Euler-Lagrange differential equation

To seek an optimal function, it is necessary to find the stationary value of the functional about the optimal solution y  

The stationary value is obtained by solving = 0, where h x) is an arbitrary continuously differentiable function  

Case 1 The variational equation is subjected to the fixed end-point conditions 

These conditions are known as the natural boundary conditions. In either case, the optimal function satisfies the Euler-Lagrange equation. 

---

By functional differentiation, Equation 4.22 leads us to the Euler-Lagrange deterministic equation for the electron density, viz.,... 

By applying the first-order condition for optimization, that is, the first derivative with respect to the control variable, and Lagrange multipliers should disappear resulting in Euler-Lagrangian differential equations given below... 

Variational calculus, Dreyfus (1962), may be employed to obtain a set of differential equations with certain boundary condition properties, known as the Euler-Lagrange equations. The maximum principle of Pontryagin (1962) can also be applied to provide the same boundary conditions by using a Hamiltonian function. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Note that the equations are given in explicit algebraic form. Comparing to problem (35), note that all additional constraints have disappeared and the differential equations have had a simple, low-order Euler discretization applied to them. However, as with (35) we note that the Lagrange function for this problem. 

However, the Euler-Lagrange equation then becomes an integro-differential equation [12]... 

A Euler-Lagrange equation for the variational problem of 8 J LqdV may be obtained by considering the differential heat conduction equation, and we have... 

Now that we have introduced coordinates and velocities, the next question is how to predict the time evolution of a mechanical system. This is accomplished by solving a set of ordinary differential equations, the equations of motion, which can be derived from the principle of least action. It was discovered by Maupertuis and was further developed by Euler, Lagrange and Hamilton (d Abro (1951)). 

The set of equations (3.1.6) is a special case of the Euler equations of the calculus of variations (see, e.g., Arnold (1989)). They are referred to as the Euler-Lagrange equations in the literature. The Euler-Lagrange equations are ordinary second order differential equations for the generalized coordinates qa-... 

In order to construct higher-order approximations one must use information at more points. The group of multistep methods, called the Adams methods, are derived by fitting a polynomial to the derivatives at a number of points in time. If a Lagrange polynomial is fit to /(t TO, V )i. .., f tn, explicit method of order m- -1. Methods of this tirpe are called Adams-Bashforth methods. It is noted that only the lower order methods are used for the purpose of solving partial differential equations. The first order method coincides with the explicit Euler method, the second order method is defined by ... 

The variation principle is more fundamental than the differential equa tion to which it is equivalent the differential equation is simply the Euler-Lagrange equation generated by application of the variation principle within the boundaries of applicability. 

This Euler-Lagrange equation reduces to the following second-order partial differential equation for v x, y) ... 

Noether proved a theorem subsequently known as Noether s theorem, which assures the conservation laws of energy and momentum from the Euler-Lagrange equation with the uniformities of time and space, respectively (Noether 1918). First, from the uniformity of time, the Lagrangian of independent particle systems does not explicitly depend on time. The total differentiation of the Lagrangian is therefore... 

F7 /8 y = Oij in the form of Euler-Lagrange equations (ojk is the stress tensor) along with the boundary conditions (see e.g. Refs. [47, 58, 59]). This system of differential equations should be solved along with the equations of mechanical equilibrium daij x) /9x, = 0 and compatibility equations equivalent to the mechanical displacement vector , continuity [100]. 

Classical methods of calculus of variations are attractive from the point of view of the opportunity to obtain solutions in analytical form. But this is feasible in simple cases, which often are far from the demands of the state-of-art practice. In complicated cases, at a large number of optimization parameters, numerical approaches are used to solve the appropriate Euler-Lagrange equations. The main obstacle arising here is related to the fact that the numerical solution of the system of differential equations may turn out to be more complicated than the solution fi-om the very beginning of the optimization problem by numerical methods of mathematical programming. 

The Boltzmann integro-differential kinetic equation written in terms of statistical physics became the foundation for construction of the structure of physical kinetics that included derivation of equations for transfer of matter, energy and charges, and determination of kinetic coefficients that entered into them, i.e. the coefficients of viscosity, heat conductivity, diffusion, electric conductivity, etc. Though the interpretations of physical kinetics as description of non-equilibrium processes of relaxation towards the state of equilibrium are widespread, the Boltzmann interpretations of the probability and entropy notions as functions of state allow us to consider physical kinetics as a theory of equilibrium trajectories. These trajectories as well as the trajectories of Euler-Lagrange have the properties of extremality (any infinitesimal part of a trajectory has this property) and representability in the form of a continuous sequence of states of rest. These trajectories can be used to describe the behavior of (a) isolated systems that spontaneously proceed to final equilibrium (b) the systems for which the differences of potentials with the environment are fixed (c) and non-homogeneous systems in which different parts have different values of the same intensive parameters. 

The methods of Euler and Lagrange consider a continuous sequence of surfaces, contained by the boundary, which deviate from the extremum surface. For the soap film problem the extremum surface is the minimum area surface. A differential equation is then derived for the extremum surface. The solution to the differential equation will give the required surface. 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

---