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Wikipedia / Calculus of variations

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

History

The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, which was the first problem in the field to be formulated and correctly solved, and was also one of the most difficult problems tackled by variational methods prior to the twentieth century. This problem was followed by the brachistochrone curve problem raised by Johann Bernoulli (1696), which was similar to one raised by Galileo Galilei in 1638, but he did not solve the problem explicitly nor did he use the methods based on calculus. Bernoulli solved the problem using the principle of least time in the process, but not calculus of variations. In 1697 Newton solved the problem using variational techniques, and as a result, he pioneered the field with his work on the two problems. The problem would immediately occupy the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Joseph-Louis Lagrange was influenced by Euler's work to contribute greatly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.

Adrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject. To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Pierre Frédéric Sarrus (1842) which was condensed and improved by Augustin-Louis Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), John Hewitt Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of Karl Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development.

In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The dynamic programming of Richard Bellman is an alternative to the calculus of variations.

Extrema

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars of a given function space defined over a given domain has the same sign is called an extremal function or extremal. there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not. Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.

Euler–Lagrange equation

Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero. This leads to solving the associated Euler–Lagrange equation.

Consider the functional

where

  • are constants,

  • is twice continuously differentiable,

attains a local minimum close to 0,

is called the variation

and thus,

Taking the total derivative yields

Therefore,

and we have used integration by parts by definition. Also, as previously mentioned the left side of the equation is zero so that

According to the fundamental lemma of calculus of variations implies that the part of the integrand in parentheses is zero, i.e.

which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative

In general this gives a second-order ordinary differential equation The Euler–Lagrange equation is a necessary, but not sufficient A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

Example

The arc length of the curve is given by

with

Note that assuming is a function of loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes.

with

and thus,

and taking the derivative,

Thus

. Then

where

Solving, we get

which implies that

is

In other words, the shortest distance between two points is a straight line.

Beltrami's identity

does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity

is a constant. The left hand side is the Legendre transformation

implies that the Lagrangian is time-independent. 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.

Recently, methods have been developed to incorporate symmetry information directly into the inverse problem of the calculus of variations. By combining the Helmholtz conditions involved in the inverse problem with relations derived from Noether's identity, it is possible to reconstruct Lagrangians that reproduce given equations of motion while ensuring the presence of prescribed symmetries.

Euler–Poisson equation

, that is, if

must satisfy the Euler–Poisson equation,

Du Bois-Reymond's theorem

requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form has continuous first and second derivatives with respect to all of its arguments, and if

has two continuous derivatives, and it satisfies the Euler–Lagrange equation.

Lavrentiev phenomenon

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.

However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934:

gives a value bounded away from the infimum.

but Ball and Mizel There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals.

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.

Functions of several variables

plane, then its potential energy is proportional to its surface area:

Plateau's problem; the solutions are called minimal surfaces. The Euler–Lagrange equation for this problem is nonlinear:

See Courant (1950) for details.

Dirichlet's principle

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by

must vanish:

has two derivatives, we may apply the divergence theorem to obtain

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

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. 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

Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998).

Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is

. The function that minimizes the potential energy with no restriction on its boundary values is given by

If we apply the divergence theorem, the result is

the boundary integral vanishes, and we conclude as before that

must satisfy the boundary condition

: it is not imposed beforehand. Such conditions are called natural boundary conditions.

is a constant. For such a trial function,

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 (1953).

Eigenvalue problems

Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems.

Sturm–Liouville problems

The Sturm–Liouville eigenvalue problem involves a general quadratic form

is restricted to functions that satisfy the boundary conditions

be a normalization integral

is

is the quotient

. This variational characterization of eigenvalues leads to the Rayleigh–Ritz method 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.

under the additional constraint

This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.

vanish at the endpoints, we may not impose any condition at the endpoints, and set

is

as previously. After integration by parts,

only if

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 several dimensions

in three dimensions we may define

and

is

where

must also satisfy the natural boundary condition

This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) 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 (1953).

Applications

Optics

Fermat's principle along the path, then the optical length is given by

then the first variation) is

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.

Snell's law

There is a discontinuity of the refractive index when light enters or leaves a lens. Let

may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form

axis. 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.

Fermat's principle in three dimensions

be its tangent vector. The optical length of the curve is given by

The Euler–Lagrange equations for a minimizing curve have the symmetric form

where

satisfies

Therefore, the integral may also be written as

. In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

Connection with the wave equation

The wave equation for an inhomogeneous medium is

. Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy

We may look for solutions in the form

satisfies

. According to the theory of first-order partial differential equations satisfies

along a system of curves (the light rays) that are given by

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification

as a function of the upper end point. 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.

Mechanics

. The Lagrangian is the difference of energies,

is the kinetic energy its potential energy. 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:

and they are equivalent to Newton's equations of motion (for such systems).

are defined by

For example, if

Hamiltonian mechanics defined by

. This function is a solution of the Hamilton–Jacobi equation:

Further applications

Further applications of the calculus of variations include the following:

Variations and sufficient condition for a minimum

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 is defined as the linear part of the change in the functional, and the second variation is defined as the quadratic part.

, then the corresponding change in the functional is

is said to be differentiable if

is a linear functional, and is denoted by,

is said to be twice differentiable if

is a quadratic functional, and is denoted by,

is said to be strongly positive 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.

See also

Notes

References

Further reading