Lagrangian, nonlinear

We now consider the formulation of the equations of motion for a rigid body pinned at its center of mass and acted on by a (possibly nonlinear) potential field. The Lagrangian in this case is... 

Successive quadratic programming (SQP) methods solve a sequence of quadratic programming approximations to a nonlinear programming problem. Quadratic programs (QPs) have a quadratic objective function and linear constraints, and there exist efficient procedures for solving them see Section 8.3. As in SLP, the linear constraints are linearizations of the actual constraints about the selected point. The objective is a quadratic approximation to the Lagrangian function, and the algorithm is simply Newton s method applied to the KTC of the problem. 

Murtagh, B. A. and M. A. Saunders. A Projected Lagrangian Algorithm and Its Imple-w mentation for Sparse Nonlinear Constraints. Math Prog Study 16 84-117 (1982). 

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

Vol. 9. Augmented Lagrangian and Operator-Splitting Method in Nonlinear Mechanics... 

Murtagh, B. A., and Saunders, M. A., A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints, and MINOS/AUGMENTED user s manual, Technical Reports SOL 80-1R and SOL 80-14, Systems Optimization Laboratory, Dept, of Operations Research, Stanford Univ., CA (1981). 

Some Lagrangian-based Algorithms for Sparse Nonlinear Constraints", UNSW Report 1979/OR/5, University of New South Wales, 1979. 

For both mathematical and physical reasons, there are many instances in which the spatial variations in the field variables are sufficiently gentle to allow for an approximate treatment of the geometry of deformation in terms of linear strain measures as opposed to the description including geometric nonlinearities introduced above. In these cases, it suffices to build a kinematic description around a linearized version of the deformation measures discussed above. Note that in component form, the Lagrangian strain may be written as... 

Andrews, D.G., and M.E. McIntyre, An exact theory of nonlinear waves on a Lagrangian mean flow. J Fluid Mech 89, 609, 1978. 

The dependence of the average concentration on the Damkohler number can also be interpreted within the Lagrangian formulation. For example, the logistic growth function of the plankton population dynamics (Eq. (6.7)) is concave near the steady state P = K, i.e. the plankton population reacts more quickly when the carrying capacity is below the actual plankton density, than in the opposite case when higher carrying capacity allows for increase of the plankton concentration. Due to this asymmetric nonlinear response the... 

We note that the above results are not limited to the case of linear decay, but also apply to any kind of decay-type or stable reaction dynamics in a flow with chaotic advection (Chertkov, 1999 Hernandez-Garcfa et ah, 2002). In such systems where the reaction dynamics is nonlinear, the decay rate b should be replaced by the absolute value of the negative Lyapunov exponent of the Lagrangian chemical dynamics given by the second equation in (6.25), that represents the average decay rate of small perturbations in the chemical concentration along the trajectory of a fluid element. 

The numerical procedure used to solve the final equations The analytical method leads to a system of equations linear in the unknowns (i.e., the Lagrangian multipliers and their time derivatives up to order Therefore standard numerical techniques for solving such systems can be employed. The method of undetermined parameters leads to an additional system of equations generally nonlinear in the unknowns (i.e., the derivatives of the Lagrange multipliers of order s ,3x)- The order of nonlinearity depends on the particular... 

The most general form of holonomic constraint is nonlinear in the particle positions. Even the simple bond-stretch constraint is nonlinear. Consequently, Eq. [39] is in general a system of / coupled nonlinear equations, to be solved for the / unknowns (7). This nonlinear system of equations must be contrasted with the linear system of equations Eqs. [10] and [11] (which is also in general part of the method of undetermined parameters) used in the analytical method to solve for the Lagrangian multipliers and their derivatives. A solution of Eq. [39] can be achieved in two steps ... 

Clearly, approximation 1 leads to an Eq. [85] that is linear in the Lagrangian multipliers. Not surprisingly, its solution by the TB method is found to be inaccurate.- The reason for the inaccuracy is obvious in light of the steps of the matrix method the solution in Eq. [89] is just a linearization first estimate of the true solution, and no further iterations, using at least the lowest nonlinear term in the expansion, are carried out to refine this first estimate, unlike the procedure followed in the matrix method. To deal with this problem, Tobias and Brooks decouple the constraints and iterate over them until convergence is reached to within a certain tolerance. ... 

Problem Type Large-scale linear and nonlinear programs Method Projected Lagrangian... 

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