Lagrangian Solution Research Articles

Weak stability of Lagrangian solutions to the semigeostrophic equations

In (Cullen and Feldman 2006 SIAM J. Math. Anal. 37 137–95), Cullen and Feldman proved the existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in Lp, p > 1. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in L1 converges strongly in Lq, q < ∞, to a Lagrangian solution, in particular extending the existence result of Cullen and Feldman to the case p = 1. We also present a counterexample for Lagrangian solutions corresponding to a sequence of initial potential vorticities converging in . The analytical tools used include techniques from optimal transportation, Ambrosio's results on transport by BV vector fields and Orlicz spaces.

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.

Gravitational Wave Forms for Two- and Three-Body Gravitating Systems

Different numbers of self-gravitating particles (in different types of periodic motion) are most likely to generate very different shapes of gravitational waves, some of which, however, can be accidentally almost the same. One such example is a binary and three-body system for Lagrange's solution. To track the evolution of these similar wave forms, we define a chirp mass to the triple system. Thereby, we show that the quadrupole wave forms cannot distinguish the sources. It is suggested that wave forms with higher lth multipoles will be important for classification of them (with a conjecture of l<or=N for N particles).

Heterogeneous gas chemistry in the CPFD Eulerian–Lagrangian numerical scheme (ozone decomposition)

Heterogeneous catalytic chemistry is used throughout the chemical and petro-chemical industry. In predicting the performance of a reactor, knowing the gases and solids flow dynamics is as important as having good chemical rate expressions. This paper gives the solution of ozone decomposition in a bubbling bed using the CPFD numerical scheme which is a Eulerian–Lagrangian solution method for fluid–solid flows. The ozone decomposition can be described by a single stoichiometric equation and has a first order reaction rate. The ozone decomposition is a standard problem for chemical analysis and has been used to characterize gas–solid contacts in fluidized beds. The accuracy of predicting the ozone decomposition comes from correctly predicting the bed dynamics. The solution in this study is three-dimensional and predicts the coupled motion of both solids and gas. The chemical rate equation uses solids volume fraction, but the numerical method could calculate chemistry on the discrete catalyst, including a variation in size (surface area) if such a rate equation was available. The numerical results compare well with an analytic solution of the decomposition rate, and calculated results compare well with the experiment by Fryer and Potter [Fryer, C. and Potter, O.E, (1976), “Experimental investigation of models for fluidized bed catalytic reactors,” AIChE J., 22.].

Studying properties of Lagrangian bounds for many-to-many assignment problems

Classical and modified Lagrangian bounds for the optimal value of optimization problems with a double decomposable structure are studied. For the class of many-to-many assignment problems, this property of constraints is used to design a subgradient algorithm for solving the modified dual problem. Numerical results are presented to compare the quality of classical and modified bounds, as well as the properties of the corresponding Lagrangian solutions.

Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body

In this paper, we have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when more massive body is a source of radiation and the smaller primary is an oblate body. We have determined periodic orbits for fixed values of μ, σ and different values of p and h (μ mass ratio of the two primaries, σ oblate parameter, p radiation parameter and h energy constant). These orbits have been determined by giving displacements along the tangent and normal to the mobile co-ordinates as defined by Karimov and Sokolsky (in Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of radiation pressure on the periodic orbits by taking some fixed values of μ and σ.

Perturbation analysis of short-crested waves in Lagrangian coordinates

A new second-order asymptotic solution that describes short-crested waves is derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition and there being zero pressure at the free surface, in contrast with the Eulerian solution, in which there is residual pressure at the free surface. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level. The mass transport velocity and Lagrangian mean level associated with particle displacement can also be obtained directly. In particular, the mean level of the particle motion in a Lagrangian form differs that of the Eulerian form. The new formulation reduces to second-order standing or progressive wave solutions in Lagrangian coordinates at the limiting angles of approach. Expressions for kinematic quantities are also presented.

Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.

A third-order asymptotic solution of nonlinear standing water waves in Lagrangian coordinates

Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor's series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.

Properties of the Eulerian–Lagrangian method using linear interpolators in a three-dimensional shallow water model using z-level coordinates

We investigate the impact of the inexact interpolation on the Eulerian–Lagrangian solution of the advection equation by combining numerical experiments and formal analysis. The simulations, respectively, using the Eulerian–Lagrangian method (ELM) and the upwind scheme are compared. The artificial resistance of the ELM is observed which is characterised by the higher free-surface elevation and the distorted turbulent properties at a smaller time step. Through analysis, we find that the abnormalities are caused by the fact the conventional linear interpolation does not adapt well to the nonlinear velocity distribution, which produces an advection computation error that increases with a decreasing time step. The phenomena are explained and an improved method ELM is proposed based on the illustrations and analysis. The new method combines the face-controlled interpolation and the adjustable sub time steps to skip the large computation error domain in the backtracking, and it is validated by the original test case.

The transformation between the third-order Eulerian and Lagrangian solutions for irrotational progressive gravity waves

This study reports the transformation between the third-order Eulerian and Lagrangian solutions for the progressive water gravity waves propagating on water of uniform depth. Regarding to the motion of a marked fluid particle, the instantaneous velocity, mass conservation and free surface must be the same for solufions of either Eulerian or Lagrangian method. Using a successive Taylor series expansion to the path and the period of particle motion, the given conventional Eulerian solutions can be transformed into the completely unknown Lagrangian solutions and the reversible process is also identified. In the asymptotic solution, the explicit parametric equation of water particles can be obtained. In particular, the Lagrangian mean level and the Lagrangian wave frequency which differ from those in the Eulerian approach are found as part of the solutions. It shows that the present technique provides a modified method to obtain the third-order Lagrangian solution from the known Eulerian solutions.

The transformation between the Eulerian and Lagrangian solutions for irrotational standing gravity waves

This study reports the transformations between the third-order Eulerian and Lagrangian solutions for the standing gravity waves on water of uniform depth. Regarding the motion of a marked fluid particle, the instantaneous velocity, the mass conservation and the free surface must be the same for either Eulerian or Lagrangian methods. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, we obtain the third-order asymptotic expressions for the Lagrangian particle trajectories and the period of particle motion directly in Lagrangian form. In particular, the Lagrangian mean level which differs from that in the Eulerian approach is also found as a part of the solutions. Therefore, the given Eulerian solutions can be transformed into the completely unknown Lagrangian solutions and the reversible process is also identified.

NONLINEAR WATER WAVES ON UNIFORM CURRENT IN LAGRANGIAN COORDINATES

This paper presents a new third-order trajectory solution in Lagrangian form for the water particles in a wave-current interaction flow based on an Euler–Lagrange transformation. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level and within a vertical water column, which were calculated previously by an approximation method using Eulerian approach. Mass transport associated with a particle displacement can now be obtained directly in Lagrangian form. The angular frequency and Lagrangian mean level of the particle motion in Lagrangian form differ from those of the Eulerian. The variations in the wave profile and the water particle orbits resulting from the interaction with a steady uniform current of different magnitudes are also investigated. Comparison on the wave profiles given by the Eulerian and Lagrangian solution to a third-order reveals that the latter is more accurate than the former in describing the shape of the wave profile. Moreover, the influence of a following current is found to increase the relative horizontal distance traveled by a water particle, while the converse is true in the case of an opposing current.

A NEURAL NETWORK OPTIMIZER FOR SCHEDULING HYDROPOWER GENERATIONS

An approach based on neural network optimizer is proposed for determining the optimal short-term scheduling of multi-reservoir hydropower system. The proposed method is based on the Lagrange multiplier theory and search for solutions satisfying the necessary conditions of optimality in the state space. The equilibrium point of the network optimizer corresponds to the Lagrange solution of the problem and satisfies the Kuhn-Tucker condition for the problem. Here the main objective is to determine the optimal amounts of water to be released from each reservoir during each interval so as to minimize the overall energy shortages over the complete planning horizon. The method takes into account the water transportation delays between upstream and downstream reservoirs and a nonlinear hydropower generation function. An algorithm based on the proposed neural network optimizer has been developed and implemented on a multi-chain cascade of reservoir type hydropower system. Results so achieved have been compared with those obtained using conventional augmented Lagrange multiplier method. From the results, it is concluded that the proposed method is very effective in providing a good optimal solution along with constraint satisfaction.

Implementing the Retinex algorithm with Wilson–Cowan equations

It is shown that a form of the Wilson–Cowan equations representing large-scale activity in interacting neural populations can implement a form of the Retinex algorithm for color vision. It has also been shown recently that a color enhancement algorithm closely related to Retinex can be derived from a variational principle. It follows that a variational principle exists for equations of Wilson–Cowan type. Thus the Wilson–Cowan equations are the Euler–Lagrange solution of the minimization of an energy functional. This result suggests many interesting neural applications.

ON THE ORBITAL STABILITY OF TRIANGULAR LAGRANGIAN MOTIONS IN THE THREE-BODY PROBLEM

For the spatial three-body problem, the orbital stability of stationary triangular Lagrangian solutions is investigated. To the end of stability analysis, new equations coupling the mutual distances between bodies (mass points) are deduced. A theorem on the orbital instability of stationary triangular solutions related to circular orbits of mass points is proved.

Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body

We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem.

LanHEP—a package for the automatic generation of Feynman rules in field theory. Version 3.0

The LanHEP program version 3.0 for Feynman rules generation from the Lagrangian is described. It reads the Lagrangian written in a compact form, close to the one used in publications. It means that Lagrangian terms can be written with summation over indices of broken symmetries and using special symbols for complicated expressions, such as covariant derivative and strength tensor for gauge fields. Supersymmetric theories can be described using the superpotential formalism and the 2-component fermion notation. The output is Feynman rules in terms of physical fields and independent parameters in the form of CompHEP model files, which allows one to start calculations of processes in the new physical model. Alternatively, Feynman rules can be generated in FeynArts format or as LaTeX table. One-loop counterterms can be generated in FeynArts format. Program summary Program title: LanHEP Catalogue identifier: ADZV_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECH_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 83 041 No. of bytes in distributed program, including test data, etc.: 1 090 931 Distribution format: tar.gz Programming language: C Computer: PC Operating system: Linux RAM: 2 MB (SM), 12 MB (MSSM), 120 MB (MSSM with counterterms) Classification: 4.4 Nature of problem: Deriving Feynman rules from the Lagrangian Solution method: The program reads the Lagrangian written in a compact form, close to the one used in publications. It means that Lagrangian terms can be written with summation over indices of broken symmetries and using special symbols for complicated expressions, such as covariant derivative and strength tensor for gauge fields. Tools for checking the correctness of the model, and for simplifying the output expressions are provided. The output is Feynman rules in terms of physical fields and independent parameters in the form of CompHEP model files, which allows one to start calculations of processes in the new physical model. Alternatively, Feynman rules can be generated in FeynArts format or as a LaTeX table. Running time: 1 sec (SM), 8 sec (MSSM), 8 min (MSSM with counterterms)

Guiding center plasma models in three dimensions

Guiding center plasma models describe the fast charged particle gyration around magnetic field lines by an angle coordinate, defined relative to local orthogonal coordinate axes (ê1,ê2,b̂=B∕B) at each guiding center location. In three dimensions (3D), unlike uniform straight two-dimensional (2D) fields, geometrical effects make the small gyroradius expansion nonuniform in velocity phase space in first order O(ρi∕L). At second order, Hamiltonian and Lagrangian solutions may be undefined even when good magnetic flux surfaces exist; existence requires the magnetic field torsion τ=b̂⋅∇×b̂=0 and τg≡b̂⋅(∇ê1)⋅ê2=0, unless the magnetic field has a 2D symmetry, such as toroidal axisymmetry. Keeping complete 3D geometrical effects also requires the magnetic vector potential term to appear in the electric field at the same order as the electrostatic potential. These problems express properties of magnetic vector potentials, Lagrangians, and the curvature of manifolds, and have analogies to attempts to connect small scale Lagrangian theories to higher dimensional, large scale ones in the grand unification theories of physics.

Lagrangian based heuristics for the multicommodity network flow problem with fixed costs on paths

We study the multicommodity network flow problem with fixed costs on paths, with specific application to the empty freight car distribution process of a rail operator. The classification costs for sending a group of cars do not depend on the number of cars in the group, as long as the group is kept together as one unit. Arcs correspond to trains, so we have capacity restrictions on arcs but fixed costs on the paths corresponding to routes for groups of cars. As solution method, we propose a Lagrangian based heuristic using dual subgradient search and primal heuristics based on path information of the Lagrangian subproblem solutions. The method illustrates several ways of exploiting the specific structures of the problem. Computational tests indicate that the method is able to generate fairly good primal feasible solutions and lower bounds on the optimal objective function value.