Moving boundary problems with more than one dependent variables, transversality condition in a more general case, examples, Extremals with corners, refraction of extremals, examples, One-sided variations, conditions for one sided variations,. Field of extremals, central field of extremals, Jacobi's condition, The Weierstrass function, a weak extremum, a strong extremum, The Legendre condition, examples, Transforming the Euler equations to the canonical form, Variational problems involving conditional extremum, examples, constraints involving several variables and their derivatives, Isoperimetric problems, examples.

Decomposition, direct computation, Successive approximation, Successive substitution methods for Fredholm Integral Equations,. A domain decomposition, series solution, successive approximation, successive substitution method for Volterra Integral Equations, Volterra Integral Equation of first kind, Integral Equations with separable Kernel,.

Fredholm's first, second and third theorem, Integral Equations with symmetric kernel, Eigenfunction expansion, Hilbert-Schmidt theorem,. Curant, R. Interscience Press, Porter, D. Calculus of Variations and Integral Equations. The arc length of the curve is given by. The Euler—Lagrange equation will now be used to find the extremal function f x that minimizes the functional A [ y ]. Since f does not appear explicitly in L , the first term in the Euler—Lagrange equation vanishes for all f x and thus,.

In other words, the shortest distance between two points is a straight line. In that case, the Euler—Lagrange equation can be simplified to the Beltrami identity : [14]. By Noether's theorem , there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which often coincides with the energy of the system. This is minus the constant in Beltrami's identity.

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral J requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form of the Euler—Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If L has continuous first and second derivatives with respect to all of its arguments, and if. Hilbert was the first to give good conditions for the Euler—Lagrange equations to give a stationary solution.

Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler—Lagrange equations in the interior.

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However Lavrentiev in showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. For instance the following:. Here a zig zag path gives a better solution than any smooth path and increasing the number of sections improves the solution. Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of D ; the solutions are called minimal surfaces.

The Euler—Lagrange equation for this problem is nonlinear:. It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by.

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Since v vanishes on C and the first variation vanishes, the result is. The proof for the case of one dimensional integrals may be adapted to this case to show that.

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The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy.

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Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize. The function that minimizes the potential energy with no restriction on its boundary values will be denoted by u. Provided that f and g are continuous, regularity theory implies that the minimizing function u will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment v.

Then if we allow v to assume arbitrary boundary values, this implies that u must satisfy the boundary condition. This boundary condition is a consequence of the minimizing property of u : it is not imposed beforehand. Such conditions are called natural boundary conditions. For such a trial function,. By appropriate choice of c , V can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless.

This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems. It is shown below that the Euler—Lagrange equation for the minimizing u is. It can be shown see Gelfand and Fomin that the minimizing u has two derivatives and satisfies the Euler—Lagrange equation.

This variational characterization of eigenvalues leads to the Rayleigh—Ritz method : choose an approximating u as a linear combination of basis functions for example trigonometric functions and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate. The next smallest eigenvalue and eigenfunction can be obtained by minimizing Q under the additional constraint.

This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem. The variational problem also applies to more general boundary conditions. After integration by parts,. If we first require that v vanish at the endpoints, the first variation will vanish for all such v only if. If u satisfies this condition, then the first variation will vanish for arbitrary v only if. These latter conditions are the natural boundary conditions for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case.



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For example, given a domain D with boundary B in three dimensions we may define. The Euler—Lagrange equation satisfied by u is. This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li—Jost for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert Fermat's principle states that light takes a path that locally minimizes the optical length between its endpoints.

After integration by parts of the first term within brackets, we obtain the Euler—Lagrange equation.

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The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics. After integration by parts in the separate regions and using the Euler—Lagrange equations, the first variation takes the form.

Snell's law for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length. The optical length of the curve is given by. Note that this integral is invariant with respect to changes in the parametric representation of C. The Euler—Lagrange equations for a minimizing curve have the symmetric form.

In order to find such a function, we turn to the wave equation, which governs the propagation of light. The wave equation for an inhomogeneous medium is. Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy. These equations for solution of a first-order partial differential equation are identical to the Euler—Lagrange equations if we make the identification.

That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem.

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This is the essential content of the Hamilton—Jacobi theory , which applies to more general variational problems. In classical mechanics, the action, S , is defined as the time integral of the Lagrangian, L. The Lagrangian is the difference of energies,. Hamilton's principle or the action principle states that the motion of a conservative holonomic integrable constraints mechanical system is such that the action integral.

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The Euler—Lagrange equations for this system are known as Lagrange's equations:. Analogy with Fermat's principle suggests that solutions of Lagrange's equations the particle trajectories may be described in terms of level surfaces of some function of X. This function is a solution of the Hamilton—Jacobi equation :. Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [Note 11] is defined as the linear part of the change in the functional, and the second variation [Note 12] is defined as the quadratic part.

The functional J [ y ] is said to be differentiable if.