Abstract
We start with an introductory part, including the Legendre transformation, a brief discussion of linear nonholonomic systems and a presentation of three important classes of examples: the geodesic flow, Hamiltonian systems on Lie group manifolds and Hamiltonian systems on coadjoint orbits. We finish this part by showing how to deal with time-dependent Hamiltonian systems. Next, we investigate the structure of regular energy surfaces and discuss the problem of the existence of periodic integral curves for autonomous Hamiltonian systems. This leads us to the famous Weinstein conjecture and to symplectic capacities. Thereafter, we investigate the behaviour of a Hamiltonian system near a critical integral curve. We show that periodic integral curves generically come in orbit cylinders and prove the Lyapunov Center Theorem. We derive the Birkhoff normal form both for symplectomorphisms near an elliptic fixed point and for the Hamiltonian of a system near an equilibrium. The normal form of the Hamiltonian induces a foliation of the phase space into invariant tori so that, in the normal form approximation, the theory becomes integrable. Moreover, we prove the Birkhoff-Lewis Theorem, which states that under a certain nonresonance condition, near a periodic integral curve there exist infinitely many periodic points which lie on the same energy surface. Thereafter, we discuss some aspects of stability, with the main emphasis on systems with two degrees of freedom. In the final two sections we study time-dependent Hamiltonian systems. This includes a discussion of the stability problem of time-periodic systems with emphasis on parametric resonance and an introduction to the famous Arnold conjecture about the number of fixed points of Hamiltonian symplectomorphisms.
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- 1.
That is, the Hamiltonian does not depend explicitly on time. We will show in Sect. 9.3 that the framework discussed here can be easily extended to the non-autonomous case.
- 2.
As mentioned in Sect. 4.8, in the theory of nonholonomic constraints many interesting branches are studied. Here, we limit ourselves to showing that the Lagrangian formulation presented there has a counterpart on the Hamiltonian level.
- 3.
Sometimes one is led to go beyond the cotangent bundle model though, notably in the study of systems with symmetries (Chap. 10) where coadjoint orbits of Lie groups play an important role as phase space models.
- 4.
- 5.
Also referred to as an integral of motion or a first integral.
- 6.
More precisely, they are equivalent as submanifolds of M, because they are integral manifolds of the same integrable distribution, cf. Proposition 3.5.15 and Remark 1.6.13/5.
- 7.
Cf. Remark 9.1.2/1 and recall that is the Legendre transformation induced by the Lagrangian .
- 8.
- 9.
- 10.
Sometimes also called the suspended Hamiltonian.
- 11.
More generally, we could assume that the symplectic form ω is exact.
- 12.
In this book, Arnold develops an almost philosophical attitude towards this integral invariant by showing that it can be taken as a starting point for building Hamiltonian mechanics. Then, also symplectic geometry is a derived structure. From a more modern point of view, the reader can find a lot of interesting thoughts about the unifying power of symplectic and especially of contact geometry in another paper by Arnold [22], which ends with the statement that “contact geometry is all geometry”.
- 13.
This problem has its origin in celestial mechanics. In particular, it is interesting to ask whether our planetary systems admits periodic orbits, that is, whether there exist initial conditions to which the planets would return after a finite time.
- 14.
See Remark 8.2.11/2.
- 15.
This is a standard tool from bifurcation theory also called the blowing up technique, see [1] for historal references.
- 16.
Cf. Remark 8.2.11.
- 17.
It is enough to assume the Ψ is of class C 3 [225].
- 18.
- 19.
Those which fulfil an additional, stronger non-resonance condition.
- 20.
They are only getting slightly deformed. More precisely, in each step of the perturbation procedure the variables (I K ,ϑ k ) are getting modified.
- 21.
This assumption can be weakened. It is enough to assume that at least one of the Floquet multipliers lies on the unit circle and is different from 1 [225, Thm. 1].
- 22.
See Lemma 3 in Sect. 2 of [225].
- 23.
This problem is due to the occurrence of “small divisors” in these power series. We met such divisors in the proof of Theorem 9.6.2, cf. Eq. (9.6.9). There, in order to solve for the coefficients F ρσ one has to divide by the number λ ρ−σ−1, which is different from zero under the non-resonance assumption. However, for the convergence of the above power series, non-vanishing of such numbers is not sufficient. They have to be sufficiently large.
- 24.
This was already observed by Poincaré.
- 25.
Of course, this also follows from the Liouville Theorem 9.1.4.
- 26.
Defined by an isoenergetic Poincaré mapping for γ E , cf. Remark 9.6.5/ 1.
- 27.
See also Sect. 4.4 of the book of Thirring [286].
- 28.
Named after the astronomer who first observed them in 1866.
- 29.
Cf. Remark 3.8.19.
- 30.
Compare with Fig. 7.1.
- 31.
Note that we do not assume ω ε (t) to be continuous in t.
- 32.
The reader should try to set a swing in motion by leaning forward and backward over a whole period each time.
- 33.
Obviously, ν is determined up to addition of 2kπ, k∈ℤ.
- 34.
One can also show the converse, that is, every Hamiltonian symplectomorphisms can be represented as the period mapping of some periodic Hamiltonian, see Exercise 11.8 in [206].
- 35.
See also [46] for a generalization to ring shaped regions with arbitrary boundary curves.
- 36.
References
Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin-Cummings, Reading (1978)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55 (1964)
Arnold, V.I.: Proof of A.N. Kolmogorov’s theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18, 9–36 (1963)
Arnold, V.I.: Small divisor problems in classical and celestial mechanics. Russ. Math. Surv. 18, 85–192 (1963)
Arnold, V.I.: Mathematische Methoden der klassischen Mechanik. Birkhäuser, Basel (1988)
Arnold, V.I.: Symplectic geometry and topology. J. Math. Phys. 41(6), 3307–3343 (2000)
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics. In: Arnold, V.I. (ed.) Dynamical Systems III. Springer, Berlin (1988)
Bates, L., Śniatycki, J.: Nonholonomic reduction. Rep. Math. Phys. 32(1), 99–115 (1993)
Birkhoff, G.D.: Proof of Poincaré’s geometric theorem. Trans. Am. Math. Soc. 14, 14–22 (1913)
Birkhoff, G.D.: An extension of Poincaré’s last geometric theorem. Acta Math. 47, 297–311 (1925)
Birkhoff, G.D.: Dynamical Systems. American Mathematical Society Colloquium Publications, vol. IX (1927)
Birkhoff, G.D., Lewis, D.C.: On the periodic motions near a given periodic motion of a dynamical system. Ann. Mat. Pura Appl. 12, 117–133 (1933)
Chierchia, L.: Kolmogorov-Arnold-Moser (KAM) theory. In: Meyers, R.A. (editor-in-chief) Encyclopedia of Complexity and Systems Science, pp. 5064–5091. Springer, Berlin (2009)
Conley, C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math. 73, 33–49 (1983)
Douady, R.: Une démonstration directe de l’équivalence des théorèmes de tores invariants pour difféomorphismes et champs des vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 295, 201–204 (1982)
Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry—Methods and Applications. Springer, Berlin (1992)
Ekeland, I., Hofer, H.: Symplectic topology and Hamiltonian dynamics. Math. Z. 200, 355–378 (1990)
Eliashberg, Y.: Estimates on the number of fixed points of area preserving transformations. Syktyvkar University preprint (1979)
Fadell, E., Rabinowitz, P.: Generalized cohomological index theories for the group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)
Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. 41, 775–813 (1988)
Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120, 513–547 (1989)
Fukaya, K., Ono, K.: Arnold conjecture and Gromov-Witten invariants. Topology 38, 933–1048 (1999)
Ginzburg, V.L.: Some remarks on symplectic actions of compact groups. Math. Z. 210, 625–640 (1992)
Ginzburg, V.L.: An embedding S 2n−1→ℝ2n, 2n−1≥7, whose Hamiltonian flow has no periodic trajectories. Int. Math. Res. Not. 2, 83–98 (1995)
Ginzburg, V.L.: The Weinstein conjecture and the theorems of nearby and almost existence. In: Marsden, J.E., Ratiu, T.S. (eds.) The Breadth of Symplectic and Poisson Geometry. Festschrift in Honor of Alan Weinstein, pp. 139–172. Birkhäuser, Basel (2005)
Hofer, H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, 515–563 (1993)
Hofer, H.: Dynamics, topology and holomorphic curves. In: Proceedings of the International Congress of Mathematicians, Berlin, 1998. Doc. Math., extra vol. I pp. 255–280 (1998)
Hofer, H., Salamon, D.: Floer homology and Novikov rings. In: The Floer Memorial Volume. Progress in Mathematics, vol. 133, pp. 483–524. Birkhäuser, Basel (1995)
Hofer, H., Zehnder, E.: A new capacity for symplectic manifolds. In: Rabinowitz, P., Zehnder, E. (eds.) Analysis et Cetera, pp. 405–428. Academic Press, San Diego (1990)
Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser Advanced Texts. Birkhäuser, Basel (1994)
José, J.V., Saletan, E.J.: Classical Dynamics. A Contemporary Approach. Cambridge University Press, Cambridge (1998)
Klingenberg, W.: Lectures on Closed Geodesics. Grundlehren Math. Wiss., vol. 230. Springer, Berlin (1978)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Wiley-Interscience, New York (1963)
Kolmogorov, A.N.: On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR 98(4), 527–530 (1954), in Russian
Kolmogorov, A.N.: General theory of dynamical systems and classical mechanics. In: Proc. Int. Congr. Math., vol. 1, Amsterdam, 1954, pp. 315–333 (1957), in Russian. English translation in Abraham, R., Marsden, J.E.: Foundations of Mechanics, pp. 741–757. Benjamin-Cummings, Reading (1978)
Koon, W.S., Marsden, J.E.: The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems. Rep. Math. Phys. 40(1), 21–62 (1997)
Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Reidel, Dordrecht (1987)
Liu, G., Tian, G.: Floer homology and Arnold conjecture. J. Differ. Geom. 49, 1–74 (1998)
Lyapunov, A.M.: Problème général de la stabilité du mouvement. Ann. Fac. Sci. Toulouse 2, 203–474 (1907)
McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Clarendon Press, Oxford (1998)
Meixner, J., Schaefke, W.: Mathieusche Funktionen und Sphäroidfunktionen. Grundlehren Math. Wiss., vol. 71. Springer, Berlin (1954), in German
Morbidelli, A.: Modern Celestial Mechanics. Taylor & Francis, London (2002)
Moser, J.: On invariant curves of area preserving mappings of the annulus. Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. II, 286–294 (1962)
Moser, J.: A rapidly convergent iteration method and nonlinear partial differential equations I. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20(2), 265–315 (1966)
Moser, J.: A rapidly convergent iteration method and nonlinear partial differential equations II. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20(3), 499–533 (1966)
Moser, J.: Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29, 727–747 (1976)
Moser, J.: Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff. In: Geometry and Topology. Lecture Notes in Mathematics, vol. 597, pp. 464–494. Springer, Berlin (1977)
Moser, J.: Addendum to “Periodic orbits near an equilibrium and a theorem by Alan Weinstein”. Commun. Pure Appl. Math. 31, 529–530 (1978)
Poincaré, H.: Méthodes Nouvelles de la Méchanique Céleste. Gauthier-Villars, Paris (1899)
Pöschel, J.: A lecture on the classical KAM theorem. Proc. Symp. Pure Math. 69, 707–732 (2001)
Rabinowitz, P.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31, 157–184 (1978)
Salamon, D.: Lectures on Floer homology. In: Eliashberg, Y., Traynor, L. (eds.) Symplectic Geometry and Topology. IAS/Park City Mathematics Series, vol. 7, pp. 143–230 (1999)
Schwarz, M.: Introduction to symplectic Floer homology. In: Thomas, C.B. (ed.) Contact and Symplectic Geometry. Publ. Newton Inst., vol. 8, pp. 151–170. Cambridge University Press, Cambridge (1996)
Sevryuk, M.B.: Some problems of KAM-theory: conditionally-periodic motion in typical systems. Russ. Math. Surv. 50(2), 341–353 (1995)
Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Grundlehren Math. Wiss., vol. 187. Springer, Berlin (1971)
Thirring, W.: Lehrbuch der Mathematischen Physik. Band 1: Klassische Dynamische Systeme. Springer, Berlin (1977), in German
Viterbo, C.: A proof of the Weinstein conjecture in ℝ2n. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 337–357 (1987)
Weinstein, A.: Normal modes for nonlinear Hamiltonian systems. Invent. Math. 20, 47–57 (1973)
Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108, 507–518 (1978)
Weinstein, A.: On the hypothesis of Rabinowitz’s periodic orbit theorems. J. Differ. Equ. 33, 353–358 (1979)
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Rudolph, G., Schmidt, M. (2013). Hamiltonian Systems. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_9
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