Euler variational calculus

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. 

Our interest is in determining the energy minimizing configuration of the bowed-out segment. To do so, we note that this has become a simple problem in variational calculus, with the relevant Euler-Lagrange equation being... 

Since the 8i may be arbitrary, the equation. ..) =0 (called the Euler equation in variational calculus) results. This will be our next goal. 

In the variational calculus the equation for the optimum 4>, or the conditional minimum of a functional e, is called the Euler equation. As one can see in this case the Euler equation is identical with the Schrodinger one. 

The eqiiilibrium structures are detenriined by the condition that J FddV should be minimal. This is the problem of variation calculus, and the solution is given by the Euler-Lagrange equation. Assuming uniformity in y direction ... 

Derivation of a ray path for the geometrical optics of an inhomogeneous medium, given v(r) as a function of position, requires a development of mathematics beyond the calculus of Newton and Leibniz. The elapsed time becomes a functional T [x(f)] of the path x(r), which is to be determined so that ST = 0 for variations Sx(t) with fixed end-points Sxp = Sxq = 0. Problems of this kind are considered in the calculus of variations [5, 322], proposed originally by Johann Bernoulli (1696), and extended to a full mathematical theory by Euler (1744). In its simplest form, the concept of the variation Sx(t) reduces to consideration of a modified function xf (t) = x(f) + rw(f) in the limit e — 0. The function w(f) must satisfy conditions of continuity that are compatible with those of x(r). Then Sx(i) = w(l)dc and the variation of the derivative function is Sx (l) = w (f) de. 

Variational principles for classical mechanics originated in modem times with the principle of least action, formulated first imprecisely by Maupertuis and then as an example of the new calculus of variations by Euler (1744) [436], Although not stated explicitly by either Maupertuis or Euler, stationary action is valid only for motion in which energy is conserved. With this proviso, in modem notation for generalized coordinates,... 

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-... 

These are called the Euler-Lagrange equations, and they may be derived formally by use of the calculus of variations. We refer to A, which is a function of time, as a Lagrange multiplier. 

The equilibrium density profile p z) is the one that minimises the surface tension functional thus by the standard methods of calculus of variations we can write down an Euler-Lagrange equation that the density profile must satisfy ... 

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. 

It should be emphasized once again the important point confining considerably the application field of the calculus of variations. When determining extrema of the target functional the Euler-Lagrange method does not take into account the possibility for the existence of limitations imposed on the control parameters and phase coordinates. 

We conclude that the relationship between dynamic programming and the optimum control theorem has been established on the basis that — dCjSNii the costate variable N. Dynamic programming is seen to be the Hamilton-Jacobi form of the calculus of variations, whereas the optimum control theorem was the Euler-Lagrange form. 

Newton and Leibnitz. The foundations of calculus of variations were laid by Bernoulli, Euler, Lagrange and Weierstrass. The optimization of constrained problems, which involves the addition of unknown multipliers, became known by the name of its inventor Lagrange. Cauchy made the first application of the steepest descent method to solve unconstrained minimization problems. In spite of these early contributions, very little progress was made until the middle of the 20th century, when high-speed digital computers made the implementation of the optimization procedures possible and stimulated further research in new methods. 

Chapter 3 gives the fundamental mathematical principles of the calculus of variations used for the optimization of dynamic systems. Classical results of the Euler equation for functional extrema and those of constrained optimization given by the Euler-Lagrange equation are developed. 

Application of Hamilton s method in the calculus of variations to an electrostatic field model of a suspension of charged colloidal particles with selection of a Lagrangian to yield Poisson s law as the Euler equation has led to the following results (1)... 

The fundamental lemma of the calculus of variations and the postulate that the functional shall be stationary require dl = 0, from which follows the Euler equation... 

What have been the significant historical developments in the mathematics of minimum area surfaces John Bernoulli and his student Leonhard Euler were amongst the earliest workers to apply the methods of the calculus to the solution of these problems, thus laying the foundations for a new branch of the calculus, the Calculus of Variations. In a comprehensive work published in 1744 Euler derived his well known equation for the determination of minimum area surfaces and other variational problems that require the examination of a sequence of varied surfaces. The equation, in its simplest one dimensional form, is... 

The methods developed by Euler and Lagrange in the eighteenth century come under the general title of The Calculus of Variations. These methods are best illustrated by considering the simplest minimization problem, the problem of determining the minimum length of path joining two points. 

Lagrange, Comte Joseph Louis (1736-1813) An Italian-born French mathematician and astronomer noted for his work in mechanics, harmonics, and in the calculus of variations. He also established the theory of differential equations. He succeeded Swiss mathematician and physicist Leonhard Euler (1707-83) as the director of mathematics at the Prussian Academy of Sciences in Berlin, during which time he published his work in MicaniqueAnalytique(n88), that covered every area of pure mathematics. 

Formulate the maximum distillate problem using the calculus of variations. Solution Since this problem contains equality constraints, we need to use the Euler-Lagrangian formulation. First, all three equality constraints (Equations 5.47 to 5.49) are augmented to the objective function to form a new objective function... 

Recall that the fulfilment of the Euler equation (sometimes referred to as the Euler-Lagrange equation when it is discussed in more general settings) is a necessary condition for equilibrium, as is well known from the calculus of variations [218, 238]. Applying the boundary conditions (3.3) reveals that... 

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