Determination of eigenvectors with Lagrange multipliers

2021 ◽  
Author(s):  
Wooyong Han ◽  
Dong-Won Jung ◽  
Jungil Lee ◽  
Chaehyun Yu
Keyword(s):  
Lagrange Multipliers

The Determination of Maximum Range for an Unpowered Atmospheric Vehicle

1964 ◽  
Vol 68 (638) ◽  
pp. 111-116 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Multipliers ◽  
Digital Computer ◽  
Necessary Conditions ◽  
Lagrange Equations ◽  
Initial Portion ◽  
End Conditions ◽  
Maximum Range ◽  
Isothermal Atmosphere

SummaryThe problem of maximising the range of a given unpowered, air-launched vehicle is formed as one of Mayer type in the calculus of variations. Eulers’ necessary conditions for the existence of an extremal are stated together with the natural end conditions. The problem reduces to finding the incidence programme which will give the greatest range.The vehicle is assumed to be an air-to-ground, winged unpowered vehicle flying in an isothermal atmosphere above a flat earth. It is also assumed to be a point mass acted upon by the forces of lift, drag and weight. The acceleration due to gravity is assumed constant.The fundamental constraints of the problem and the Euler-Lagrange equations are programmed for an automatic digital computer. By considering the Lagrange multipliers involved in the problem a method of search is devised based on finding flight paths with maximum range for specified final velocities. It is shown that this method leads to trajectories which are sufficiently close to the “best” trajectory for most practical purposes.It is concluded that such a method is practical and is particularly useful in obtaining the optimum incidence programme during the initial portion of the flight path.

Analysis of Constraint Forces in Multibody Systems Based on the Differential Null Space Matrix Method

2019 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Governing Equations ◽  
Constraint Forces ◽  
Numerical Examples ◽  
Space Matrix ◽  
Rigid Multibody Systems ◽  
Rigid Multibody

Abstract This paper treats a problem to determine constraint forces in rigid mutibody systems. One of the most often applied algorithms for determination of constraint forces is based on the use of recursive Newton-Euler formalism. Another algorithm often applied for determination of constraint forces is based on the use of Lagrange multipliers. This paper presents a new method to determine constraint forces in rigid multibody systems. First relative displacements which violate the constraints, called anti-constraint relative displacements, are introduced, and governing equations which involve the constraint forces explicitly are derived. In the derived equations, the constraint forces appear independently from the Lagrange multipliers. Then, a method is proposed to determine the constraint forces by eliminating the Lagrange multipliers based on the methods proposed in previous papers by the author. The method is extended to have ability to treat systems with redundant constraints. Finally, validity of the proposed method is confirmed via numerical examples.

scholarly journals Nonlinear Closure Relations for Electron Transport in Hydrodynamical Models

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
A. Salhoumi
Keyword(s):  
Distribution Function ◽  
Fourth Order ◽  
Lagrange Multipliers ◽  
Hermite Polynomials ◽  
Boltzmann Distribution ◽  
Hydrodynamical Models ◽  
Calculation Results ◽  
Closure Relations ◽  
Theory Calculation

Closure relations problem of hydrodynamical models in semiconductors is considered by expressing third- and fourth-order closure relations for the moments of the distribution function in terms of second-order Lagrange multipliers using a generalized Maxwell-Boltzmann distribution function within information theory. Calculation results are commented and compared with others to justify the accuracy of the approach developed in this paper. The comparison involves, in the first part with good agreements, the closure relations results obtained within extended thermodynamics which were checked by means of Monte Carlo simulations, in the second part, the results obtained by Grad's method which expands the distribution function up to fourth-order in Hermite polynomials. It is seen that the latter method cannot give any restriction on closure relations for higher-order moments, within the same conditions proposed in our approach. The important role of Lagrange multipliers for the determination of all closure relations is asserted.

scholarly journals Simultaneous determination of mass and volume of a set of weights in group weighing

ACTA IMEKO ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 17
Author(s):  
Ch. Wuethrich ◽  
K. Marti
Keyword(s):  
Least Squares ◽  
Simultaneous Determination ◽  
Lagrange Multipliers ◽  
New Technique ◽  
Reference Weight ◽  
A New Technique

This work introduces a new technique for the determination of mass and volume of a set of weights based on closed series (group weighing). A traditional closed series is repeated at two different air densities. A least squares set of equations, involving two Lagrange multipliers, is used to determine the mass and the volume of each weight simultaneously with a traceability on the mass and the volume of the reference weight.

Determination of Holonomic and Nonholonomic Constraint Reactions in an Index-3 Augmented Lagrangian Formulation With Velocity and Acceleration Projections

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Daniel Dopico ◽  
Francisco González ◽  
Javier Cuadrado ◽  
József Kövecses
Keyword(s):  
Lagrange Multipliers ◽  
Augmented Lagrangian ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Dynamics Simulation ◽  
Correct Reaction ◽  
Lagrangian Formulations ◽  
Reaction Forces ◽  
Augmented Lagrangian Algorithms

Index-3 augmented Lagrangian formulations with projections of velocities and accelerations represent an efficient and robust method to carry out the forward-dynamics simulation of multibody systems modeled in dependent coordinates. Existing formalisms, however, were only established for holonomic systems, for which the expression of the constraints at the position-level is known. In this work, an extension of the original algorithms for nonholonomic systems is introduced. Moreover, projections of velocities and accelerations have two side effects: they modify the kinetic energy of the system and they contribute to the constraint reaction forces. Although the effects of the projections on the energy have been studied by several authors, their role in the calculation of the reaction forces has not been described so far. In this work, expressions to determine the constraint reactions from the Lagrange multipliers of the dynamic equations and the Lagrange multipliers of the velocity and acceleration projections are introduced. Simulation results show that the proposed strategy can be used to expand the capabilities of index-3 augmented Lagrangian algorithms, making them able to deal with nonholonomic constraints and provide correct reaction efforts.

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