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

## Variational Principle | Applications of Quantum Mechanics

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. 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.

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. 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 9] is defined as the linear part of the change in the functional, and the second variation [Note 10] is defined as the quadratic part. The functional J [ y ] is said to be differentiable if. The functional J [ y ] is said to be twice differentiable if. Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.

From Wikipedia, the free encyclopedia. For the use as an approximation method in quantum mechanics, see Variational method quantum mechanics. Limits of functions Continuity.

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## Calculus of variations

Mean value theorem Rolle's theorem. Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem. Fractional Malliavin Stochastic Variations. Glossary of calculus. Main article: Euler—Lagrange equation. See also: Sturm—Liouville theory.

Main article: Applications of the calculus of variations. Main article: Action physics. First variation Isoperimetric inequality Variational principle Variational bicomplex Fermat's principle Principle of least action Infinite-dimensional optimization Functional analysis Ekeland's variational principle Inverse problem for Lagrangian mechanics Obstacle problem Perturbation methods Young measure Optimal control Direct method in calculus of variations Noether's theorem De Donder—Weyl theory Variational Bayesian methods Chaplygin problem Nehari manifold Hu—Washizu principle Luke's variational principle Mountain pass theorem Category:Variational analysts Measures of central tendency as solutions to variational problems Stampacchia Medal Fermat Prize Convenient vector space.

Indeed, it was only Lagrange's method that Euler called Calculus of Variations. An extremal is a function that makes a functional an extremum. Methods of Mathematical Physics.

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I First English ed. New York: Interscience Publishers, Inc. See e. Kelland, Philip Lectures on the principles of demonstrative mathematics. The argument y has been left out to simplify the notation. A bilinear functional is a functional that depends on two argument functions and is linear when each argument function in turn is fixed while the other argument function is variable. Sufficient Conditions for a Weak Extremum" — Sufficient conditions for a weak minimum are given by the theorem on p. Sufficient Conditions for a Strong Extremum" — Sufficient conditions for a strong minimum are given by the theorem on p.

It is worth mentioning that both products in 2. The following properties are true :. The following relation holds true :. It is easy to show that system 3. Applying Theorem 2. Consider equation 3. It is equivalent to 3. Hence, by Theorem 3. Dividing by a 0 t , we get. Furthermore, A j t is bounded on I. We can see that f satisfies the Lipschitz condition with Lipschitz constant A.

Hence, there exists a unique solution of 3. If the function y is a solution of the homogeneous equation 3. By Corollary 3.

## Quantum Variational Calculus

Hence we have the desired result. Then every solution y of 3. Thus the results hold. Suppose that y 1 and y 2 are linear dependent, so there exist constants c 1 and c 2 not both zero, such that. Thus the solutions y 1 and y 2 are linearly independent in J. Then there exists a fundamental set of the two solutions y 1 and y 2 of equation 3. Let y 1 t , y 2 t be functions defined on I.

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Assume that y 1 t and y 2 t are two solutions of equation 3. Since y 1 and y 2 are solutions of equation 3. Using Theorem 3. Equation 3. The characteristic polynomial of equation 3. Since equation 3. Case 1: real and different roots of the characteristic equation 3. Case 2: complex roots of the characteristic equation 3. The general solution of equation 3. Then the general solution is.

### Recensie(s)

The characteristic equation of 3. If the characteristic equation 3. It follows that.