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  1. Real Functions in One Variable - Taylor's
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  5. Calculus of variations

  • Calculus of variations - Wikipedia.
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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.

Real Functions in One Variable - Taylor's

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. 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.

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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.

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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.

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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. 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. 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.

Calculus of variations

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.


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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.