both the Eulerian and Lagrangian formulations of particle methods for solving the governing integral equations are derived and then combined to propose the hybrid method. In the proposed method, the Eulerian and Lagrangian formulation is utilized to solve the governing equations in the inner computational domain and in areas with free surfaces, respectively. The results of these two domains are then merged to arrive at the solution. In such a way, the hybrid particle method effectively balances the computational demands and applicability. The numerical stability and interpolation consistency of the proposed method are shown by introducing the laminar viscosity model. A detailed numerical procedure is provided involving the boundary conditions and time-stepping strategy. The proposed method is validated by several benchmark problems. Furthermore, the method was applied to several benchmark problems including the hydrostatic test, Taylor–Green vortex, and numerical wave generation. The results of these numerical examples suggest that the proposed method is computationally less demanding while maintaining accuracy.

Nonlinear Deformation of Cylinders from Materials with Different Behavior in Tension and Compression

A new numerical-analytical method for solving physically nonlinear deformation problems of axisymmetrically loaded cylinders made of materials with different behavior in tension and compression has been developed. To linearize the problem, the uninterrupted parameter continuation method was used. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible displacement rates, is constructed. To find the main unknowns of the problem of physically nonlinear cylinder deformation, the Cauchy problem for the system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta-Merson method with automatic step selection. The initial conditions were established by solving the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson’s scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method. The test problem for the nonlinear elastic deformation of a thin cylindrical shell is solved. Coincidence of the spatial solution with the shell solution was obtained. Physically nonlinear deformation of a thick-walled cylinder was studied. It is shown that failure to take into account the different behavior of the material under tension and compression leads to significant errors in the calculations of stress-strain state parameters.

Optimal Vibration Fields in Problems of Modeling Dynamic States of Technical Objects

Introduction. Vibration interaction control is timely in production processes related to liquid and bulk media, systems of solids experiencing kinematic or force disturbances. At the same time, there is no single methodological basis for the formation of vibrational interactions. The issues of constructing optimal vibration fields of technical objects have not been addressed. The objective of the study is to develop a structural approach to the development of mathematical models in the problems of formation, evaluation, and correction of vibration fields of technical objects under conditions of intense force and kinematic loads. The task is to build vibration fields that are optimal in terms of the set of requirements, with the possibility of selecting the criterion of optimality of the vibration field of a technical object.Materials and Methods. A structural approach was used as the basic methodology. It was based on a comparison of mechanical vibratory systems used as computational schemes of technical objects, and structural schemes of automatic control systems, which are equivalent in dynamic terms. Lagrange formalism, elements of operational calculus based on Laplace integral transformations, sections of vibration theories, algebraic methods, and the theory of spline functions were used for structural mathematical modeling.Results. An approach to the selection of criteria for the optimality of vibration fields based on minimizing the residual of vibration fields for various required conditions was proposed. The problem was considered within the framework of a mechanical vibratory system formed by solids. It was shown that the optimal vibration field was determined by an external disturbance and was to satisfy condition Ay̅ = b. There, A — matrix mapping the operator of conditions to the shape of the vibration field at control points; b — vector of values of vibration field characteristics; “–” above y meant the vibration amplitude of the steady-state component of the coordinate. To evaluate the field with account for noisy or unreliable requirements for dynamic characteristics, the smoothing parameter was used, indicating the priority of the criterion of optimality of the vibration field shape. The construction of a field for a mechanical vibratory system showed that the value of the vibration amplitudes of generalized coordinates remained constant when the frequency of external kinematic disturbances changed. Two approaches to the correction of the field optimality criteria were considered: equalization of the vibration amplitudes of the coordinates of a technical object and the selection of an energy operator.Discussion and Conclusion. The development of the applied theory of optimal vibration fields involved, firstly, the correlation of the energy operator and the operator of the requirements for the shape of the vibration field in the theory of abstract splines. The second pair of comparable elements was the criterion of optimality of the vibration field and a system of requirements for the characteristics of the field at control points. The structural theory of optimal vibration fields improved in this way will find application in various industries. Accurate calculations in the formation, assessment, and correction of the states of systems under vibration loading are required in the tasks of increasing the durability of structures, improving measurements in complex vibratory systems, and developing new technologies and materials.

A δ-free approach to quantization of transmission lines connected to lumped circuits

The quantization of systems composed of transmission lines connected to lumped circuits poses significant challenges, arising from the interaction between continuous and discrete degrees of freedom. A widely adopted strategy, based on the pioneering work of Yurke and Denker, entails representing the lumped circuit contributions using Lagrangian densities that incorporate Dirac δ-functions. However, this approach introduces complications, as highlighted in the recent literature, including divergent momentum densities, necessitating the use of regularization techniques. In this work, we introduce a δ-free Lagrangian formulation for a transmission line capacitively coupled to a lumped circuit without the need for a discretization of the transmission line or mode expansions. This is achieved by explicitly enforcing boundary conditions at the line ends in the principle of least action. In this framework, the quantization and the derivation of the Heisenberg equations of the network are straightforward. We obtain a reduced model for the lumped circuit in the quantum Langevin form, which is valid for any coupling strength between the line and the lumped circuit. We apply our approach to an analytically solvable network consisting of a semi-infinite transmission line capacitively coupled to an LC circuit and study the behavior of the network as the coupling strength varies.

Sensitivity of polarizations and spin correlations of Z boson to anomalous neutral triple gauge couplings at lepton collider with polarized beams

We investigate the effects of anomalous neutral triple gauge couplings in ZZ and Zγ production processes, followed by the leptonic decay of the Z boson, at a lepton collider with center-of-mass energy s=250 GeV and polarized beams. We use an effective Lagrangian formalism to parametrize the anomalous couplings in terms of dimension-8 operators cB˜W, cBW, cWW, and cBB, and study the sensitivity of observables such as cross section, polarization, and spin correlation as functions of these couplings. We perform a Bayesian statistical analysis using Markov Chain Monte Carlo methods to determine simultaneous limits on the anomalous couplings, taking into account various luminosities L∈{0.1 ab−1,0.3 ab−1,1 ab−1,3 ab−1,10 ab−1} and systematic uncertainties. We find that polarization and spin correlation observables significantly enhance the sensitivity to anomalous couplings, providing stringent constraints on these couplings. Published by the American Physical Society 2024

Berthing proximity maneuvers and trajectory planning of a space manipulator via Kane’s formulation

On-orbit servicing is one of the most challenging tasks of the recent space era. Space Manipulator Systems (SMS) can represent viable technological solutions for several space operations, such as in-situ refueling, repairing of operating spacecraft, or debris grappling. Regardless of the specific task, SMS must approach the target object and then establish a physical connection with it using a single or more robotic arms. This work considers a space vehicle equipped with a pair of two-link robotic arms, with the intent of efficiently designing and simulating the overall spacecraft dynamics until contact takes place. The links that compose each arm are modeled as rigid bodies, assuming relatively slow dynamics. Instead, flexibility is introduced in the model of the rotary joints. The overall dynamics is investigated using Kane’s methodology, which is general, highly systematic, and joins the advantages of both the Newton/Euler and the Lagrange formulation. Both SMS and the target, assumed as noncooperative, are placed in low Earth orbit, in close proximity. Trajectory planning of the robotic arm is investigated, while considering the effect of the arm motion on the overall attitude dynamics. Precise attitude maneuvering is mandatory in order that grappling be successfully completed. To do this, a quaternion-based, globally stable attitude feedback law is designed, implemented, and tested. Actuation is demanded to an arrangement of control momentum gyroscopes. These devices are modeled with the inclusion of gyroscopic coupling effects, and are driven by means of suitable steering laws, for the purpose of pursuing the desired attitude control action. Accurate modeling of SMS, with also the inclusion of disturbances due to joint flexibility, in conjunction with the use of suitable feedback attitude control and steering laws for both the robotic arms and the actuation devices, allows obtaining the desired variables, i.e. (i) the motor torques (for steering both the joints and the attitude devices) and (ii) the constraint actions (forces and torques). A Monte Carlo analysis points out effectiveness of the control architecture proposed in this study in the challenging mission scenario of interest.

Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering

Abstract This research article analytically investigates a soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences and engineering. The soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences along with engineering, is examined with a view to securing various pertinent results of interest. For the first time, the conserved currents of an integrodifferential equation (especially those of higher dimensions) are calculated using a detailed optimal system of one-dimensional subalgebras. Infinitesimal generators of diverse structures ascribed to Lie point symmetries of the understudy model are first calculated via Lie group analysis technique. Additionally, we construct various commutations along Lie-adjoint representation tables connected to the nine-dimensional Lie algebra achieved. Further to that, detailed and comprehensive computation of the optimal system of one-dimensional subalgebras linked to the algebra is also unveiled for the under-investigated model. This, in consequence, engenders the calculation of abundant conserved currents for the soliton equation through Ibragimov’s conserved vector theorem by utilizing its formal Lagrangian. Later, the applications of our results are highlighted.

Soliton dynamics and stability in the ABS spinor model with a PT -symmetric periodic complex potential

We investigate the effects on solitons dynamics of introducing a PT -symmetric complex potential in a specific family of the cubic Dirac equation in (1+1)-dimensions, called the Alexeeva–Barashenkov–Saxena model. The potential is introduced taking advantage of the fact that the nonlinear Dirac equation admits a Lagrangian formalism. As a consequence, the imaginary part of the potential, associated with gains and losses, behaves as a spatially periodic damping (changing from positive to negative, and back) that acts at the same time on the two spinor components. A collective coordinates (CCs) theory is developed by making an ansatz for a moving soliton where the position, rapidity, momentum, frequency, and phase are all functions of time. We consider the complex potential as a perturbation and verify that numerical solutions of the equation of motions for the CCs are in agreement with simulations of the nonlinear Dirac equation. The main effect of the imaginary part of the potential is to induce oscillations in the charge and energy (they are conserved for real potentials) with the same frequency and phase as the momentum. We find long-lived solitons even with very large charge and energy oscillations. Additionally, we extend to the nonlinear Dirac equation an empirical stability criterion, previously employed successfully in the nonlinear Schrödinger equation.

Computing Nash Equilibria in Potential Games with Private Uncoupled Constraints

We consider the problem of computing Nash equilibria in potential games where each player's strategy set is subject to private uncoupled constraints. This scenario is frequently encountered in real-world applications like road network congestion games where individual drivers adhere to personal budget and fuel limitations. Despite the plethora of algorithms that efficiently compute Nash equilibria (NE) in potential games, the domain of constrained potential games remains largely unexplored. We introduce an algorithm that leverages the Lagrangian formulation of NE. The algorithm is implemented independently by each player and runs in polynomial time with respect to the approximation error, the sum of the size of the action-spaces, and the game's inherit parameters.

An Integrated Lagrangian Modeling Method for Mechanical Systems with Memory Elements

Mechanical memory elements cannot be accurately modeled using the Lagrangian method in the classical sense, since these elements are nonconservative in the plane of their non-constitutive relationships, and the system differential equations are not self-adjoint and therefore do not allow a Lagrangian formulation. To overcome this problem, the integrated Lagrangian modeling method is introduced, in which the associated conventional energies in the system are replaced by the corresponding memory state functions of the memory elements. An example, a vehicle shimmy system equipped with fluid mem-inerters, is presented to verify the improvement of modeling accuracy of mechanical systems with memory elements via the integrated Lagrangian method. The simulation results show that under pulse and random excitation, using the Lagrangian method to model the system, the values of system response indicators exhibit significant errors ranging from 5.17% to 24.54% compared with the values obtained by the integrated Lagrangian method, namely, the accurate values. In addition, the influencing factors of the error and are discussed and the fractional-order memory elements and their modeling are also briefly generalized.

Structural, topological, vibrational, and electronic analysis, and ADMET study of methyl-2-(4-isobutylphenyl)propanoate

In this work, the methyl ibuprofen ester, named methyl-2-(4-isobutylphenyl)propanoate (C14H20O2, hereinafter MET-IBU).was synthesized and characterized by 1D- and 2D-NMR, ATR-FTIR, FT-Raman, and UV–Vis spectroscopy, while their structural, QTAIM topological and spectroscopic properties were investigated using DFT calculations. Through the analysis of the topological properties of the electron density computed at bonding critical points (BCP's) for MET-IBU molecule were determined the electron density, the Laplacian of electron density, the eigenvalues of Hessian matrix, the ellipticity of electron density, the potential energy density, the kinetic energy density in the Lagrangian and Hamiltonian forms, the total energy density, the Lagrangian density, and the electron delocalization index. Density Functional Theory (DFT) calculations were also used to determine the vibrational normal modes, the frontier molecular orbitals (FMOs), the dipole moment, and the molecular electrostatic potential (MEP) of MET-IBU. Theoretical UV absorption spectrum was also evaluated from time-dependent density functional theory (TD-DFT) providing the electronic transitions, which are expected for this compound. The molecular descriptors of the properties of absorption, distribution, metabolism and excretion, and toxicity (ADMET) were obtained. Empirical models for estimating ADME attributes showed that the MET-IBU has a distribution volume that allows permeability and activity in the Central Nervous System - CNS, with structural contributions from the side chain of isobutyl and methoxy that favor liver metabolism and excretion with a low toxic response. Furthermore, the MET-IBU does not have ionizable centers compared to its analog Ibuprofen constituting a more lipophilic chemical product.

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

Coupled heat and water transfer in heterogeneous and deformable soils: Numerical model using mixed finite element method

We present a generic model framework for coupled heat and water transfer (CHWT) in deformable (non-rigid) soils with spatial variations of soil properties. The model backbone is a mixed finite element method (FEM), which solves the Philip and de Vries (1957) CHWT model and achieves conservation of mass and energy on both local and global scales. Spatial variations occur in soil hydraulic and thermal properties due to transient water content and temperature distributions. Based on the mixed FEM scheme, a gradient measure and a clustering model (“k-means”) are proposed to trace the regions with large spatial variations of soil properties, and an adaptive mesh refinement technique is developed to improve the spatial resolution and simulation accuracy. Deformation perturbates local soil topography and alters transient soil water and temperature regimes in the deformed regions. A quasi-static deformation model is presented, and the deformation effects are incorporated into the mixed FEM scheme. When external load exists, soil deformation is simulated with an updated Lagrangian formulation, and the local water content and temperature variations due to soil volume changes are also updated in the CHWT model. Numerical examples, including thermally induced soil water transfer and water infiltration, illustrate the model ability to provide plausible CHWT results, especially the refined solutions near the wetting fronts and the water content and temperature distributions when the soil is deformable. In conclusion, the proposed model framework provides an effective pipeline to incorporate and process the spatial variations of soil properties and soil deformation in CHWT simulations.

Every finite graph arises as the singular set of a compact 3‐D calibrated area minimizing surface

AbstractGiven any (not necessarily connected) combinatorial finite graph and any compact smooth 6‐manifold with the third Betti number , we construct a calibrated 3‐dimensional homologically area minimizing surface on equipped in a smooth metric , so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly twisted special Lagrangian form. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.

Cell-centered indirect Arbitrary Lagrangian-Eulerian numerical strategy for solving 3D gas dynamics equations

Solving the Euler equations under the Lagrangian formalism enables to simulate various complex engineering applications. However, the use of this formalism can lead to significant mesh deformations as the mesh follows the fluid velocity. The mesh quality may be considerably deteriorated requiring a regularization procedure. In the present document, it is shown that the ideas presented by J. Yao (2013)[68] and J. Yao and D. Stillman (2016) [30] may be used efficiently in a 3D hydrodynamics code to perform reliable regularization steps. The flexibility of the methodology in addition to its simplicity, since it only relies on trivial geometrical considerations, opens the way for various extensions. The remapping step then considered, uses the geometrical splitting procedure of the Lagrangian phase to perform effective projections. This coherence ensures the compactness of the overall algorithm. At last but not least, a 3D Flux-Corrected-Remapping method is presented. This yields a particularly robust remapping algorithm while also leading the way for higher order projection extensions.

Dynamics modeling and path following controller of tractor-trailer-wheeled robots considering wheels slip

To improve the payload capacity and maneuverability of a Differentially-Driven Wheeled Robot (DDWR), a wheeled vehicle, which is called trailer, is connected to the DDWR. In all of the previous studies of DDWRs with a trailer, the robot wheels are subject to pure rolling constraints. However, when these multibody systems move with high velocities/accelerations, transfer a heavy payload, or travel on a slippery surface, they experience slipping and/or skidding. In the present study, a Tractor-Trailer Wheeled Mobile Robot (TTWMR) is modeled whose wheels may slip in lateral and longitudinal directions. To this end, Lagrange formalism is employed along with the LuGre friction model to derive dynamics of the considered multibody system. Next, the problem of path following for the trailer is addressed. Toward this goal, the partial feedback linearization technique will be utilized. The obtained simulation results prove the superiority of the performance of the suggested method in comparison to the previous studies. Additionally, the response of the system in the presence of the external disturbances and uncertainties in system parameters will be examined.

Non-Linear Behavior of Double-Layered Grids

Abstract This study presents a numerical approach to an analysis of the mechanical behavior of double-layered tensegrity grids. We present a comparative study on the behavior of tensegrity grids through geometric nonlinear analysis (GNA) and combined nonlinear analysis (CNLA) (geometric and material), considering the possible effect of evolution in the elasto-plastic domain of the cable elements. The effect of the relaxation of cable on the amplification of the displacement of these grids was taken into account. The updated Lagrangian formulation, which modifies the Newton-Raphson iterative scheme with incremental loading, was adopted. We have developed a numerical computational model specific to tensegrity structures that simulates the geometric and material nonlinear behavior. The reliability of the calculation tool developed has been validated. Additionally, the results of the application of the numerical model on a grid, which was generated based on demi-cuboctahedral tensegrity cells, are presented.