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Published in:
Notes on Differential Geometry Read chapter Read first chapter

2015 | OriginalPaper | Chapter

7. Notes on Lusternik-Schnirelman and Morse Theories

Author : Franco Cardin

Published in: Elementary Symplectic Topology and Mechanics

Publisher: Springer International Publishing

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Abstract

In this chapter (longer than the others) we will present techniques that are close to other modern research themes: existence theorems for critical points of functions. We will move in many directions, but mainly using constructions that are taken from the techniques in symplectic geometry described up to now. In particular, those on generating functions for Lagrangian submanifolds.

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previous chapter A Short Introduction to the Asymptotic Theory of Rapidly Oscillating Integrals
next chapter Finite Exact Reductions
Footnotes
1
Even though today there are ‘weak’ formulations of Morse theory extending it to degenerate critical points (maybe first in [86]), the universally known definition of Morse function is, up to today, that of functions whose critical points are all non-degenerate.
 
2
That is, it is defined for \(t \in \mathbb{R}\).
 
3
That is, any Cauchy sequence does converge in it.
 
4
That are clearly Palais-Smale.
 
5
The set of critical points is a closed set, its complement is an open set, and X b ∖ X a is in the interior of such complement: it is then easy to define two open neighborhoods A 1 and A 2 as above.
 
6
Using Urysohn lemma, given two closed, disjoint sets cl(A 1) and X ∖ A 2, it is possible to define a C ∞ -function ϕ, that has value 1 in cl(A 1) and has value 0 in X ∖ A 2. So, for example, \(\bar{Y }:= -\phi \frac{\nabla f(x)} {\vert \nabla f(x)\vert ^{2}}\).
 
7
This means that there is no constant c > 0 such that \(\vert f^{{\prime}}(x)\vert > c\) on S.
 
8
In other words, c cannot be an accumulation point of critical values.
 
9
Once again, Palais-Smale condition implies this fact (Condition (C)): \(f^{-1}([c -\varepsilon _{0},c +\varepsilon _{0}])\setminus V _{0}\) is closed, so we cannot find in it sequence {x j } with df (x j ) → 0, in fact, in such a case there exists some subsequence converging to a critical point x ∗, which should belong to the closed set \(f^{-1}([c -\varepsilon _{0},c +\varepsilon _{0}])\setminus V _{0}\), absurd.
 
10
Recall that H 0(X) represents the constant functions on the connected components of X, if X = B (ball) then \(H^{0}(B) = \mathbb{R}\), while if k > 0: H k (B) = { 0}.
 
11
That is, with non degenerate Hessian \(f^{{\prime\prime}}\big\vert _{f^{{\prime}}=0}\).
 
12
Observe that, from the definition of GFQI, c is uniform in x.
 
13
See the unpublished work of Ottolenghi-Viterbo [100] and the beautiful book of Siburg [107].
 
14
Here we denote by \(\mathcal{F}: \text{Sym}(k \times k) \times \mathbb{R}^{n} \rightarrow \text{Sym}(k \times k)\ \ \text{the map}\ \ (R,x)\mapsto R^{T}\mathit{QR} - B(x)\).
 
15
\(f_{x}^{-\infty } = f^{-\infty },\ \forall x \in U\).
 
16
I thank Gian Maria Dall’Ara which pointed out to me this nice fact.
 
17
Arnol’d conjecture is itself an extension, more or less natural, of the last geometric theorem of Poincaré.
 
18
In a non trivial way, see [119], Prop. 3.3 p. 693.
 
19
Again, by Lusternik-Schnirelman Theorem 7.7.
 
20
c stands for capacity, see [55].
 
21
This is not restrictive, since by Whitney theorem it is sufficient N paracompact.
 
22
By coordinates, \(v_{g} = \sqrt{\det g}\,\mathit{dx}^{1} \wedge \ldots \wedge \mathit{dx}^{n}\).
 
23
By coordinates, for any scalar function Φ, the related gradient vector field X is defined by \(X^{i} = (\sharp d\varPhi )^{i} = (\nabla _{g}\varPhi )^{i} = g^{\mathit{ij}} \frac{\partial \varPhi } {\partial x^{j}}\).
 
24
These formulas were first presented by Hamilton in the “Second essay on a general method in dynamics”, 1835. The author thanks Sergio Benenti for pointing out this remarkable fact.
 
25
In the Section 28 of Gantmacher [66], this example is really quoted as ‘perturbation theory’.
 
26
It is sufficient to see that contractive perturbations of the identity are bi-Lipschitz homeomorphisms: (Rem: \(\vert x - y\vert \geq \big\vert \vert x\vert -\vert y\vert \big\vert \geq \vert x\vert -\vert y\vert \)) Since f: X → X is contractive, | f(x 1) − f(x 2) | ≤ λ | x 1 − x 2 | ,   λ < 1, then I − f is injective: \(\vert (x_{1}-f(x_{1}))-(x_{2}-f(x_{2}))\vert = \vert (x_{1}-x_{2})-(f(x_{1})-f(x_{2}))\vert \geq \vert x_{1}-x_{2}\vert -\vert f(x_{1})-f(x_{2})\vert \geq (1-\lambda )\vert x_{1}-x_{2}\vert \). Thus the inverse g di I − f is Lipschitz with \(\mathrm{Lip}(g) = \frac{1} {1-\lambda }\). The surjectivity of I − f is gained from the fixed point: for any y ∈ X, the map x ↦ y + f(x) is obviously contractive, then there exists an unique x such that x = y + f(x), so that x − f(x) = y. □ 
 
27
This new definition of Generating Function Weakly Quadratic at Infinity, is equivalent to the definition of GFQI; this equivalence was established by Viterbo and Theret [113, 114].
 
Literature
3.
B. Aebischer, et al., Symplectic Geometry. Progress in Mathematics, (Boston, Mass.), vol. 124 (Birkhäuser, Basel, 1994), xii, 239p
7.
V.I. Arnol’d, Mathematical Methods of Classical Mechanics, 2nd edn. (Springer, New York, 1989)
11.
M. Audin, M. Damian, Morse Theory and Floer Homology. Universitext, vol. XIV (Springer, London, 2014), 596pp. Original French edition published by EDP Sciences, Les Ulis Cedex A, France, 2010
12.
A. Banyaga, On the group of strong symplectic homeomorphisms. CUBO, Math. J. 12(03), 49–69 (2010)
14.
S. Benenti, Hamiltonian Optics and Generating Families (Bibliopolis, Neaples, 2004)
15.
S. Benenti, Hamiltonian Structures and Generating Families. Universitext (Springer, New York, 2011)CrossRefMATH
17.
O. Bernardi, F. Cardin, On Poincaré-Birkhoff periodic orbits for mechanical Hamiltonian systems on \(T^{{\ast}}\mathbb{T}^{n}\). J. Math. Phys. 47, 072701-1–072701-15 (2006)
21.
R. Bott, L. Tu, Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82 (Springer, New York/Berlin, 1982), xiv+331pp
30.
31.
33.
M. Chaperon, Une idée du type “géodésiques brisées” pour les systèmes Hamiltoniens. C. R. Acad. Sci. Paris Sér. I Math. 298(13), 293–296 (1984)MATHMathSciNet
35.
M. Chaperon, Familles génératrices. Cours l’école d’été Erasmus de Samos (Publication Erasmus de l’Université de Thessalonique, 1993)
46.
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry—Methods and Applications. Part III. Introduction to Homology Theory. Graduate Texts in Mathematics, vol. 124 (Springer, New York, 1990), x+416pp
53.
Y. Eliashberg, A theorem on the structure of wave fronts and applications in symplectic topology. Funct. Anal. Appl. 21, 227–232 (1987)CrossRefMathSciNet
55.
65.
S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry. Universitext, 3rd edn. (Springer, Berlin, 2004), xvi+322pp
66.
F. Gantmacher, Lectures in Analytical Mechanics (Mir Publications, Moscow, 1975), 264pp
67.
C. Godbillon, Éléments de topologie algébrique (Hermann, Paris, 1971), 249ppMATH
68.
C. Golé, Symplectic Twist Maps. Global Variational Techniques. Advanced Series in Nonlinear Dynamics, vol. 18 (World Scientific, River Edge, 2001), xviii+305pp
69.
72.
H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser Advanced Texts (Birkhäuser, Basel, 1994), xiv+341pp
75.
V. Humilière, On some completions of the space of Hamiltonian maps. Bull. Soc. Math. Fr. 136(3), 373–404 (2008)MATH
76.
85.
L. Lusternik, L. Schnirelman, Méthodes topologiques dans les problèmes variationnels (Hermann, Paris, 1934)MATH
86.
A. Marino, G. Prodi, Metodi perturbativi nella teoria di Morse. Boll. Un. Mat. Ital. (4) 11(3), 1–32 (1975)
91.
D. McDuff, D. Salamon, Introduction to Symplectic Topology. Oxford Mathematical Monographs, 2nd edn. (The Clarendon/Oxford University Press, New York, 1998)
93.
J. Milnor, Morse Theory, vol. 51 (Princeton University Press, Princeton, 1963), vi+153pp
100.
A. Ottolenghi, C. Viterbo, Solutions generalisees pour l’equation de Hamilton-Jacobi dans le cas d’evolution (pre-print)
101.
107.
K.F. Siburg, The Principle of Least Action in Geometry and Dynamics. Lecture Notes in Mathematics, 1844 (Springer, Berlin, 2004)
108.
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 34, 2nd edn. (Springer, Berlin, 1996), xvi+272pp
113.
D. Theret, Utilisation des fonctions génératrices en géométrie symplectique globale. PhD thesis, Université Paris 7, 1996
114.
117.
W.M. Tulczyjew, Geometric Formulation of Physical Theories, Statics and Dynamics of Mechanical Systems (Bibliopolis, Neaples, 1989)
119.
120.
C. Viterbo, Solutions of Hamilton-Jacobi Equations and Symplectic Geometry. Addendum to: Séminaire sur les Équations aux Dérivžées Partielles. 1994–1995 (École Polytech., Palaiseau, 1995)
121.
C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 217 (Springer, Dordrecht, 2006), pp. 439–459
122.
C. Viterbo, Symplectic Homogenization (2014). arXiv:0801.0206v3
124.
A. Weinstein, Lectures on Symplectic Manifolds. CBMS Conference Series (AMS 29, Providence, 1977)
Metadata
Title
Notes on Lusternik-Schnirelman and Morse Theories
Author
Franco Cardin
Copyright Year
2015
Book
Elementary Symplectic Topology and Mechanics

Print ISBN: 978-3-319-11025-7

Electronic ISBN: 978-3-319-11026-4

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-11026-4_7

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