Variational calculus in the large

The branch of mathematics involving the use of topological concepts and methods in qualitative studies of variational problems: the existence and estimation of the number of extremals, a study of certain of their qualitative properties and the relations between the amounts of extremals of various types (cf. Variational problem). The field is sometimes also deemed to include the "global" theory of stationary (critical) points of functions on manifolds, where similar problems are considered for these points. (In any event, the last-named theory is closely connected with the calculus of variations in the large and has common origins with it, while its problems often serve as simplified models of proper variational problems. The latter are sometimes studied by examining their approximations by the former.) All extremals (stationary points) existing in the given problem are of interest, irrespective of whether they have a corresponding real extremal (i.e. maximal or minimal) value of the functional (or of the function) or if they are stationary only. This is one of the differences between the calculus of variations in the large and the earlier branches of variational calculus, in which, after a relatively simple deduction of stationarity conditions which are the same for all extremals, the study concentrates on a (usually local) extremum — usually a minimum. In addition, a large part of "classical" subjects involve the study of a small neighbourhood of an extremal, while in variational calculus in the large use is made of the topological properties of the entire functional space of the variational problem, i.e. of the entire space of curves (functions, surfaces, etc.) on which the functional under consideration is defined (or any manifold on which the function under consideration is defined). These properties are connected in turn with the topology of this space (domain, manifold, etc.) where these curves (surfaces) ought to lie in or on which these functions ought to be defined and/or assume values (and also with the nature of the boundary conditions or any supplementary conditions). Such a "global" character of variational calculus in the large proper is stressed by the appellation "in the large" . (In the course of development of variational calculus in the large it proved necessary to study properties of the second variation [1], which is purely local. These properties were previously studied only in the context of the application to the conditions for the minimum of a functional.)

The calculus of variations in the large crystallized in the third decade of the 20th century in attempts to solve the problem on the estimation of the number of closed geodesics (cf. Closed geodesic) on a closed Riemannian (and, more generally, a Finsler) manifold. This problem is also known as the periodic problem of the calculus of variations in the large [1], [2], [4].

The general method of study in variational calculus in the large may be described as follows. For a given function, including a functional, considered as a function on a corresponding infinite-dimensional functional space, one studies the variation of various topological properties of the domain of smaller values

$$ f ^ { - 1 } (- \infty , C] = \{ {x } : {f ( x) \leq C } \} $$

with the variation of the level $ C $ of the function. One attempts to show that these properties vary only when $ C $ passes through stationary values (corresponding to the stationary points of the function) and to describe the connection between the changes accompanying such transitions and the properties of the respective stationary points. Certain connections are obtained between the stationary points of $ f $ on one hand, and the topological properties of the domain of smaller values $ f ^ { - 1 } (- \infty , C] $ with sufficiently large $ C $, or even of the entire space on which $ f $ is defined, on the other. If the latter properties are known, the connections established make it possible to draw certain conclusions regarding the stationary points. In variational problems proper one more step (which may be trivial or very difficult) remains to be performed: The results obtained, which concern auxiliary objects (points of some functional space), must be interpreted in the terms in which the original problem was posed. It is this last step which is the most difficult one in problems on closed geodesics.

The program just described can be readily realized for a smooth function $ f $ on a closed manifold $ M $. In comparing the domains of smaller values $ f ^ { - 1 } (- \infty , C] $ for various $ C $, gradient descent is commonly employed, i.e. the motion of points in accordance with the gradient dynamical system defined by $ f $ and any auxiliary Riemannian metric on $ M $. The situation near the stationary points is considered separately. If they are all non-degenerate, the change in the domain of smaller values $ f ^ { - 1 } (- \infty , C] $ when $ C $ passes through a stationary level may be described in great detail — accurately up to a diffeomorphism. Such a description, just like a similar description of the changes in the level manifolds $ f ^ { - 1 } ( C) $, has proved to be important in topology , [6], while in the calculus of variations in the large a less complete information in terms of some numerical invariant — the Lyusternik–Shnirel'man category $ \mathop{\rm cat} M $— and in terms of homology is still sufficient. Homological information results in Morse inequalities, which usually yield a much better estimate of the number of non-degenerate critical points than does the above category. Nevertheless, both this category and Morse inequality estimates are suitable, even without assuming that the stationary points are non-degenerate, though Morse inequalities yield an estimate in which degenerate stationary points are considered with specially assigned multiplicities; in this context one speaks of the estimate of the number of analytically-different stationary points or of the algebraic number of these points. The category, on the other hand, yields an estimate of the number of stationary points in the ordinary sense (this is emphasized by speaking of the estimate of the number of geometrically-different stationary points). Moreover, certain additional information may be given: Either the number of stationary levels is not smaller than $ \mathop{\rm cat} M $, or else one deals with a continuum of stationary points.

In order to obtain similar results for functions on infinite-dimensional manifolds, certain additional assumptions as to the analytical properties of these functions (other than smoothness) must be made. The most complete analogy with the finite-dimensional case is obtained by using the so-called Palais–Smale condition $ C $, [7], but this condition is not satisfied in certain cases of interest, and if weaker conditions are met the results obtained may be weaker as well. The study of the trajectories of gradient descent or any one of its analogues may well cause difficulties. For instance, for certain problems of geometrical origin — minimal closed submanifolds of a Riemannian manifold, harmonic mappings — the problem is reduced to the behaviour of the solutions of parabolic systems of non-linear partial differential equations [8]. Satisfactory results may sometimes be obtained only for minimum points. The principal applications are to eigenvalues of non-linear operators [7], [8], [9] and to multi-dimensional problems in variational calculus — i.e. problems in which the functional under consideration is expressed as some multiple integral — including the problems of geometrical origin mentioned above.

For one-dimensional problems (i.e. problems in which the functional being considered is expressed as an integral with one independent variable) specific methods which are more elementary than the method of gradient descent may be employed; in fact, many results were obtained by the use of such methods [1], [2], [3]. The problem of closed geodesics and the problem of estimating the number of geodesic arcs connecting two points on a complete connected Riemannian manifold $ M $ are both one-dimensional. The latter problem has been completely solved: If $ M $ can not be contracted into a point, the number of such arcs is infinite [10].

It would appear that the first application of variational calculus in the large in other branches of mathematics was the computation of the homology of the classical Lie groups . The most important modern application — along with the use of the theory of stationary points of functions in topology, see above — is the computation of stationary homotopy groups of Lie groups (the so-called Bott theory [12]). Variational calculus in the large is also employed in global differential geometry [13].

References

[1]	M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934) MR1451874 MR1501555 MR1561686 MR1501489 MR1501428 Zbl 0011.02802 Zbl 60.0450.01
[2]	L.A. Lyusternik, L.G. Shnirel'man, "Topological methods in variational problems and their application to the differential geometry of surfaces" Uspekhi Mat. Nauk , 2 : 1 (17) (1947) pp. 166–217 (In Russian) MR0029532
[3]	H. Seifert, W. Threlfall, "Variationsrechnung im Groszen (Morsesche Theorie)" , Teubner (1938)
[4]	G.D. Birkhoff, "Dynamical systems with two degrees of freedom" Trans. Amer. Math. Soc. , 18 (1917) pp. 199–300 MR1501070 Zbl 46.1174.01
[5a]	J.W. Milnor, "Topology from the differential viewpoint" , Univ. Virginia Press (1965)
[5b]	A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) MR0436148 MR0224103 Zbl 0164.23805
[6]	J. Milnor, "Lectures on the -cobordism theorem" , Princeton Univ. Press (1965) MR190942
[7]	J. Eells, "A setting for global analysis" Bull. Amer. Math. Soc. , 72 (1966) pp. 751–807 MR0203742 Zbl 0191.44101
[8]	S.I. Al'ber, "The topology of functional manifolds and the calculus of variations in the large" Russian Math. Surveys , 25 : 4 (1970) pp. 51–117 Uspekhi Mat. Nauk , 25 : 4 (1970) pp. 57–122 Zbl 0222.58002
[9]	M.S. Berger, "A bifurcation theory for nonlinear elliptic partial differential equations and related systems" J.B. Keller (ed.) S. Antman (ed.) , Bifurcation theory and nonlinear eigenvalue problems , Benjamin (1969) pp. 113–216 Zbl 0181.11603
[10]	J.-P. Serre, "Homologie singulière des espaces fibrés" Ann. of Math. , 54 (1951) pp. 425–505 MR0045386 MR0039255 MR0039254 Zbl 0045.26003 Zbl 0042.17402 Zbl 0042.17401
[11a]	L.S. Pontryagin, "Homology in compact groups" Mat. Sb. , 6 (48) : 3 (1939) pp. 389–422 (In Russian)
[11b]	L.S. Pontryagin, "Homologies in compact Lie groups" Mat. Sb. , 6 (1938) pp. 389–422 (In Russian) Zbl 0165.04104 Zbl 0022.31602
[12]	J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401
[13]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) MR0229177 Zbl 0155.30701

Comments

E. Witten [a2] remarked that to the critical points and connecting descent manifolds one can assign a complex, the homology of which is isomorphic to the homology of the space. This implies the Morse inequalities. A refinement of the Morse index, with useful applications, is the Conley index, see [a3].

For a modern treatment using Hilbert manifolds see [a1].

References

[a1]	W. Klingenberg, "Lectures on closed geodesics" , Springer (1978) MR0478069 Zbl 0397.58018
[a2]	E. Witten, "Supersymmetry and Morse theory" J. Diff. Geom. , 17 (1982) pp. 661–692 MR0683171 Zbl 0499.53056
[a3]	C. Conley, E. Zehnder, "Morse type index theory for Hamiltonian equations" Comm. Pure Appl. Math. , 37 (1984) pp. 207–253 Zbl 0559.58019
[a4]	M. Morse, S.S. Cairns, "Critical point theory in global analysis and differential topology" , Acad. Press (1969) MR0245046 Zbl 0177.52102