Lagrangian linear systems

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 ... 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

Since the resulting system after radial collocation is still too complex for direct mathematical solution, the next step in the solution process is discretization of the two-dimensional system by orthogonal collocation in the axial direction. Although elimination of the spatial derivatives by axial collocation increases the number of equations,8 they become ordinary differential equations and are easily solved using traditional techniques. Since the position and number of points are the only factors affecting the solution obtained by collocation, any set of linearly independent polynomials may be used as trial functions. The Lagrangian polynomials of degree N based on the collocation points... 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

Any multicomponent system whose dynamical behavior is governed by coupled linear equations can be modelled by an effective Lagrangian, quadratic in the system variables. Hamilton s variational principle is postulated to determine the time behavior of the system. A dynamical model of some system of interest is valid if it satisfies the same system of coupled equations. This makes it possible, for example,... 

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... 

Murray, R. (1995). Non linear control of mechanical systems A Lagrangian perspective. In Proceedings of the IFACSymposium on Nonlinear Control Systems Design (NOLCOS). IFAC. 

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